Optical properties of cylindrical nanowires

Optical properties of cylindrical nanowiresL.A. Haverkate; L.F. Feiner15th December 2006

AbstractA theoretical analysis is presented of the optical absorption of III-V semiconductorcylindrical nanowires. The optical properties are described by meansof the dielectric function, calculated for band-to-band transitions close tothe band gap.We have treated the electronic structure using effective mass theory,taking the degeneracy of the valence band into account.A strong polarization anisotropy is found which is due to quantum confinement,in agreement with atomistic methods. We show that the effectivemass approach provides a fast and flexible tool to analyze the diameter dependentproperties of nanowires for a wide range of semiconductor materials.In addition we discuss the effect of classical Mie scattering and show thatit is negligible in the quantum regime.

ContentsConclusions 107A Hole wavefunctions for different k z 112B Polarization selection rules 116C Interband matrix elements 119D Reference articles 122Bibliography 1255

IntroductionIn the last several years semiconductor nanowires have attracted much interest,due to the tunability of their fundamental optical and electronicproperties. Techniques for the growth of nanostructures have been developedand high quality III-V semiconductor nanowires with a length of severalmicrons and a lateral size of only a few nanometers have been obtained. Recentexperiments have shown a large polarization anisotropy in such wires[1][2]. For example, Figure 1 shows the photoluminescence and excitationspectra of an InP nanowire on a flat gold surface [2]. The radius of this wirewas ∼ 15 nm and the measured polarization anisotropy is fully explained bythe dielectric mismatch between the wire and the surrounding.Figure 1: Experimental results for optical absorption [2]. a) Photoluminescenceimage (CCD camera, incident laser light polarized parallelto the wire axis) and b) Excitation spectra for parallel (‖) and perpendicular(⊥) polarized incident light of an InP nanowire on a flat goldsurface. The length of the wire is ∼ 2 µm, its radius ∼ 15 nm and thewavelength of the exciting laser beam is 457.9 nm. The emitted lightwas unpolarized in order to take only the polarization anisotropy in theabsorption process into account.However, next to this classical effect of dielectric contrast it is expectedthat quantum confinement starts to contribute significantly for decreasing6

wire radius. This quantum effect has already been observed in the shift inthe fundamental band gap [3], but based on atomistic theories[4][5] it is alsopredicted that quantum confinement causes drastic changes in the polarizationanisotropy of nanostructures.In this paper we will analyze the optical absorption properties of III-Vsemiconductor cylindrical nanowires using effective mass theory. Within thisapproach it is possible to describe the optical and electronic properties forvarying wire thickness and for a wide range of semiconductor materials. Contrary,ab initio methods using a fully atomistic description (tight-binding,pseudo-potential,...) are limited with respect to the dimensions of the nanosystemsince with increasing number of atoms the calculations become moreand more complex, or even impossible. Although the effective mass approachgenerally is less accurate, it thus provides a relative fast and flexible toolto simulate real nanowires, with dimensions which are technically feasibleat the present day.Despite the large amount of papers on the subject of nanowires, or evennanostructures in general, little attention has been paid to the effects ofclassical scattering. Usually it is assumed that the wavelength of the incidentlight is sufficiently larger than the wire radius in order to neglect the spatialvariance of the electromagnetic (EM) field within the wire, which justifiesconsidering the response of the nanowire to the incident light in the dipolelimit. For increasing wire radius, however, the wave behavior of the EMfield cannot simply be neglected any more. Therefore, it is one of the mainquestions in this thesis if this so called Mie-scattering already starts to playa significant role in the quantum confinement regime.For this purpose we have to know the local response field inside the wire.But historically most of the work in classical scattering theory was dedicatedto measurable quantities far from the scattering objects, driven by the largeinterest from application fields as astronomy and meteorology. In Part I,Chapters 1 and 2, we therefore start with a classical theory describing thescattering of light by an infinite cylindrical structure. In particular we deriveexplicit expressions for the EM field inside the wire by using a procedureoriginally developed by Mie [6].In Part II we subsequently focus on the effects of quantum confinementby means of a corrected description of the dielectric function of cylindricalnanowires. In Chapter 3 the electronic properties of nanowires made fromIII-V compounds are discussed, Chapter 4 treats the EM matrix element forband-to-band transitions between the top Γ 8 valence bands and the lowestlying Γ 6 conduction band in III-V semiconductor nanowires and finally inChapter 5 the dielectric function and polarization anisotropy of a nanowireare obtained including the quantum confinement corrections by the bandto-bandtransitions.7

Part IClassical theory of lightscattering by a cylinder8

Chapter 1General solutionIn this chapter classical theory is treated which describes the scattering oflight by an infinite cylinder at arbitrary angle of incidence and wire radius.In the first sections general theory is discussed and specified to the case ofan infinite cylinder by using a procedure originally developed by Mie [6]. Insection the theory will be put in an applicable form by deriving measurablequantities (cross sections, efficiency factors) in the far field region.1.1 General theory1.1.1 Maxwell equationsThe scattering of light at oblique incidence by an infinite cylinder needs afull, formal treatment, in particular when the solution has to be expandedin the wire radius.The starting-point of the full problem is Maxwell’s theory. Assuming thelight waves to be periodic with time dependence e −iωt , the charge densityρ equal to zero and the magnetic permeability µ equal to 1, the Maxwellequations are:where∇ × H = −ik 0 m 2 E, (1.1)∇ × E = ik 0 H, (1.2)∇ · H = 0, (1.3)∇ · (m 2 E) = 0, (1.4)k 0 = ω c= 2πλ 0, (1.5)is the wave number in vacuum andm 2 = ε + 4πiσω . (1.6) 9

Chapter 1.General solutionThe parameter m is the complex refractive index of the medium at thefrequency ω of the light waves and consists of an optic part and an electricpart. The former is associated with ε, the dielectric constant, the latterwith the conductivity σ, which is taken to be zero since the electrical partis beyond the scope of this paper. Both parts are complex and depend onthe circular frequency ω of the light waves.It should be noted that in general m is a tensor and moreover depends onthe position in the medium. For the applications considered in this paper themedium is assumed to be homogeneous and in that case m is a constant. Wewill also assume here that m is a scalar. As a consequence, from (1.1)-(1.4),the field vectors E and H satisfy the vector wave equation:∆A + k 2 0m 2 A = 0. (1.7)As a consequence the rectangular components of E and H satisfy the scalarwave equation∆ψ + k 2 0m 2 ψ = 0, (1.8)which has plane wave solutions with the propagation constant equal to k 0 m.This shows that the wave is damped if m has a negative imaginary part andin that case absorption takes place.1.1.2 Boundary conditionsIn case of a sharp boundary between two homogeneous media (1 and 2) theintegral representation of the Maxwell equations (1.1) and (1.2) gives theboundary conditions on the tangential components of the fields, after a wellknown limiting process (Jackson [12], page 16):n × (H 2 − H 1 ) = 0, (1.9)n × (E 2 − E 1 ) = 0, (1.10)where n is the normal to the boundary.In the same way the Maxwell equations (1.3) and (1.4) lead to the boundaryconditions on the normal components:n · (m 2 2 E 2 − m 1 2 E 1 ) = 0, (1.11)n · (H 2 − H 1 ) = 0. (1.12)The tangential and normal boundary conditions are not independent. Forinstance, boundary condition (1.10) can be derived from (1.12), Maxwellequation (1.3) and applying the limiting procedure. In the same way it canbe shown that (1.9) and (1.11) are dependent on each other. Therefore it issufficient to look only at the tangential components.10

1.2. Mie’s formal solution for circular cylinders1.2 Mie’s formal solution for circular cylindersIn order to solve the boundary value problem exactly the coordinate systemshould be the one in which the scalar wave equation is separable in thecoordinates. In case of circular cylinders these coordinates are (ρ, φ, z),where the cylinder axis coincides with the z-axis (see Figure 1.1). As acondition for this separability the cylinder length L has to be assumed muchlarger then its diameter:L ≫ 2R, (1.13)where R denotes the cylinder radius. In this case the cylinder can be seenas infinitely long and then it is possible to use the following formal solutiondeveloped by Mie [6]. If ψ satisfies the scalar wave equation (1.8), defineM ψ and N ψ asM ψ = ∇ × (ê z · ψ), (1.14)mk 0 N ψ = ∇ × M ψ . (1.15)Then both M ψ and N ψ satisfy the vector wave equation (1.7), and theelementary solutions of Maxwell’s equations can be expressed asE = M v + iN u , (1.16)H = mM u − imN v , (1.17)where u and v are the two independent solutions of the scalar wave equation.The scalar wave equation (1.8) in cylindrical coordinates for a homogeneousmedium with complex refractive index m is( ∂2∂ρ 2 + 1 ∂ 2ρ ∂ρ + 1 ∂ 2)ρ 2 ∂φ 2 + ∂2∂z 2 + m2 k02 ψ = 0, (1.18)and its solutions can be found by separating the variables. The resultingdifferential equation for the ρ coordinate is the Bessel equation, which hastwo independent solutions: J n , the integral order Bessel function and N n ,the integral order Neumann function. This means that the solutions of (1.8)can be found by an appropriate superposition of:√ϕ n = Z n (ρ m 2 k0 2 − g2 )e i(gz−ωt) e inφ , (1.19)with n an integer, Z n any Bessel function of order n and g arbitrary. Incylindrical coordinates M ϕn and N ϕn are then derived as:⎛M ϕn = ⎝inρ− ∂ ∂ρ0⎞⎠ ϕ n , mk 0 N ϕn =⎛⎜⎝⎞ig ∂ ∂ρ−ng ⎟ρm 2 k0 2 − g2⎠ ϕ n , (1.20)11

Chapter 1.General solutionon the basis of cylindrical unit vectors ê ρ , ê φ , ê z . Consequently, with u n =A n ϕ n and v n = B n ϕ n for certain A n , B n and taking the sum over all n, thecomponents of E and H areE ρ =H ρ =∞∑n=−∞∞∑n=−∞n=−∞inρ v n −g ∂u nmk 0 ∂ρ , (1.21)inmρu n + g ∂v nk 0 ∂ρ , (1.22)normal to the cylinder surface and∞∑E φ = − ∂v n∂ρ − ingmk 0 ρ u n, (1.23)E z =H φ =H z =∞∑n=−∞∞∑n=−∞∞∑n=−∞tangential to the cylinder surface.1.3 Scattering problemi(m 2 k 2 0 − g2 )mk 0u n , (1.24)−m ∂u n∂ρ + ingk 0 ρ v n, (1.25)− i(m2 k 2 0 − g2 )k 0v n (1.26)With the above formal solution it is now possible to solve the general scatteringproblem of an arbitrary polarized plane electromagnetic wave incidentobliquely on a circular cylinder of infinite length.For oblique incidence the direction of propagation of the incident wave makesan angle θ with the normal to the z-axis, see Figure 1.1. Furthermorethe cylinder is assumed to be surrounded by vacuum and the refractive indexof the cylinder is equal to m. In case of a surrounding homogeneousmedium with refractive index m 1 the solutions are of the same form if mis considered as the refractive index of the cylinder relative to the medium:m = m 2m 1. With the above definitions the incident wave, depicted inFigure 1.1 is represented by the scalar wave functionwhere12ψ 0 = Ẽ0 e −i(k 0x cos θ+k 0 z sin θ+ωt)= Ẽ0 e −i(hz+ωt) ∞ ∑n=−∞(−i) n J n (lρ)e ınφ , (1.27)h ≡ k 0 sin θ, (1.28)√l ≡ k 0 cos θ = k0 2 − h2 . (1.29)

1.3. Scattering problemH OIIkE OIIE sca H OIkEintE OIXZYFigure 1.1: Definition of the coordinates for scattering by a circularcylinder. The incident waves are showed, including the correspondingincident fields: E 0 I , H 0 I in Case I and E 0 II , H 0 II in Case II. Theangle of incidence is defined by θ .Equation (1.27) represents a wave travelling in the −ê x direction if θ equalszero. Note that the last expression in (1.27) is an expansion in Bessel functionsand has the required form of (1.19). In this way also the scatteredwave and internal wave (inside the cylinder) can be formed from a superpositionof functions of the form (1.19). Finiteness at the origin requires thatJ n (ρ √ m 2 k0 2 − h2 ) is the radial function describing the internal wave, whereh is given by (1.28) because of continuity at the boundary ( (1.9) and (1.10)).The last argument also holds for the scattered wave, which is described bythe first Hankel function H n(1) (ρ √ k0 2 − h2 ), describing an outgoing wave atlarge distances from the cylinder.Following the procedure of Van de Hulst [7] and Kerker [8], the polarizedincident wave has to be resolved into two components:• Case I: a Transverse Magnetic (TM) mode. The magnetic field ofthe incident wave is perpendicular to the cylinder axis (Figure 1.1).This mode is described by choosing u n = 1 ilẼ 0 (−i) n ϕ n (withg = −h, Z n = J n ) and v n = 0 in (1.21)- (1.26). This choice alsofixes the orientation of the incident electric field: E 0 I = Ẽ0 (cos θê z −sin θê x ) e −i(hz+lx+ωt) . The factor 1 ilis just a normalization constant,for further details see Bohren and Huffman [11].• Case II: a Transverse Electric (TE) mode.The electric field is perpendicularto the cylinder axis. Now u n = 0 and v n = 1 ilẼ0(−i) n ϕ n andthe incident field is given by E 0 II = Ẽ0 ê y e −i(hz+lx+ωt) .For an arbitrary elliptically polarized incident wave the solutions can befound by an appropriate superposition of Case I and Case II. The decompo-13

Chapter 1.General solutionsition of the incident wave does not necessarily mean that the scattered andinternal waves resolve in the same way. This can be explained by lookingclosely at the general expressions of the scalar fields inside and outside thecylinder. These are:Case ICase IIwhereρ > R u nI = Ẽ0 F n {J n (lρ) − b nI H n (1) (lρ)} , (1.30)v nI = Ẽ0 F n {a nI H n (1) (lρ)} , (1.31)ρ < R u nI = Ẽ0 F n {d nI J n (jρ)} , (1.32)v nI = Ẽ0 F n {c nI J n (jρ)} , (1.33)ρ > R u nII = Ẽ0 F n {b nII H n (1) (lρ)} , (1.34)v nII = Ẽ0 F n {J n (lρ) − a nII H n (1) (lρ)} , (1.35)ρ < R u nII = Ẽ0 F n {d nII J n (jρ)} (1.36)v nII = Ẽ0 F n {c nII J n (jρ)} , (1.37)F n ≡ 1 il e−i(hz+ωt) (−i) n e inφ (1.38)andj≡√m 2 k 2 0 − h2 . (1.39)Unlike the incident waves, which are chosen to be TM or TE, the solutionsfor the scattered and internal scalar waves are in general decomposed intotwo components:• A solution with the same orientation as the incident wave (TM or TE),contained in u nI (Case I) and v nII (Case II) respectively.• A ”cross mode” with an opposite orientation, TE (v 1 ) in Case I andTM (u II ) in Case II.Only in case of normal incidence, θ = 0, the cross terms turn out to be zeroand the scattered and internal waves resolve in the same way as the incidentwave (see paragraph below).As stated in section 1.2, the scalar wave expressions (1.30)-(1.37) alsodetermine the fields inside and outside the cylinder in the various cases. Theincident, scattered and internal fields are denoted with E 0 , E sca and E int ,respectively.As an example, combining the expression for the scattered scalar wave inCase I (the second term in (1.30)) with the equations for the field components(1.21)-(1.26) one finds for the scattered electric field E sca in cylindercoordinates:14

1.3. Scattering problemE sca I = Ẽ0∞∑n=−∞⎡ ⎛⎢F n ⎣a nI⎝inρ− ∂ ∂ρ0⎞⎛⎠ H n (1)⎜(lρ) + i(−b nI ) ⎝−ih∂k 0 ∂ρnhk 0 ρl 2 k⎞ ⎤⎟⎠ H (1) ⎥(lρ) ⎦ .n(1.40)The scattered electric field E sca II in Case II can be obtained from (1.40)by replacing a nI by −a nII and −b nI by b nII .1.3.1 Scattering coefficients, general solutionThe coefficients a nI , b nI , c nI and d nI (a nII , b nII , c nII and d nII ) are in generalfunctions of the angle of incidence θ and the wire radius R. They can bedetermined by the fact that the boundary conditions (1.11)-(1.12) requirecontinuity of the tangential components of E and H. As a consequence theequations (1.23)-(1.26) have to be continuous at R = ρ. These conditionslead in both cases to four linear algebraic equations which can be solved forthe four coefficients:Case Ia nI (R, θ) =ı sin θ n(m 2 − 1){Nn−1 − On −1 }lR{( jk 0) 2 L n − (m 2 + 1) jk 0D n + L −1 n (C n − m 2 Dn)} , 2′b nI (R, θ) = H(1) n (lR){( jk 0) 2 K n − m 2 jc nI (R, θ) = l2j 2 {H n(1) (lR)M n {( ja nI (R, θ)H n(1)J n (jR)k 0D n } + H n(1) (lR){− jk 0D n K n + m 2 Dn 2 − C n },k 0) 2 L n − (m 2 + 1) jk 0D n + L −1 n (C n − m 2 Dn)}2}(lR),{}d nI (R, θ) = ml2 J n (lR)j 2 J n (jR) − b nI(R, θ)H n(1) (lR), (1.41)J n (jR)Case II:′a nII (R, θ) = H(1) n (lR){( jk 0) 2 K n − jk 0D n } + H n(1) (lR){m 2 jk 0D n K n + m 2 Dn 2 − C n }H n(1),(lR)M n {( jk 0) 2 L n − (m 2 + 1) jk 0D n + L −1 n (m 2 Dn 2 − C n )}b nII (R, θ) = −a nI (R, θ),{}c nII (R, θ) = l2 J n (lR)j 2 J n (jR) − b nII(R, θ)H n(1) (lR),J n (jR){}d nII (R, θ) = ml2 a nII (R, θ)H n(1) (lR)j 2 , (1.42)J n (jR)15

Chapter 1.General solutionwhereC n ≡ (m2 − 1) 2 n 2 tan 2 θj 2 R 2 , (1.43)D n ≡ cos θ J ′ n(jR)J n (jR) , (1.44)K n ≡ J n(lR)′ ′J n (lR) , L n ≡ H(1) n (lR)H n (1) (lR) , (1.45)′M n ≡ H(1) n (lR)J n (lR) , N n ≡ H(1) n (lR)J n (lR) , (1.46)′O n ≡ H(1) n (lR)J n(lR)′(1.47)and the functions l (1.29) and j (1.39) depend on θ. Despite of the differentform, equations (1.30)-(1.37) are the same as derived by Bohren [11]. Theygive the complete, formal solution for the scattering problem of a planeelectromagnetic wave incident obliquely on a circular cylinder of infinitelength. In principle the electromagnetic fields and the intensities can beobtained by calculating the full expansion of (1.30)-(1.33) for Case I ((1.34)-(1.37) for Case II) and subsequently use these expressions to calculate thefields (1.21-1.26). However, in practice it is impossible to get an exactanalytic solution and a numerical procedure is the only way to solve thefull problem.In the special case of a normal incident wave (θ = 0), the scatteringcoefficients (1.30)-(1.37) reduce to:Case I16a nI (R, 0) = 0,b nI (R, 0) =mJ n (k 0 R)J n(mk ′ 0 R) − J n(k ′ 0 R)J n (mk 0 R)mH n (1) (k 0 R)J n(mk ′ 0 R) − H (1) ′n (k 0 R)J n (mk 0 R) ,c nI (R, 0) = 0,d nI (R, 0) =Case IIH (1) ′n (k 0 R)J n (k 0 R) − H n (1) (k 0 R)J n(k ′ 0 R)mH (1) ′n (k 0 R)J n (mk 0 R) − m 2 H n (1) (k 0 R)J n(mk ′ 0 R) (1.48) ,a nII (R, 0) =J n (k 0 R)J n(mk ′ 0 R) − mJ n(k ′ 0 R)J n (mk 0 R)H n (1) (k 0 R)J n(mk ′ 0 R) − mH (1) ′n (k 0 R)J n (mk 0 R) ,b nII (R, 0) = 0,H (1) ′n (k 0 R)J n (k 0 R) − H n (1) (k 0 R)J ′c nII (R, 0) =n(k 0 R)m 2 H (1) ′n (k 0 R)J n (mk 0 R) − mH n (1) (k 0 R)J n(mk ′ 0 R) ,d nII (R, 0) = 0. (1.49)

1.4. Far field theoryAs stated before, the cross terms disappear, which means that all waves inCase I are TM and all waves in Case II TE.1.4 Far field theoryIn principle the scattering theory derived in the previous sections is completeand everything one wants to know can be derived from it. However, inorder to make predictions about measurable quantities, in this section thetheory will be put in an applicable form.It is important to realize that usually the experimental measurements are doneat a large distance from the scattering object(s), so in the first paragraphgeneral expressions for the fields in this region are derived. Subsequentlymeasurable quantities (cross sections, efficiency factors) are defined andapplied to the situation of scattering by an infinite cylinder. The theorydepicted here is derived in a detailed form by Bohren and Huffman [11].More intuitive approaches are found in [7],[8].1.4.1 Far field approximationAs stated in section 1.3 the scattered wave is associated with the first Hankelfunction, based on the fact that the wave has to be an outgoing wave. At largedistances from the cylinder the first Hankel function can be approximatedby its asymptotic expression:H n(1) (z) ∼√2πz eiz (−i) n e −iπ/4 , | z | ≫ n 2 . (1.50)This is the only ingredient needed to approximate the scattered part of thefields at large distances from the wire.Consider for this purpose equation (1.40) for the scattered electrical field.In the far field approximation (lρ ≫ 1) the Hankel functions in this expressionare approximated by (1.50). After elaboration of the derivatives andneglecting all terms ∼ 1lρ √ , which fall of much faster then the terms ∼ √1lρ lρ,this results in:E sca I ∼ −Ẽ0e −iπ/4 √ 2πlρ ei(lρ−hz−ωt)∞ ∑n=−∞(−1) n e inφ [a nI ê φ + b nI (sin θê ρ + cos θ)ê z ] .This is the result for an incident wave with the magnetic field perpendicularto the wire axis (Case I). For Case II, when the incident field is TE(the electric field perpendicular to the wire axis) the asymptotic expressionof the scattered field has the same form, apart from changing a nI into −a nIIand −b nI into b nII .(1.51)17

Chapter 1.General solutionEquation (1.51) shows that the surfaces of constant phase, or wavefronts,of the scattered wave obeyρ cos θ − z sin θ = C, C ∈ R, (1.52)which represents cones of half-angle θ and apexes at z = −C/ sin θ. Includingthe e −iωt factor, the scattered wave can be visualized as a cone slidingdown the cylinder [11].1.4.2 Poynting vector and electromagnetic energy ratesOne of the most important properties of electromagnetic (EM) waves is theflux of EM energy through a certain area. In the case of light scattering ata particle not only the magnitude of this flux has to be specified, but alsoits direction. This is given by the Poynting vector S =c8π Re{E × H∗ },which defines the time-averaged flux of energy crossing a unit area. As aconsequence the rate of EM energy crossing a plane surface A, with normalunit vector ˆn, is equal to ∫ S · ˆn dA.For a surface A which encloses a volume V the net rate W at which EMenergy crosses the boundary A is defined as∮W = − S · ˆn dA, ˆn ≡ unit normal outward to A. (1.53)AThis is a definition in the sense that the minus sign ensures that W is positiveif there is a net rate of EM energy flowing into the volume V (S · ˆn < 0),so in the case of absorption of EM energy in the volume.Denoting the incident and scattered EM fields in the same way as before,the Poynting vector at any point outside the particle can be written in thesefields as:S = c8π Re{(E 0 + E sca ) × (H ∗ 0 + H ∗ sca)} = S 0 + S sca + S ext ,whereS 0 = c8π Re{E 0 × H ∗ 0} , S sca = c8π Re{E sca × H ∗ sca} , (1.54)S ext = c8π Re{E 0 × H ∗ sca + E sca × H ∗ 0)} .The decomposition in (1.54) nicely shows that, next to the expected Poyntingvectors of the incident (S 0 ) and scattered (S sca ) fields, a term S extarises which describes the interaction between the incident and scatteredwaves.To be more precise, it turns out that S ext represents the removal of energyfrom the incident light waves, the extinction. Consider for this purposean imaginary sphere of radius a and surface A around a particle of finitesize . The rate of energy W abs absorbed within the sphere equals the energy18

1.4. Far field theoryrate absorbed by the particle because the surrounding medium is supposedto be non-absorbing. W abs is given by equation (1.53), now with ê a theoutward unit normal to the sphere and may be decomposed in:W abs = W 0 − W sca + W ext ,where∮W 0 = −∮W ext = −AA∮S 0 · ê a dA , W sca =S ext · ê a dA .AS sca · ê a dA , (1.55)The choice of the minus signs here again ensures that all the energy ratesare positive, note in particular S sca · ê a > 0. Furthermore the energy rateW 0 associated with the incident wave vanishes for a non-absorbing medium,soW ext = W abs + W sca , (1.56)which shows that W ext indeed represents the extinction, namely the sum ofthe energy scattering rate and energy absorbing rate.In case of an infinite cylinder the imaginary sphere in the precedingargumentation has to be replaced by an imaginary surrounding cylinder ofinfinite length. Now it is convenient to look at the rate of EM energy flow perunit length , since this quantity is finite. Furthermore, infinite cylinders don’texist except as an idealization, so the statements here have to be carefullyapplied to the situation of a cylinder long compared with is diameter, asused in the previous sections. This is possible if edge effects are negligible,such that there is no net contribution to W abs from the ends of the imaginarycylinder.In that case, denoting r as the radius of the constructed cylinder and a lengthL equal to the length of the cylinder, the expressions for the scattering andextinction rates of EM energy per unit length becomeW sca /L =W ext /L =∫ 2π0∫ 2π0(S sca ) ρ ρ dφ | ρ = r ,(S ext ) ρ ρ dφ | ρ = r , (1.57)with (S sca ) ρ and (S ext ) ρ the (positive) radial components of the expressionsderived in (1.54). On physical grounds the absorption energy rate has tobe independent of r if the medium outside the cylinder is non absorbing.Indeed, with the far field solution of (1.51), the r dependence in (1.57) dropsout.19

Chapter 1.General solution1.4.3 Cross sections and efficienciesIn stead of using the energy rates it is more convenient to take the normalizedforms of them: cross sections, or, better, efficiency factors. The former aresurfaces, defined asC abs = W absI 0, C sca = W scaI 0, C ext = W extI 0, (1.58)where I 0 =c8π |Ẽ0| 2 is the incident intensity. Dividing these optical crosssections by the geometrical cross section G, dimensionless efficiency factorsare found:Q abs = C absG , Q sca = C scaG , Q ext = C extG . (1.59)Note that equation (1.56) has a synonym in terms of efficiency factors:Q ext = Q sca + Q abs . (1.60)For a circular cylinder with radius R and length L the geometrical cross sectionequals 2RL. Note that the efficiency factors indeed are dimensionless.With the far field scattered electric field (1.51) and a similar expressionfor the scattered magnetic field now it is a matter of patience to derive:∫ 2πQ sca I = 1 (|T 11 (π − φ)| 2 + |T 12 (π − φ)| 2 ) dφπx 0{}= 2 ∞∑|b 0I | 2 + 2 (|b nI | 2 + |a nI | 2 ) , (1.61)xn=1Q ext I = 2 x Re{T 11(π = φ)}}= 2 ∞∑{bx Re 0I + 2 (b nI ) , (1.62)n=120∫ 2πQ sca II = 1 (|T 22 (π − φ)| 2 + |T 21 (π − φ)| 2 ) dφπx 0{}= 2 ∞∑|a 0II | 2 + 2 (|b nII | 2 + |a nII | 2 ) , (1.63)xn=1Q ext II = 2 x Re{T 22(π = φ)}}= 2 ∞∑{ax Re 0II + 2 (a nII ) , (1.64)n=1

1.4. Far field theorywhereT 11 (π − φ)T 22 (π − φ)T 12 (π − φ)T 21 (π − φ)≡≡≡≡∞∑n=−∞∞∑n=−∞∞∑n=−∞∞∑n=−∞b nI e −in(π−φ) ,a nII e −in(π−φ) ,a nI e −in(π−φ) ,b nII e −in(π−φ) (1.65)are the four components of the amplitude scattering matrix T, as defined byKerker [8] and Bohren & Huffman [11]. 1 2Two general features can be mentioned from (1.61)-(1.64), the efficiencyfactors for light falling obliquely on a cylinder long compared to its radius:• The efficiencies are expansions in the size parameter kR of the particlein question.• The extinction quantities only depend on the scattering amplitudes inthe forward direction (φ = π), while it contains the effect of scatteringin all directions by the particle. This is a particular form of the opticaltheorem and a intuitive explanation is given in [7], [11].The efficiencies Q abs , Q sca and Q ext are the main quantities which can bemeasured in optical experiments. If the resolution in a particular experimentis high enough, also the differential efficiencies dQ sca /dφ can be estimated,which are given bydQ sca I /dφ = 1πx (|T 11(π − φ)| 2 + |T 12 (π − φ)| 2 ),dQ sca II /dφ = 1πx (|T 22(π − φ)| 2 + |T 21 (π − φ)| 2 ). (1.66)They specify the angular distribution of the scattered light.It is important to note that the efficiencies defined here in principle cantake values larger then unity, contrary to what one should expect from themeaning of the word ”efficiencies”. In particular it can be shown that in thegeometrical limit, i.e. if all the dimensions of the scattering object are much1 The expressions for Q ext are derived after quite a lot of algebraic work [11], it can bedone faster by using the optical theorem in advance [7].2 The transformation of φ to π − φ comes from the definition of the incident wave: asin [11] the incident wave is in the −ê x direction, while Van de Hulst [7] and Kerker [8]use the opposite and no transformation is needed.21

Chapter 1.General solutionlarger then the wavelength, the extinction efficiency approach the limitingvalue two. This is rather peculiar, because it suggests that the object removestwice the energy that is incident on it. This so called extinction paradoxis resolved by taking also diffraction into account: the edge deflects rays inits neighborhood which from a geometrical view would have passed undisturbed.In this way the incident wave is influenced beyond the geometricalsize of the scattering particle.22

Chapter 2Small dielectric cylindersAs stated in chapter 1, it is not possible to express the general solution of thescattering problem explicitly as a function of the material properties (dielectricconstant, wire radius), geometric configuration (angle of incidence,radial distance from cylinder) and the wave number of the incident light. Inthis chapter the special case of cylindrical wires with radius small comparedto the wavelength of the incident light will be treated. It will be shown thatin this approximation it is possible to get an analytic solution. In section 2.4numerical results are given for InP.2.1 Coefficients in Rayleigh approximationWhen the radius of the cylinder is sufficiently small compared to the wavelengthof the incident light, the Bessel functions appearing in the scatteringcoefficients can be expanded in terms of kR. To be precise, sufficiently smallmeans the following condition:where|m|x ≪ 1, (2.1)x ≡ k 0 R (2.2)is defined as the size parameter of the circular cylinder. This condition isphysically based on the two Rayleigh assumptions:• The wave behavior of the incident field can be neglected with respectto the size of the particle: x ≪ 1. This implies the external field canbe considered as an homogeneous field.• The applied field should penetrate so fast into the particle that thestatic polarization is established in a time t short compared to theperiod T , so t/T ≪ 1. Since the velocity inside the cylinder is c/mand the wave period T = 1/ck this assumption is satisfied by (2.1).23

Chapter 2.Small dielectric cylindersWith condition (2.1) the following expressions for the Bessel and firstHankel functions can be used:J 0 (z) ≃ 1 − z24 , J ′ 0(z) ≃ − z 2 ,J 1 (z) ≃ −J 0 ′ (z) , J 1(z) ′ ≃ 1 2 − 316 z2 ,H 0 (z) ≃ 1 + 2iπ γ + 2iπ log z 2 , H′ 0(z) ≃ 2i 1π z ,H 1 (z) ≃ −H 0 ′ (z) , H′ 1(z) ≃ 2i 1π z 2 , (2.3)where | z | ≪ 1, H denotes the first Hankel function and γ is Euler’s constant.The Hankel functions in 2.3 are expanded to zeroth order in z, because thisis sufficient for a second order approximation of the scattering coefficients.With these expansions it can be easily shown that the scattering coefficients(1.41) and (1.42) up to second order in the size parameter x are approximatedby:Case Ia 0I (x, θ) = 0,a 1I (x, θ) = πx24(m 2 − 1)(m 2 + 1) sin θ + O(x4 ),b 0I (x, θ) = − iπx24 (m2 − 1) cos 2 θ + O(x 4 ),b 1I (x, θ) = − iπx24Case IIa 0II (R, θ) = O(x 4 ),a 1II (R, θ) = − iπx2 (m 2 − 1)4 (m 2 + 1) + O(x4 ),b 0II (R, θ) = 0,b 1II (R, θ) = − πx24(m 2 − 1)(m 2 + 1) sin2 θ + O(x 4 ), (2.4)(m 2 − 1)(m 2 + 1) sin θ + O(x4 ), (2.5)This result is in agreement with the earlier work of Wait [9], [10] andalso gives the expressions derived by Van de Hulst [7] and Kerker [8] fornormal incidence (θ = 0) . The internal coefficients for the fields inside thewire, c nI , d nI , c nII and d nII are not explicitly shown here because they havea complex form. They follow directly from equations (1.41) and (1.42).In principle now it is possible to proceed further and use equations (2.4) and(2.5) for the approximation of the fields (equations (1.21)-(1.26)). However,it is really important to be careful, because an expansion of the fields tosecond order in the size parameter needs more and further expanded coefficientsthen showed above.24

2.2. Fields inside the wire2.2 Fields inside the wireTo our best knowledge the expressions for the fields inside a dielectric cylinderhave only been derived in the dipole limit x → 0 [7][13]. This is arelatively small result compared to the the huge amount of research donein the far field region outside the scattering object, where a lot of interestin particular for applications in meteorology and astronomy worked as adriving force.In the dipole approximation, also used by Wang and Lieber [1], theincident field is really taken to be homogeneous. It is a special, strongerform of the Rayleigh approximation discussed above, since the second ordercorrections now are completely neglected. In this way the expressions forthe fields are independent of the size parameter and are derived in terms ofthe incident fields as [7][13]:E int ‖ = E 0 ‖ , (2.6)E int ⊥ =21 + m 2 E 0 ⊥ . (2.7)Here we will extend this solution to finite values of the size parameterx. Before doing this care has to be taken by expanding the coefficients, asnoted before. This is because the internal fields also depend implicitly on thesize parameter via ρ, apart from the explicit dependence via the coefficients.This implicit dependence can be split up in two parts:• The internal fields are expressed in terms of J n (jρ), see (1.30)-(1.37).In second order this results in a ρ dependence by (2.3).• Some of the components of the fields (1.21)-(1.26) have an extra 1/ρdependence.Taking this into account for a second order approximation of the internalfields one needs the internal coefficients c nI , d nI , c nII and d nII up to thefollowing orders in x:n 0 1 2 3 ...order 3 2 1 0 ...Now it is a matter of mathematics to get the solution of the fields insidethe wire up to second order. Since the expressions for oblique incidenceare too complex to show in an illuminating way, only the results for normalincidence are showed below.Starting with Case I, where for normal incidence Ẽint I ρ(x, 0) and Ẽint I φ(x, 0)are directly zero (see end of section 1.3), the z component of the electric field25

Chapter 2.Small dielectric cylindersbecomesE int ‖ z (x, 0) = E 0 e −iωt { 1 − imk 0 ρ cos φ − m2 k0 2ρ2cos 2 φ +214 (m2 − 1)(1 − ρ2R 2 ) x2 + (2.8)1(4 (m2 − 1) −2γ + iπ − 2 log x ) }x 2 + O(x 3 ).2As required, this solution reduce to (2.6) in the dipole limit x → 0(note: mk 0 ρ = x ρ R→ 0). Also the limit m → 1 provides the desired resultE int I = E 0 I : for m = 1 there is no optical difference between inside andoutside any more.Furthermore, the part between brackets in equation (2.8) can be divided inthree parts, each written on a different line here and each with a differentphysical background:• The first part (including the Ẽ0 e −iωt term in front of the brackets)expresses the original wave behavior of the incident field: it is theexpansion of e −imkρ cos φ up to second order.• As the cylinder radius increases, the effect of optical focusing gets amore important role. This is described by the second part: it has itsmaximum in the middle of the cylinder and falls off quadratically tozero at ρ = R. Note that this term is quadratic in the size parameter.• Also the third part is quadratic in x, but in contrast to the second termconstant over the wire. It has its origin in the expansion of H 0 (kρ), see(2.3). It is a rather striking expression: the iπ part in it can be seen asa constant phase shift of the field. Roughly speaking it is responsiblefor a correction on the absorption: taking the absolute value squaredthis iπ part mixes with the complex part of m and decreases the fluxof energy crossing the cylinder.Note that the log x 2part together with the x2 after the brackets isfinite: lim x→0 x 2 log x = 0. It gives positive contribution to I int =|Ẽint I z (x, 0)| 2 that can be large enough to get a value for I int /I 0larger then unity. This is an example of the extinction paradox, aswill be explained further in the next sections where a link will bemade between internal quantities and the external efficiency factors.The components of the electric field in Case II show mainly the samefeatures as mentioned above. Remember from section 1.3 that the internalelectric field is perpendicular to the wire axis (TE) at normal incidence, soẼ int II z (x, 0) = 0. The other two components become26

2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation{2Ẽ int II ρ (x, 0) = sin φ(m 2 + 1) Ẽ0 e −iωt 1 − imk 0 ρ cos φ − m2 k0 2ρ22cos 2 φ +18 (m2 − 1)(1 − ρ2R 2 ) x2 + (2.9)1 (m 2 (− 1) 14 (m 2 + 1) 2 − 2γ + iπ − 2 log x ) }x 2 + O(x 3 ),2Ẽ int II φ (x, 0) ={2cos φ(m 2 + 1) Ẽ0 e −iωt 1 − imk 0 ρ cos φ − m2 k0 2ρ2cos 2 φ +238 (m2 − 1)(1 − ρ2R 2 ) x2 + (2.10)1 (m 2 (− 1)4 (m 2 −(m 2 + 1 + 1) 2 ) − 2γ + iπ − 2 log x ) }x 2 + O(x 3 ).2Again, the terms on the first line describe the original wave behavior ofthe incident field. The incident TE field was taken to be in the positive ê ydirection and decomposing this in cylindrical coordinates one gets indeedthe first terms in the expressions (2.9) and (2.10).2.3 Efficiency, polarization anisotropy and - contrastin Rayleigh approximationContrary to the quantities inside the wire, in the far field region it is possibleto get simple analytic expressions in Rayleigh approximation at obliqueincidence. For expansion of the scattering and extinction efficiencies (1.61)-(1.64) to third order in the size parameter x, the coefficients have to beestimated to second and fourth order for Q sca and Q ext , respectively.The third order term for Q ext I and Q ext II are too extended to showhere, but are included in all calculations of the next section.With this in mind the efficiencies (1.61)- (1.64) are approximated by:{Q sca I (x, θ) = |m 2 − 1| 2 cos 4 θ + 2 sin2 θ(1 + sin 2 }θ) π 2 x 3|m 2 + 1| 2 + O(x 5 ) ,8(2.11)Q sca II (x, θ) = |m2 − 1| 2 {2 − cos 2|m 2 + 1| 2 θ } π 2 x 3+ O(x 5 ) , (2.12)4{Q ext I (x, θ) = Im{m 2 − 1} cos 2 θ +4 } sin2 θ πx|m 2 + 1| 2 2 + O(x3 ), (2.13)Q ext II (x, θ) = Im{m2 − 1}|m 2 + 1| 2 2πx + O(x3 ) . (2.14)27

Chapter 2.Small dielectric cylindersAs required, in the limit m → 1 all efficiencies become zero: the refractiveindex is the same everywhere and there is no scattering any more. Also inthe dipole limit x → 0 the efficiencies become zero: in the far field regionthe scattered field can be neglected with respect to the incident field.The solutions (2.11)-(2.14) depend in a specific way on the angle ofincidence θ, which will be illustrated in the next paragraph. It has its originin the boundary conditions (1.9)-(1.12) on the fields.Apart from this the limit θ → π 2gives an extra requirement: in this limitthe difference between Case I and Case II has to vanish as can be arguedwith symmetry arguments. This can be seen by taking θ = π 2in Figure 1.1.In both cases the incident fields E 0 and H 0 become perpendicular to thewire axis and by rotational symmetry around this axis Case I and Case IIdescribe the same situation. Indeed, taking the limit θ → π 2the efficienciesin Case I and Case II are equal:Q sca I (x, π 2 ) = Q sca II (x, π 2 ) = |m2 − 1| 2 π 2 x 3|m 2 + 1| 2 , (2.15)2Q ext I (x, π 2 ) = Q ext II (x, π 2 ) = Im{m2 − 1}|m 2 + 1| 2 2πx + O(x3 ) . (2.16)It is a limit in the sense that an incident wave in the same direction as thewire axis (θ = π 2) needs a special treatment. How to consider light incidenton the endpoints of an (relatively) infinite cylinder? In fact this is the situationof wave guiding, which will not be treated in this paper.Furthermore, the solutions (2.11)-(2.14) can be compared to literaturefor θ = 0. At normal incidence they are in agreement with the efficiencyfactors derived by Van de Hulst [7] and Kerker [8]:Q sca I (x, 0) = |m 2 − 1| 2 π2 x 38+ O(x 5 ) , (2.17)Q sca II (x, 0) = |m2 − 1| 2 π 2 x 3|m 2 + 1| 2 + O(x 5 ) , (2.18)4Q ext I (x, 0) = Im{m 2 − 1} πx2 + O(x3 ), (2.19)Q ext II (x, 0) = Im{m2 − 1}|m 2 + 1| 2 2πx + O(x3 ) . (2.20)2.3.1 Polarization anisotropy, polarization contrastIn order to express the difference between incident TM (Case I) and TE(Case II) waves properly, it is common to define a polarization anisotropyρ [1] [26]. Up to now, for dielectric cylinders this quantity has mainly been28

2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximationestimated by looking at the internal fields at normal incidence and in thedipole limit. Denoting the polarization anisotropy in this special case byρ int , it is defined by:ρ int ≡ |E int I| 2 − |E int II | 2|E int I | 2 + |E int II | 2 , (2.21)where |E int | 2 = I int indicates the internal intensity in dipole approximation.By using the fields in dipole approximation (2.6) and (2.7) this yieldsρ int = |m2 + 1| 2 − 4|m 2 + 1| 2 + 4 . (2.22)In principle one could proceed further by using the expressions (2.8), (2.9)and (2.10) to get the expanded intensities and so an expansion of the polarizationanisotropy ρ int (x, 0) inside the wire , but in fact this last step requiresfar too much calculations: also the direction of the field has to be taken into account properly. Even harder, the intensity (or Poynting vector) startsto depend on the position in the wire.Instead, it is much easier to calculate the polarization anisotropy bymaking use of the efficiency factors in the far field region. In terms of theextinction efficiencies, the extinction polarization anisotropy is defined byρ ext ≡ Q ext I − Q ext IIQ ext I + Q ext II, (2.23)A formal prove that this ratio in general equals the polarization anisotropyinside the wire, ρ ext = ρ int , is complicated and will not be given here. Instead,the equality can be explained by the following argument: the internalfields are modified with respect to the incident field both by scattering andabsorption, so ρ int contains the relative difference of the total removal ofenergy. This is nothing else than the relative difference in extinction betweenthe two cases, which is described by ρ ext .Contrary to the general case, it is easy to prove the equality at normalincidence in the dipole limit. Using equations (2.19) and (2.20), ρ ext isapproximated byρ ext (x, 0) = |m2 + 1| 2 − 4|m 2 + 1| 2 + 4 + O(x2 ) = ρ int + O(x 2 ) , (2.24)so taking the dipole limit x → 0 on both sides yields ρ ext = ρ int .Next to the extinction polarization anisotropy defined above, it is alsoinsightful to define scattering and absorption polarization anisotropies. Sincescattering, absorption and extinction are related to each other by (1.60)only the scattering polarization anisotropy will be treated here. It is definedbyρ sca ≡ Q sca I − Q sca IIQ sca I + Q sca II. (2.25)29

Chapter 2.Small dielectric cylindersUsing (2.17) and (2.18) the scattering polarization anisotropy at normalincidence is expanded in the size parameter byρ sca (x, 0) = |m2 + 1| 2 − 2|m 2 + 1| 2 + 2 + O(x2 ) , (2.26)Although often used, the polarization anisotropy is not a quite usefulquantity to work with in practice. This will be illustrated in the next section,but at this stage it is anticipated by introducing a new quantity thatdescribes the difference between the case of incident TM waves and incidentTE waves. It is called the polarization contrast and defined byC ext ≡ Q ext IQ ext II, (2.27)C sca ≡ Q sca IQ sca II, (2.28)for the total removal of EM energy and for scattering, respectively.Again, using the expanded efficiencies (2.17)-(2.20) the approximationsat normal incidence areC ext (x, 0) = |m2 + 1| 2+ O(x 2 ) , (2.29)4C sca (x, 0) = |m2 + 1| 2+ O(x 2 ) . (2.30)2Note that the limit m → 1 does not work for the scattering polarizationanisotropies and contrasts any more.30

2.4. Results2.4 ResultsAs an illustration of the results in the previous sections it is insightful tochoose a particular material, InP for example. In particular it is interestingfor which radius (or size parameter) the Rayleigh approximation is valid.Recall that the complex refractive index is the only material property appearingin the classical theory derived here, apart from R. It depends onthe circular frequency ω of the incident light, see section 1.1 : the materialresponds to the incident periodic EM field and this response depends on thefrequency and so on the wavelength of the incident light.For InP this is illustrated in Figure 2.1. Actually the figure shows the realand imaginary parts ɛ ′ and ɛ ′′ of the complex dielectric function ɛ [11], whichis related to the complex refractive index bym 2 ≡ ɛ ≡ ɛ ′ + i ɛ ′′ . (2.31)Absorption and scattering are more simply described by these optical ”constants”,so from now on all quantities are discussed in terms of ɛ.Ε',Ε''forInP17.51512.510Ε'7.55Ε''2.50350 400 450 500 550 600ΛFigure 2.1: Bulk values of the real and imaginary part ɛ ′ and ɛ ′′ of thecomplex dielectric function ɛ for InP, as a function of λ 0 . It is calculatedby interpolating between 32 (optic) experimental values in this interval.From Figure 2.1 it becomes directly clear that showing the efficienciesas a function of the dimensionless size parameter x is misleading: it reallymatters if x is changed by varying the wave number or the radius. At differentwave numbers also ɛ has changed, only over a narrow range at smallvalues of k the optical constants can be considered as constant.However, in literature it is common to show the efficiency as a functionof x at a fixed ɛ, mainly because it is the most convenient way. This is also31

Chapter 2.Small dielectric cylindersthe starting point in this paper, see Figure 2.2. For six fixed values of λ 0 , sofor six different values of ɛ, the extinction efficiencies in Case I and Case IIup to third order in x at normal incidence are plotted as a function of x.QIext1.41.210.80.60.4608 nm508 nm422 nm395 nm376 nm354 nma:QIext,Θ ⩵ 00.200 0.01 0.02 0.03 0.04 0.05xQIIext0.0150.010.005608 nm508 nm422 nm395 nm376 nm354 nmb:QIIext,Θ ⩵ 000 0.01 0.02 0.03 0.04 0.05xFigure 2.2: Extinction efficiencies Q ext I and Q ext II at normal incidenceas a function of the size parameter x = kR. In every plot k, and so ɛ,is fixed. The corresponding values for the wavelength are showed in theleft corner. The efficiencies are expanded up to third order in x.In the illustrated domain the linear terms (2.19) and (2.20) in the extinctionefficiencies dominate and the total removal of EM energy from theincident beam increases with the size parameter.The slope of this relation depends on the wavelength and in a quite differentway for the two cases. It is explained by looking closely to the dependenceon the complex dielectric function:• In Case I the slope increases to a maximum around λ 0 = 400 nmafter which it decreases for increasing wavelength. This is caused bythe factor Im{m 2 − 1} = ɛ ′′ in (2.19), see Figure 2.1.• In Case II the slope decreases in the whole domain for increasing λ 0 .The increase of the denominator of Im{m2 −1} ɛ∝ ′′in (2.20)|m 2 +1| 2 (ɛ ′ ) 2 +(ɛ ′′ ) 2dominates the increase of the numerator to ∼ 400 nm. Afterwardsɛ ′′ falls off so fast that the slope remains decreasing for increasingwavelength.It even seems from the figure that the third order terms can be neglectedin the given domain. Nevertheless, a closer look gives the opposite: forincreasing x the third order terms start to give significant corrections. Thisis illustrated in Figure 2.3: here the correction by the third order terms withrespect to (2.19) and (2.20) are shown in percentages.The figure reveals that also the magnitude of the deviation depends onthe wavelength: in both cases the influence of the third order correctionbecomes larger for increasing wavelength. This is explained with the same32

2.4. Results%2015105608 nm508 nm422 nm395 nm376 nm354 nma:dev. QIext,Θ ⩵ 0%8642608 nm508 nm422 nm395 nm376 nm354 nmb:dev.QIIext,Θ ⩵ 000 0.01 0.02 0.03 0.04 0.05x00 0.01 0.02 0.03 0.04 0.05xFigure 2.3: Deviation of linear behavior in percentages of Q ext IQ ext II at normal incidence, as a function of x.andkind of arguments as for the extinction factors itself, but it is omitted heresince the third order corrections are not shown explicitly.In the same line as for the extinction efficiencies also the scattering andabsorption efficiencies can be illustrated. Since these quantities are dependentof each other only Q sca I and Q sca II are showed here. Figure 2.4 showsthat for small x the extinction is completely dominated by the absorption:the contribution of the scattering to the total extinction (Figure 2.2) is onlyabout 1%. This is due to the absence of a first order term in the expansionQIsca0.0120.010.0080.0060.0040.002608 nm508 nm422 nm395 nm376 nm354 nma:QIsca,Θ ⩵ 000 0.01 0.02 0.03 0.04xQIIsca0.000080.000060.000040.00002608 nm508 nm422 nm395 nm376 nm354 nmFigure 2.4: Scattering efficiencies Q sca I and Q sca II at normal incidenceas a function of x. In every plot k, and so ɛ, is fixed. The correspondingvalues for the wavelength are showed in the left corner. The efficienciesare expanded to third order in x.b:QIIsca,Θ ⩵ 000 0.01 0.02 0.03 0.04xof Q sca compared to Q ext , see equations (2.17)-(2.20).The correction to the third order terms in Figure 2.4 are depicted inFigure 2.5. This is the deviation caused by the x 5 terms in percentages.The calculation of these x 5 terms is only done for scattering, since in thiscase the coefficients are needed to x 4 while for extinction one needs also thesixth order terms.33

Chapter 2.Small dielectric cylinders%432608 nm508 nm422 nm395 nm376 nm354 nma:err. QIsca,Θ ⩵ 0%5432608 nm508 nm422 nm395 nm376 nm354 nmb:err.QIIsca,Θ ⩵ 01100 0.01 0.02 0.03 0.04x00 0.05 0.1 0.15 0.2xFigure 2.5: Correction to the third order expansion of Q sca I and Q sca IIby the x 5 terms in percentages; again at normal incidence, as a functionof x.The above illustrations are useful for determining the range of the sizeparameter in which a certain approximation is valid. For instance, Figure 2.5shows that for a maximal deviation of 5% the third order approximationholds to x ∼ 0.03 and x ∼ 0.2 for Q sca I and Q sca II , respectively.However, this way of displaying is quite awkward for investigating thewavelength dependence: the different curves belong to different values of k.Actually, keeping the wire radius fixed requires looking at a smaller x valueby going to a curve at higher wavelength.Also the determination of the limiting R values will not work properly.A size parameter x ∼ 0.03 at λ 0 = 400 nm gives R ∼ 2.2 nm, but it would bemuch more convenient to get these values as a function of the wavelength.Before doing this, the next paragraph will illustrate the dependence on theangle of incidence.2.4.1 Efficiencies and polarization anisotropy at oblique incidenceThe most dominant feature that will appear by displaying the efficienciesas a function of the angle of incidence θ is the symmetry requirement forθ = π 2, as explained in section 2.3.Starting with extinction, it is important to note that also the x 3 termsare encountered in Figure 2.6. This means for instance that the first orderterm (2.14), which is independent of θ, is corrected a little bit by the thirdorder term. The scattering efficiencies are shown in Figure 2.7. In bothcases the cylinder radius is fixed, R = 2 nm. Figure 2.6 illustrates the θdependence of Q ext I and Q ext II as given in equations (2.13) and (2.14).The difference between Q ext I and Q ext II at θ = 0 is explained by the largedenominator in (2.14): |m 2 +1| 2 ≃ 200. For increasing θ, Q ext I decreases toa limiting value which is equal to Q ext II at θ = π 2: the first term in (2.13)34

2.4. Resultsfalls off to zero, while the second term increases leading to Q ext I ≃ Q ext IIat θ = π 2. Also the scattering efficiencies, shown in Figure 2.7, are the sameat θ = π 2. As discussed before this is due to the fact that the TM and TEcase describe the same physical situation at θ = π 2 .QIext0.80.60.40.20a:QIext, R ⩵ 2 nm0 Π 8Π4Θ608 nm508 nm422 nm395 nm376 nm354 nm3 Π8Π2QIIext0.0140.0120.010.0080.0060.0040.0020b:QIIext, R ⩵ 2 nm376 nm354 nm0 Π 8422 nm395 nmΠ4Θ608 nm508 nm3 Π8Π2Figure 2.6: Extinction efficiencies Q ext I and Q ext II as a function of theangle of incidence θ for a fixed cylinder radius R = 2 nm. In every plotk, and so ɛ, is fixed. The efficiencies are expanded to third order in x.QIsca0.01750.0150.01250.010.00750.0050.002500 Π 8a:QIsca, R ⩵ 2 nmΠ4Θ608 nm508 nm422 nm395 nm376 nm354 nm3 Π8Π2QIIsca0.000150.00010.000050b:QIIsca, R ⩵ 2 nm422 nm395 nm376 nm354 nm0 Π 8608 nm508 nmΠ4Θ3 Π8Π2Figure 2.7: Scattering efficiencies Q sca I and Q sca II as a function of θfor R = 2 nm. In every plot k, and so ɛ, is fixed. The efficiencies areexpanded to third order in x.In order to look more closely to the difference between TM and TEwaves, the extinction polarization anisotropy (2.23) as well as the scatteringpolarization anisotropy (2.25) corresponding to the expanded efficiencies atR = 2 nm are depicted in Figure 2.8. Indeed, ρ sca and ρ ext become zero in thelimit θ → π 2. At normal incidence the polarization anisotropies reach theirmaximum value, around 0.985 for extinction as well as for scattering. Butthe distinction between the curves for different wavelength is hard to extractfrom the figures. Also the difference between extinction and scattering isnot illustrated clearly.As stated in section 2.3 it is more convenient to look at the polarization35

Chapter 2.Small dielectric cylinders1a:ΡscaforR ⩵ 2 nm1a:ΡextforR ⩵ 2 nmΡsca0.8608 nm0.6 508 nm422 nm0.4395 nm0.2 376 nm354 nm00 Π 8Π4Θ3 Π8Π2Ρext0.8608 nm0.6 508 nm422 nm0.4395 nm0.2 376 nm354 nm00 Π 8Π4Θ3 Π8Π2Figure 2.8: Polarization anisotropy ρ ext and scattering polarization anisotropyρ sca as a function of θ for R = 2 nm. In every plot k, and so ɛ,is fixed. The factors are expanded up to second order in x.contrast, equations (2.27) and (2.27). For R = 2 nm this is displayed inFigure 2.9. Now the difference between scattering and extinction becomesclear: at normal incidence the depolarization for scattering is significantlarger then for extinction. By rotating the angle of incidence to θ = π 2 thescattering polarization anisotropy also decreases faster to the limiting valuezero. This effect is also visible in Figure 2.8, but less clearly.C sca2502001501005000 Π 8C scaforR ⩵ 2nmΠ4Θ608 nm508 nm422 nm395 nm376 nm354 nm3 Π8Π2C ext1401201008060402000 Π 8C extforR ⩵ 2nmΠ4Θ608 nm508 nm422 nm395 nm376 nm354 nm3 Π8Π2Figure 2.9: Polarization contrast C ext and scattering polarization contrastC sca as a function of θ for R = 2 nm. In every plot k, and so ɛ, isfixed. The factors are expanded up to second order in x.2.4.2 Efficiencies and polarization anisotropy as a functionof wavelengthAs stated in section 2.4, the most physically accurate picture of the scatteringprocess is obtained by showing the efficiencies and polarization anisotropiesas a function of the wavelength (or wave number).Apart from the practical points made in the previous sections, this kind36

2.4. Resultsof figures also contain far more information: for every wavelength a set ofoptical constants has to be used. This is done by interpolating betweenmeasured data points for the complex dielectric function, see Figure 2.1. Inthis way the response of the dielectric cylinder as a function of the frequencyof the incident EM field is obtained.QIext2.521.510.5a:QIext, Θ⩵05nm4nm3nm2nm1nmQIIext0.030.020.01b:QIIext,Θ⩵05nm4nm3nm2nm1nm0350 400 450 500 550 600Λ0350 400 450 500 550 600ΛFigure 2.10: Extinction efficiencies Q ext I and Q ext II at normal incidenceas a function of the wavelength at constant cylinder radii. Thecorresponding five R values are showed in the right corner. The efficienciesare expanded to third order in x. The plots are calculated byinterpolating between thirty-two (optic) experimental values of ɛ in thisinterval.QIsca0.30.250.20.150.10.05a:QIsca,Θ⩵05nm4nm3nm2nm1nm0350 375 400 425 450 475 500ΛQIIsca0.00150.001250.0010.000750.00050.00025b:QIIsca, Θ⩵05nm4nm3nm2nm1nm0350 400 450 500 550 600ΛFigure 2.11: Scattering efficiencies Q sca I and Q sca II at normal incidenceas a function of the wavelength at constant cylinder radii. The efficienciesare expanded to third order in x.For the extinction and scattering efficiencies at normal incidence this isdepicted in Figure 2.10 and Figure 2.11, respectively. The domain of thewavelength is limited: the high and low frequency regions are omitted. Thefigures show the dependence of the wavelength at five fixed values of R.The dependence of the cylinder radius is visible, especially for extinction:the extinction efficiencies increase linearly with increasing radius by goingfrom one curve to the next in Figure 2.10.37

Chapter 2.Small dielectric cylindersThe shape of the curves has been explained above: they where alreadyvisible in Figure 2.2 and Figure 2.4, but not so clearly. Now the particulardependence of (2.19) and (2.20) on the complex dielectric function is reallyvisible. For instance, Figure 2.10 nicely shows that the behavior of theimaginary part ɛ ′ is completely reflected in the extinction efficiency.%20151055nm4nm3nm2nm1nma:err. QIsca,Θ⩵00350 400 450 500 550 600Λ%1.51.2510.750.50.25b:err. QIIsca,Θ⩵05nm4nm3nm2nm1nm0350 400 450 500 550 600ΛFigure 2.12: Correction to the third order expansion of Q sca I and Q sca IIby the x 5 terms in percentages; again at normal incidence, as a functionof λ 0 at constant R.In the same way as in Figure 2.5, the correction to the third order termsin Figure 2.11 is illustrated in Figure 2.12. This is the deviation caused bythe x 5 terms given as a percentage. For a cylinder radius below 2 nm theused expansion is accurate to 5%. For larger radii it really depends on thewavelength if the approximation is acceptable.Ρsca0.9920.990.9880.9860.9840.982a:Ρsca,Θ⩵015nm0.98350 400 450 500 550 600ΛΡext0.9850.980.9750.970.9650.96b:Ρext,Θ⩵05nm4nm3nm2nm1nm350 400 450 500 550 600ΛFigure 2.13: Polarization anisotropy ρ ext and scattering polarization anisotropyρ sca at normal incidence as a function of the wavelength at constantcylinder radii. The corresponding five R values are showed in theright corners. Figure (a) shows that ρ sca is independent of the wire radius.The polarization anisotropy ρ ext is expanded including the secondorder term, ρ sca up to second order in x.The polarization ratio and scattering polarization ratio corresponding toFigure 2.10 and Figure 2.11 are shown in Figure 2.13. Figure 2.13 (a) shows38

2.4. Resultsthat ρ sca expanded up to second order in x is independent of the wire radius,see (2.26). For extinction also the second order is included. Figure 2.13 (b)shows that in that case ρ ext depends on the wire radius: in particular forwavelengths larger then 400 nm the polarization ratio becomes larger. Inother words, increasing the wire radius implies a bigger difference betweenthe case of an incident wave with the magnetic field perpendicular to thewire axis (TM) and the case where the electric field is perpendicular (TE).Remember that this result only applies for small R, results for larger radiusor size parameter are showed in [7] [8] [11].It is really interesting to compare the obtained polarization anisotropyρ ext with the depolarization in the dipole limit ρ int (2.22), also used in [1][26].This is shown in Figure 2.14. The blue curve shows the case when for ρ intin the dipole limit only the real part of the bulk ɛ is taken into account.The difference with the the red curve, indicating ρ int for the complete ɛ,becomes dramatically large for λ below 400 nm, and remains significant forthe other wavelengths.Ρ0.980.960.940.920.90.880.86a:Ρext,Ρint; ,Θ⩵0Ρint, dip.app.,Ε'Ρint, dip.app.Ρext,R⩵5nmΡext,R⩵2nm350 400 450 500 550 600ΛC ext C int140120100806040200b:C ext,C int;Θ⩵0C int, dip.app.,Ε'C int, dip.app.C ext,R⩵5nmC ext,R⩵2nm350 400 450 500 550 600ΛFigure 2.14: Comparison of the internal polarization anisotropy/contrastin the dipole limit with ρ ext / C ext . In blue curve, indicating ρ int , onlythe real part of ɛ is taken into account. The red curve shows ρ int for thecomplete ɛ .The yellow and green curve show that increasing the wire radius to R =5 nm already gives a significant difference between the solution of the dipolelimit and the expanded polarization anisotropy ρ ext . Figure 2.14 (b) showsthe same results in terms of the contrasts.39

Part IIAbsorption40

Chapter 3Electronic propertiesIn this chapter the electronic properties of nanowires made from III-V compoundsare discussed. The results are based on more detailed studies whichcan be found in basic semiconductor books [14] [15] and articles by Luttinger[16], Sercel [17] and Marechal [18].The electronic band structure and wave functions in a nanowire are calculatedusing the effective mass approximation. This method is in particularconvenient to study the optical properties of a semiconductor structure, becauseanalytic expressions for the band dispersion, effective mass and electron/holewavefunctions around high symmetry points can be obtained.Before turning to the nanowire, in section 3.1 first the band dispersion inbulk material will be derived. Next to general theory, the specific situationof the degenerate top valence band in III-V semiconductor materials will betreated. It has its specific importance in the next chapters and will thereforealso be the guideline in the other sections of this chapter: in section 3.2 theeffective mass theory for bulk systems is treated, sections 3.3 and 3.4 summarizethe envelope description in case of an infinite nanowire and explicitresults for InP and InAs are found in section 3.5.3.1 The k · p methodThere are various ways to determine the electronic bands of a semiconductor.Global dispersion relations of bulk materials are available (pseudo-potentialtechniques, tight binding) but in a lot of cases they are unnecessary.In particular, for describing the optical properties of a semiconductor structureit is often sufficient to know the band dispersion in a small range aroundthe band extremes. This is achieved by the k · p method, which differs fromthe procedures mentioned above in the fact that, next to the band gaps, alsothe oscillator strengths of the transitions are used as input. In the k · p methodthe band dispersion around any point k a is obtained by extrapolationfrom the k = k a energy gaps and optical matrix elements, using either de-41

Chapter 3.Electronic propertiesgenerate or non-degenerate perturbation theory . The input data at k = k acan be obtained from experimental results, typically at the high symmetrypoints of the crystal.Starting point is the one-electron Schrödinger equation describing themotion of an electron in an averaged potential V (r), which is obtained fromthe Hamiltonian of a perfect crystal containing N unit cells after usualassumptions such as the Born-Oppenheimer and mean field approximation.The potential V (r) is assumed to reflect the periodicity of the perfect crystal:V (r + R) = V (r), (3.1)where R are the lattice vectors. Including the spin-orbit interaction theHamiltonian describing the unperturbed semiconductor becomesH 0 = p2+ V (r) +2m 0 4c 2 m 2 (σ × ∇V ) · p, (3.2)0where m 0 denotes the free electron mass and σ are the Pauli spin matrices.The relativistic character of the spin-orbit term is reflected by the 1 dependence.c 2Note that the total Hamiltonian, including the spin-orbit interaction, is invariantunder a translation by R. H 0 thus commutes with the translationoperator of the crystal and has Bloch functions as solutions. After normalizingover the whole crystal, containing N unit cells, these are definedas:ψ nk (r) = N − 1 2 e ik·r u nk (r), (3.3)where the u nk ’s have the periodicity of the lattice, are normalized over oneunit cell and k lies in the first Brillouin zone.The Bloch functions (3.3) form a complete and orthonormal set. Nextto this, the Bloch solutions ψ n0 = N − 1 2 u n0 at k = 0 are also periodic. Oncethese, or to be more precise, the corresponding interband matrix elementsand energies ɛ n ≡ ɛ n (0) are known, the energy dispersion around the zonecenter (k = 0) can be derived using perturbation theory.In principle, this argumentation can be extended to any point k = k a ,provided the transition matrix elements and energies at k = k a are known.This result has been widely discussed in literature [14] [15] [16] [17] [18],here only the results around the zone center are summarized.Assuming the band structure has an extremum (almost) at the zonecenter and taking k sufficiently small 1 , the dispersion relation for a nondegenerateband (apart from spin) is given byɛ n (k) = ɛ n + 2 k 2, (3.4)2m ∗ n1 Sufficiently small means that the corresponding energy difference ɛ n(k) − ɛ n remainsmuch smaller then the band edge differences ɛ n − ɛ n ′ and that the terms linear k are smallenough to be neglected.42

3.1. The k · p methodwhere m ∗ n is the effective mass of the band,1m ∗ nand π nn ′= 1 m 0+ 2m 2 0 k2 ∑n ′ ≠n| π nn ′ · k | 2ɛ n − ɛ n ′are the the interband matrix elements at the zone center:(3.5)π nn ′ ≡ 〈u n0 | p + 4m 0 c 2 σ × ∇V | u n ′ 0〉 (3.6)∫(= d 3 r u ∗ n0(r) p + )4m 0 c 2 σ × ∇V u n ′ 0(r). (3.7)With the same assumptions, for a degenerate band relation (3.4) is replacedbyh j,j ′(k) = ɛ j δ j,j ′ + 2 k 221m ∗ jj ′ (3.8)with1m ∗ = 1 δ j,j ′ + 2jjm ′ 0 m 2 0 k2∑ (k · π jm )(k · π mj ′)ɛ j − ɛ n ′ɛ n ′≠ɛ j. (3.9)Here j denotes the degeneracy and the summation over n ′ describes the couplingbetween the group of degenerate states and the other bands. Contraryto the case of a non-degenerate band, one is left with a matrix h j,j′ (k) whichhas to be diagonalized in order to get the dispersion relation(s).It should be noted that within the notation used here the tensor behaviorof the effective mass is neglected. In general, the coupling between k andthe interband matrix elements causes the effective mass to be non isotropicand inclusion of this effect is achieved by the substitution1m ∗ n1m ∗ αβnk 2 −→ ∑ αβ=1m ∗ αβn1m 0δ αβ + 2 m 2 0k α k β , (3.10)∑n ′ ≠nπ α n ′ n πβ nn ′ɛ n − ɛ n ′(3.11)in (3.4) and a similar one in case of a degenerate band. The surfaces ofconstant energy belonging to this effective mass tensor are not spheres anymore, but warped in certain directions, depending on the symmetry propertiesof the band under consideration.Furthermore, there are three remarks important to be made at this stage.First, in most of the cases the summation over the bands n ′ in (3.5)and (3.9) can be executed over a limited number of values. For large energydifferences ɛ n − ɛ n ′ the contribution of n ′ to the effective mass becomes43

Chapter 3.Electronic propertiesrelatively unimportant. Also the interband matrix elements in the numeratorreduces the number of bands that contribute to the effective mass. Aswill be explicitly shown in subsection 4.4.1, the matrix elements are subjectto selection rules which are determined by the symmetry properties of thebands in question. Most of the matrix elements become zero by this kind ofsymmetry arguments.Secondly, including the spin-orbit interaction in case of a non-degenerateband makes little practical difference, since its effect is absorbed in the interbandmatrix elements which are determined by experiment. For a degenerateband this is different, because the spin-orbit interaction in general lifts thedegeneracy and will cause small splitting between the bands.A last important remark has to be made concerning the limitations of thek·p method as depicted here, up to second order in k. The above results relyon the assumption that ɛ n (k)−ɛ n remains much smaller then the band edgedifferences ɛ n − ɛ n ′ (and a similar assumption in case of degenerate bands),which is not necessarily satisfied, e.g. in semiconductor compounds with anarrow band gap. Instead of expanding beyond second order in the sameframework, a commonly used approach [14] [15] [17] initiated by Kane [19]solves this problem by diagonalizing the group of neighboring bands exactlyand afterwards treating the coupling with the well separated other bands ina second order perturbation. However, in the remaining part of this paperit is assumed that the bands under investigation are well separated from theothers, i.e. splitting terms as the band gap E g and spin-orbit splitting ∆ 0are assumed to be sufficiently large.3.1.1 Top valence bands in III-V semiconductorsIn principle the above theoretical statements now can be applied to anyband, or group of bands, once the energy and the interband matrix elementsare known. Here the band structure of the six fold degenerate (includingspin) top valence band at the Γ point (k = 0) in III-V semiconductor compoundswill be summarized. However, the explicit diagonalization of thematrix (3.8) will be performed further on in the envelope function frameworksince this is the most convenient way when the nanowire structure isanticipated.Starting with symmetry considerations, it is well known [14][15] that thetop valence bands in III-V materials have Γ 4 like symmetry, apart from spin.The corresponding spatial parts of the valence band wavefunctions at k = 0are p-like, which means that they are triply degenerate and transform underrotations like the three components of a vector. Including spin this leads tosix band edge Bloch functions, which are denoted by |X〉|σ〉, |Y 〉|σ〉, |Z〉|σ〉,with σ =↑,↓.The one-electron Hamiltonian H 0 is diagonalized by linear combinationsof these band edge Bloch functions. Rewriting the spin orbit term in (3.2)44

3.1. The k · p methodasH s.o. = λL ′ · σ, (3.12)with L ′ the angular momentum of the atomic states and treating H s.o. as asmall perturbation 2 , this term is diagonalized by the eigenfunctions of thetotal angular momentum J = L ′ + σ of the atomic states. Subsequentlythe total Hamiltonian H 0 can be expressed in the transformed zeroth ordereigenfunctions |j, j z 〉, with j the eigenvalues of J and j z the eigenvalues ofits projection J z along the z axis. These are defined as∣ 3∣ 32 , 3 2〉= −1 √2 |X + iY 〉| ↑ 〉, (3.13)√22 , 1 2〉= − √6 1|X + iY 〉| ↓ 〉 +3|Z〉| ↑ 〉, (3.14)∣ 32 , − 1 〉 √2 = √6 12|X − iY 〉| ↑ 〉 +3|Z〉| ↓ 〉, (3.15)∣ 32 , − 3 〉2 = √2 1|X − iY 〉| ↓ 〉 (3.16)for the j = 3 2quadruplet and∣ 12 , 1 2〉= − √3 1|X + iY 〉| ↓ 〉 + |Z〉| ↑ 〉, (3.17)∣ 12 , − 1 2〉= − √3 1|X − iY 〉| ↑ 〉 − |Z〉| ↓ 〉, (3.18)for the two j = 1 2 states. The last ones are split from the j = 3 2states by thespin-orbit interaction, with a magnitude ∆ 0 = 3 2 λ. For ∆ 0 sufficiently large,such that the matrix elements which couple the j = 3 2 and j = 1 2bands arenegligible compared to ∆ 0 , the 6 × 6 Hamiltonian can be decoupled into a4 × 4 and a 2 × 2 matrix.In most III-V semiconductors, the 4 × 4 matrix of the j = 3 2states correspondsto the top most valence band. Assuming the spin-orbit couplinglarge enough, in this paper the valence band dispersion will be derived bydiagonalizing the Γ 8 Hamiltonian of these j = 3 2states. This is achieved inthe same framework as used by Sercel [17] and as in [18]; the explicit resultsare given in section 3.2.As stated above, corrections to this approach can be found by including thesplit-off (Γ 7 ) band of the j = 1 2states and possibly also the lowest conductionband, which usually has Γ 6 symmetry. The last one in general isless important since in most III-V semiconductors the spin-orbit splitting ismuch smaller than the band gap E g . Including more bands will improve theresults, but makes the calculations harder. Focussing on the dispersion forsmall k around the zone center, it is assumed that these corrections can beneglected in first instance.2 λ is small because of the relativistic character of the spin-orbit interaction45

Chapter 3.Electronic properties3.2 Effective mass approximationSuppose an infinite system which is built from the perfect crystal, and adisturbance δV which has to be restricted by specific properties, as will beexplained below. The Schrödinger equation (S.E.) of the system is given by(H 0 + δV ) |Ψ〉 = E |Ψ〉. (3.19)In principle the solutions of the S.E. can be found by expanding Ψ in termsof the complete orthonormal set of Bloch functions, but without making anyfurther approximation this requires an extensive job since the disturbanceδV breaks the translational symmetry of the crystal.The problem is solved much easier by assuming δV to be slowly varyingover one unit cell and making use of the band parameters of the unperturbedsystem, equations (3.4) and (3.8). This approach is known as the effectivemass approximation. It can be derived either by utilizing Bloch functions, orin the context of the more localized Wannier functions. Here the Wannierfunctions are used. They are related to the Bloch functions by Fouriertransformation and defined by∑a nR (r) = N − 1 2 e −ik·R ψ nk (r). (3.20)kNote that the Wannier functions are indexed by the lattice vector R, reflectingthe localized character. They form a complete, orthonormal setjust as the Bloch functions and depend on the difference between r and R:a nR (r) = a n (r − R).Using the complete and orthonormal set of Wannier functions, the solutionΨ(r) of (3.19) is expanded as:Ψ(r) = ∑ jRF j (R)a jR (r), (3.21)where j sums over the j degenerate bands and thus includes only one bandn in the non degenerate case. The functions F j (R) are known as the envelopewave functions: as will be shown below, they describe wave packets,extended over (a part of) the crystal and are the envelopes of the atomisticvariations caused by the Wannier functions.In order to convert the total Hamiltonian H 0 +δV in (3.19) into operatorsacting on the Wannier functions, it is stated here that k and R are conjugateoperators in the sense thatR ←→ i∇ k and k ←→ −i∇ R , (3.22)in the limit of large N. Note that R now is treated as a continuous variable,which is justified by the large N limit, i.e. the size of the semiconductorcompound is much larger than the distance between the atoms.46

3.2. Effective mass approximationUsing this result and assuming that δV is a slowly varying function withrespect to a lattice vector, it can be shown [14][18] that the S.E. (3.19)reduces to a Schrödinger equation for the envelope functions:{ɛ n (−i∇ R ) + δV (R)}F n (R) = EF n (R) (3.23)in case of a non degenerate band n and∑j ′ {h j,j ′(−i∇ R ) + δV (R)}F j ′(R) = EF j (R) (3.24)for a degenerate band. For a given band, equation (3.23) ((3.24)) describesthe motion of a particle with effective mass m ∗ n (m ∗ jj ′ ) in a potential δV .Note that the total wave function of this particle, moving in the perturbedcrystal, is obtained from the solutions of (3.23)/(3.24) by multiplying withthe Wannier functions as in (3.21).3.2.1 Crystal Hamiltonian in envelope representationThe above envelope framework initially was derived in the context of impuritystates, but as stated by Sercel [17], the procedure can also be usedto develop a representation of the unperturbed Hamiltonian H 0 which anticipatesa centrosymmetric or cylindrical heterostructure. Instead of theWannier representation (3.21), the solution is expanded in the zone centerBloch functions |u j 〉 by the ansatz|Ψ〉 = ∑ j|F j 〉 |u j 〉, (3.25)which is justified if the energy difference ɛ j − ɛ n ′ between the degeneratebands and al others is sufficiently large such that u nk ≃ u n0 .The notation used in (3.25) stresses the fact that the envelope functions F jact in a different space as the zone center Bloch functions, this is shownin more detail in chapter 4 considering the transition matrix element. TheBloch functions are defined within a unit cell, while the envelopes are extendedover a sufficiently large group of lattice points. It should be mentionedagain that the assumption u nk ≃ u n0 is essential in this context.In addition to the assumptions in section 3.1 3 , an extra approximationhas to be made here concerning the anisotropy, in order to profit fully fromthe envelope representation. Conform the situation in most of the III-Vsemiconductor materials, it is assumed that the anisotropic terms in theHamiltonian can be neglected, at least as a first order approximation. Inthis spherical approximation the lower cubic terms causing the warping of3 I.e. k sufficiently small and energy gaps such as ∆ 0 and E g large enough to neglectthe coupling of the band(s) under investigation with the others.47

Chapter 3.Electronic propertiesthe bands are set to zero by a restriction on the involved Kohn-Luttingerparameters: γ 2 = γ 3 [17][18].As an upshot, adopting the spherical approximation amounts to replacethe space group T d of the crystal with the full rotational group. Now thecrystal Hamiltonian is invariant under rotations and additional operatorscan be found which share the same basis of eigenstates. In a cylindricalrepresentation these operators are P z and F z , where P z is the envelopemomentum along the z-axis and F z denotes the total angular momentumalong the z axis:F z = J z + L z , (3.26)with L z the z component of the envelope angular momentum L. The zcomponent of the total angular momentum is a conserved operator, or inother words, F z commutes with the crystal Hamiltonian. Consequently,the eigenvalue f z of F z is a good quantum number and the Hamiltonian isdiagonal with respect to F z .3.2.2 Top valence bands in III-V semiconductorsThis is illustrated in more detail by narrowing the focus again to the situationof the top Γ 8 valence bands in III-V semiconductors.Following the same approach as in section 3.1, j in (3.25) sums over the j zvalues of the j = 3 2 quadruplet and |u j〉 = | 3 2 , j z〉. The envelope functions|F j 〉 now are represented as |k z ; k, m〉, where k z is the eigenvalue of P z , k denotesthe radial wavenumber and m ɛ Z are the eigenvalues of the envelopeangular momentum L z . Making use of L z = F z − J z , the envelope functionsin the cylindrical representation are of the form J fz−jz (kρ)e i(f z−j z )φ e ikzz ,where J n (z) is a Bessel function. With 〈ρ φ z|k z ; k, f z − j z 〉 the envelopefunctions in cylindrical coordinates, this results in〈ρ φ z|k z ; k, f z − j z 〉 | 3 2 , j z〉 ∝ J fz −j z(kρ)e i(f z−j z )φ e ik zz | 3 2 , j z〉. (3.27)as a basis for the solution (3.25), which is is orthogonal in f z , j z , k and k z .The Hamiltonian H Γ 8F zof the the top Γ 8 valence band in III-V semiconductorsnow is expressed in this basis by [17][18]H Γ 8F z=⎛p + q s r 0⎜⎝s p − q 0 rr 0 p − q −s0 r −s p + q⎞⎟⎠ , (3.28)48

3.3. Envelope description for infinite cylinderswhere the basis is ordered with respect to j z as { 3 2 , 1 2 , − 1 2 , − 3 2} and p, q, rand s are given byp + q = − 22m 0((γ 1 + γ 2 )k 2 + (γ 1 − 2γ 2 )k 2 z) , (3.29)p − q = − 22m 0((γ 1 − γ 2 )k 2 + (γ 1 + 2γ 2 )k 2 z) , (3.30)r = 22m 0√3γ2 k 2 , (3.31)s = 22m 02 √ 3γ 2 kk z . (3.32)This Hamiltonian has two different eigenvalues, corresponding to a heavyhole (HH) and a light hole (LH) band which are degenerate at the zonecenter:ɛ HH = − 22m 0(γ 1 − 2γ 2 )(k 2 HH + k 2 z), (3.33)ɛ LH = − 22m 0(γ 1 + 2γ 2 )(k 2 LH + k 2 z). (3.34)Both bands are doubly degenerate and the unnormalized eigenvectors aregiven by|HH1〉 =|LH1〉 =⎛⎜⎝k 2 HH +4k2 z √3k 2HH2k zk HH10⎞⎛− √ 32k z⎜ k LH⎝ 10⎟⎠ ,⎞⎟⎠ ,⎛|HH2〉 = ⎜⎝⎛|LH2〉 = ⎜⎝01− 2kzk HHk 2 HH +4k2 z √3k 2HH01− 2kzk LH− √ 3⎞⎞⎟⎠ , (3.35)⎟⎠ , (3.36)with respect to the basis given in (3.27), ordered as { 3 2 , 1 2 , − 1 2 , − 3 2 } withrespect to j z .3.3 Envelope description for infinite cylindersIn principle it is possible to apply the effective mass approximation in thecontext of the geometrical configuration of a nanowire. However, as pointedout in [14][18], care has to be taken concerning the foundation of the theoreticalframework developed in the previous sections.49

Chapter 3.Electronic propertiesIn the first place, the effective mass approximation relies on the assumptionthat the potential is a slowly varying function over a unit cell. Imposingthe wire configuration by takingδV (r) = −V 0 Θ(R − ρ), (3.37)with Θ the Heaviside function, this requires the wire radius R to be sufficientlylarge. Intuitively this makes sense directly, for if there are just afew atoms within the wire, the potential change at the boundary of the wirecannot be neglected any more with respect to the interatomic distances a.To be more precise, by rewriting the potential (3.37) in Fourier space, it canbe seen [14][18] that the entire concept of an effective mass is only useful ifaR≪ 1, the limit in which only the Fourier components δV (k) around thezone center contribute significantly.Secondly, in the theory of section 3.2 the atomic wavefunctions are assumedto be the same everywhere. If the effective mass approximation is nottreated in a suitable form, it thus fails to describe in a proper way the heterostructuresituation with two or more completely different environments(e.g. a semiconductor compound in vacuum) and consequently drastic changesin the atomic wavefunctions.This problem is solved in a general way by assuming that every different environmentcan be described in a large part independently of the other(s)[18].The bands in the different systems are subsequently related to each otherby matching bands with the same symmetry. Instead of equation (3.21), thecorresponding wavefunction is assumed to be of the formΨ(r) = ∑ jRF j (R)a (s)jR(r), (3.38)where s indicates that for the atomic functions, in this case in Wannier representation,the solutions are taken far in the corresponding system s.Even if the environments are large enough to approximate them mainlyas bulk systems in this way, it still remains a problem to match the wavefunctionsof the different systems near the boundaries. For example, itcannot be expected that the atoms around the interface of different systemssimply are positioned at the lattice points of a perfect crystal. The atomicfunctions on both sides of the boundary are not orthogonal to each otherand in case of a semiconductor structure in vacuum, there are even no atomsoutside the structure any more. Another practical point is the oxidation ofthe structure, resulting in a system probably better described as a core shellstructure.However, the neglect of these effects concerning changes in the atomicwavefunctions and matching of the boundary can be justified by the spatial50

3.3. Envelope description for infinite cylindersextent of the envelope functions: by taking an infinite potential well for thenanowire geometry, the envelope functions fall off to zero at the boundary.In this model the atomic functions outside the wire are of no importanceand possible fluctuations around the boundary are neglected because theoverlapping envelope is almost zero. More problems are expected when V 0is finite. In this case the envelope function leaks with a certain extent intothe region outside the wire and the change in atomic wavefunctions plays amore important role.In the present paper the potential V 0 in (3.37) is assumed to be largeenough to consider it as representing an infinite potential well. In case ofan infinite cylinder structure, this leads to a boundary condition on theenvelopes of bulk wavefunctions:F j (ρ = R, φ, z) = 0, ∀ j, φ, z. (3.39)The solutions for this boundary condition can be labeled with a set of quantumnumbers, say λ, where λ will be specified for the valence band in paragraph3.4.1 and for the conduction band in paragraph 3.3.2. In addition,for a given band λ the wavefunctions depend on the wavenumber k z .Denoting the complete labeling with λ k z one obtains a normalizationcondition for the envelope functions F λ kz,j if the total wavefunctionΨ λ kz (r) = C λ kz∑jR F λ k z ,j(R)a jR (r) is normalized to unity:∫= C 2 λ k z∑dr |Ψ λ kz (r)| 2j, j ′ ∑= C 2 λ k z∑j, j ′ ∑R, R ′ F ∗ λ k z ,j ′(R′ )F λ kz,j(R)∫dr a ∗ j ′ R ′(r)a jR(r)Fλ ∗ k z, j ′(R′ )F λ kz , j(R)δ R ′ ,R δ j ′ ,jR, R ′∑ ∑= Cλ 2 k z|F λ kz , j(R)| 2 = 1, (3.40)jRfrom which the normalization constant C λ kz is obtained. In the third stepin equation (3.40) the orthonormality of the Wannier functions is used. Thesummation over R can be replaced by an integral in the same context asequation (3.22). The same normalization condition is obtained using theBloch representation (3.25).3.3.1 Hole in III-V semiconductor nanowiresWith the above remarks in mind consider again the top valence bands of III-V semiconductors. The wire geometry is imposed by the infinite potentialwell:{ ∞, ρ > R,δV (ρ) =(3.41)0, ρ ≤ R.51

Chapter 3.Electronic propertiesThis leads to the boundary condition on the envelopes (3.39), with j =j z = { 3 2 , 1 2 , − 1 2 , − 3 2}. Since the bulk heavy- and light hole solutions (3.35)and (3.36) have four components which cannot be zero simultaneously, thisrequirement (3.39) can be satisfied only if the total wavefunction is a superpositionof the four bulk heavy- and light hole eigenstates for a givenf z . Consequently, apart from normalization constant (3.40) the envelopewavefunctions are determined byF λ kz ,j z(ρ, φ, z) = {(v HH1 |HH1〉 jz + v HH2 |HH2〉 jz )J fz −j z(k HH ρ) + (3.42)(v LH1 |LH1〉 jz + v LH2 |LH2〉 jz )J fz −j z(k LH ρ)} e i(f z−j z )φ e ik zz ,where |HH1〉-|LH2〉 are the bulk eigenstates given in (3.35) and (3.36)and v HH1 , v HH2 , v LH1 , v LH2 are the coefficients which satisfy F λ kz,jz (ρ =R, φ, z) = 0. The boundary condition for j z = { 3 2 , 1 2 , − 1 2 , − 3 2} results in thedeterminant equation{0 = J f−3 (k LH )J2 f−1 (k LH )J2 f+1 (k HH )J2 f+3 (k HH )2}+ J f−3 (k HH )J2 f−1 (k HH )J2 f+1 (k LH )J2 f+3 (k LH )2+ 3J f−32(k LH )J f−12+ 4k2 zk LH k HH{J f−32+J f−32+ (k2 LH +4k2 z )(k2 HH +4k2 z )3k 2 L k2 HH(k HH )J f+12(k LH )J f−12(k HH )J f−12J f−32(k HH )J f+32(k HH )J f+12(k LH )J f+12(k HH )J f−12(k LH ) (3.43)(k LH )J f+3 (k HH )2}(k HH )J f+3 (k LH )2(k LH )J f+12(k LH )J f+3 (k HH ),2which is a relation for the allowed energies. Here the wire radius R is absorbedin the wave numbers by k HH → k HH R, k LH → k LH R and k z → k z R.From now on k HH , k LH and k z denote these dimensionless ”wavenumbers”,unless stated otherwise.Together with the constraint obtained from the equation for the energy,ɛ HH = ɛ LH = E, (3.44)where the bulk energies ɛ HH and ɛ LH are given by (3.33), the determinantequation (3.43) fixes the radial wavenumbers k HH (k z ) and k LH (k z ) for a givenk z . Using these solutions of k HH (k z ) and k LH (k z ), also the coefficientsv HH1 -v LH2 are obtained from the boundary equation (3.39).Before turning to more explicit results in the next sections, the followinggeneral remarks are important to keep in mind. First, the determinantequation (3.43) is invariant under the inversion f z → −f z , which reflects thetime-reversal symmetry of the Hamiltonian H Γ 8F zgiven in (3.28): the totalangular momentum reverses direction if t → −t. Consequently, the energysolutions E are doubly degenerate in f z and the corresponding wavefunctionsturn into each other under f z → −f z .52

3.3. Envelope description for infinite cylindersSecondly it should be noted that the wavenumber k z , giving the dispersionin the z direction where the electron (hole) is still free to move, cannotbe separated from the lateral terms in the envelope wavefunction (3.42).The radial wavenumbers k HH and k LH are functions of k z , so the dispersionin the z direction in general depends on the lateral distribution of thewavefunction.Furthermore, for a given k z the set of equations (3.43), (3.44) has to besolved numerically. Only in special cases the energy E and hole wavefunctionreduce to relative simple analytical expressions. In the next section, firstsome analytical results at k z = 0 are summarized. Subsequently an expressionfor the effective mass of a hole in a III-V nanowires will be derived byexpansion around k z = 0, the wire zone center.3.3.2 Electron in III-V semiconductor nanowiresUp till now only the situation of the degenerate top valence bands in III-Vsemiconductors was discussed. Since also the conduction band propertiesare needed in the remaining of this paper and because it is also illustrativeto consider a nondegenerate example which is much easier to handle, herethe electron dispersion and wavefunctions of the lowest lying conductionband in III-V semiconductors are treated shortly.Again an infinite confinement is assumed, as given in equation (3.41).Since the conduction band is non-degenerate, the S.E. for the electron isgiven by (3.23) and the envelope function for ρ < R is given byF λ kz (ρ, φ, z) = C λ J lz (k lz ρ)e ilzφ e ikzz , (3.45)with C λ the normalization constant.Assuming also the warping sufficiently small, i.e. adopting the sphericalapproximation by taking an uniform effective mass m ∗ c for the conductionband, the energy dispersion becomes 22m ∗ c(k 2 l z+ k 2 z) = E. (3.46)The boundary condition (3.39) now simply gives k lz ,n = j lz,nR, the allowedvalues of k lz which are independent of k z . Here j lz ,n is the n th zero ofthe Bessel function J lz(x)It is convenient to introduce a notation which summarizes the labeling ofthe conduction subbands, as derived in the current framework. In the presentcase, the total Hamiltonian already is diagonal in the envelope angularmomentum, so the conduction subbands are labeled with |l z |.53

Chapter 3.Electronic propertiesFollowing the notation of [4], the irreducible representation of the conductionsubbands in cylinder configuration is characterized byC (±)|l z |, n , (3.47)where (±) denotes the parity, n the n th solution at this parity and theabsolute value of the envelope angular momentum is taken because of thedegeneracy in l z .3.4 Hole dispersion around k z = 0As discussed in [17] [18], the top valence band Hamiltonian H Γ 8F z, given by(3.28), decouples into two 2 × 2 blocks at the wire zone center, k z = 0. Bothblocks have solutions which are characterized by parity: the correspondingBessel functions are only even or only odd under ρ → −ρ.Apart from a general discussion, in this section the focus will be narrowedto an exceptional case: the odd solutions for |f z | = 1 2. In this case it ispossible to derive a transparent equation for the effective mass in the zdirection by Taylor expansion around k z = 0.3.4.1 Solutions at the wire zone centerAt the wire zone center, k z = 0, one obtains from the energy equation (3.44):√ √γ1 + 2γ 2 m ∗ HHk HH = βk LH , β ≡=γ 1 − 2γ 2 m ∗ , (3.48)LHwhere m ∗ HH and m∗ LHare the effective masses of the heavy and light holebulk bands, respectively. Also the boundary condition simplifies at k z = 0.By block diagonalizing (3.28) it is found that the four heavy- and light holewavefunctions decouples into two groups: eitherorv HH2 = v LH2 = 0 , v LH1 = α 1 v HH1 , (3.49)v HH1 = v LH1 = 0 , v LH2 = α 2 v HH2 . (3.50)Here α 1 and α 2 are determined by the determinant equation (3.43) at thewire zone center, which decouple into two mutually excluding determinantsor1 J fz− 3 23J fz− 3 21 J fz + 3 23J fz + 3 2(k HH )(k LH )(k HH )(k LH )= − J f z+ 1 (k HH )2(k LH ) ≡ α 1, (3.51)J fz+ 1 2= − J f z − 1 (k HH )2(k LH ) ≡ α 2, (3.52)J fz − 1 254

3.4. Hole dispersion around k z = 0so the only possible solutions indeed are given by (3.49) and (3.50). Notethat the inversion symmetry of f z is revealed by the two determinants: usingJ −n (z) = (−1) n J n (z), (3.53)it easy to show that (3.49) turns into (3.50) under f z → −f z .Energy equality (3.48), together with either (3.49) or (3.50) determinesthe energy at the zone center. The different energy bands and correspondingwavefunctions are characterized by parity at the wire zone center: for agiven f z ɛ Z + 1 2the solution corresponding to (3.49)/(3.50) contains onlyeven/odd (odd/even) Bessel functions. Note that Bessel functions transformunder inversion in ρ in the same way as their label (i.e. under z → −z,J n (z) → J n (z) if n is even, J n (z) → −J n (z) if n is odd).As long as k z = 0, parity thus is a good quantum number and this is stillapproximately the case for k z close to 0. Consequently, the energy bandsand corresponding valence subbands are labelled with +/− for respectivelyeven/odd solutions at k z = 0. Note that for a given parity there are differentsolutions labelled by n.At this point it is convenient to specify the labeling of the valence subbandsfurther. As stated in paragraph 3.2.1, in a cylindrical representationthe total Hamiltonian H Γ 8is diagonal in F z , which implies that the subbandsare also labeled with |f z | (the absolute value is taken because of thedegeneracy in f z → −f z ). Consequently, the complete set of solutions forthe Γ 8 valence band in the infinite cylinder configuration is characterized bythe quantum numbers f z , (±), n th solution at this parity. This irreduciblerepresentation of the valence subbands is indicated withE (±)|f z |, n , (3.54)where, contrary to the notation in [4], n denotes the n th solution at a particularparity.In general, even at k z = 0 the original bulk heavy- and light hole solutionsare coupled to each other in a nanowire. However, it turns out that the oddsolutions at |f z | = 1 2form an exceptional group. The determinant equation(either (3.49) or (3.50)) in this case reduces toJ 1 (k HH )J 1 (k LH ) = 0, (3.55)with Bessel zeros j 1,n as solutions:k HH = j 1,nR , k LH = j 1,nβR(3.56)55

Chapter 3.Electronic propertiesork HH = β j 1,nR , k LH = j 1,nR , (3.57)where k HH , k LH are the original wavenumbers, so R is written out explicitely.As can be seen from (3.51) or (3.52), the relevant coefficient α 1 /α 2 is zerofor these solutions. Since the other heavy -, light hole pair already is excluded(equation (3.49) or (3.50)) the odd wavefunctions at k z = 0, |f z | = 1 2consequently are pure heavy - or light hole like. Note that for the light holesolutions both α 1 and α 2 should be inverted.3.4.2 Hole dispersion around k z = 0 for |f z | = 1 2 , (−)The confinement by the infinite wire geometry, resulting in the determinantequation (3.43), reduces the dimensions in which the electron (hole) is freeto move to one. The dispersion relation in this direction (z) becomes morecomplex than the quadratic dispersion of the two original bulk bands, due tothe fact that k z cannot be separated from the lateral terms in the envelopewavefunction in case of the degenerate III-V top valence band. However,for the odd solutions at |f z | = 1 2it is possible to approximate the dispersionaround the wire zone center with an effective mass.For this purpose, the first step is to expand k HH (k z ) and k LH (k z ) up tosecond order in k z . Recall that the Γ 8 band minimum is assumed to be atat the zone center, so∂E∂k z∣ ∣∣∣kz=0= 0. (3.58)Utilizing this assumption, one finds for the lateral wavenumbers, by Taylorexpansion around k z = 0,k 2 HH(k z ) ≃ a 2 HH+ b 2 HHk 2 z , k 2 LH(k z ) ≃ a 2 LH+ b 2 LHk 2 z. (3.59)and the correspon-Consequently, expanding up to second order in k z , k HHding Bessel function are approximated byk HH (k z ) ≃ a HH + b2 HH2a HHk 2 z, (3.60)J n (k HH (k z )) ≃ J n (a HH ) + b2 HH2a HHJ ′ n (a HH )k 2 z (3.61)and the expressions for k LHare similar.56

3.4. Hole dispersion around k z = 0Before expanding (3.43) in this way, first it can be simplified for |f z | = 1 2by using the Bessel function property (3.53):0 = 4 3 k2 LHk 2 HHJ 1 (k LH )J 1 (k HH ) {J 0 (k LH )J 2 (k HH ) + 3J 0 (k HH )J 2 (k LH )} +4k LH k HH{J21 (k LH )J 0 (k HH )J 2 (k HH ) + J 2 1 (k HH )J 0 (k LH )J 2 (k LH ) } k 2 z +43 (k2 LH+ k 2 HH)J 1 (k LH )J 1 (k HH )J 0 (k LH )J 2 (k HH )k 2 z, (3.62)where it should be mentioned that k HH = 0 (or k LH = 0) is not a solution[18]. Expanding the determinant equation up to second order with (3.59)-(3.61), the zeroth order part of the first line in (3.62) gives the solutions atk z = 0:0 = J 1 (a LH )J 1 (a HH ) {J 0 (a LH )J 2 (a HH ) + 3J 0 (a HH )J 2 (a LH )} , (3.63)where the term between the brackets in (3.63) corresponds to the even solutions.In order to simplify the quadratic term in the expansion of (3.62), a HHand a LH should be fixed by either the even or the odd solutions in (3.63).For the odd solutions this results in a transparent equation for the dispersionaround the wire zone center. Imposing J 1 (a HH )J 1 (a LH ) = 0 and utilizing aproperty of the Bessel functions,J ′ n(z) = ∓J n±1 (z) ± n z J n(z), (3.64)the quadratic term becomes{ []0 = a LH a 1 b 2 LHHH 3 2a LHJ 0 (a LH ) ∓J 1±1 (a LH ) ± 1a LHJ 1 (a LH ) J 1 (a HH )J 2 (a HH )+[]1 b 2 HH3 2a HHJ 0 (a LH )J 1 (a LH ) ∓J 1±1 (a HH ) ± 1a HHJ 1 (a HH ) J 2 (a HH ) +[]b 2 HH2a HHJ 0 (a HH ) ∓J 1±1 (a HH ) ± 1a HHJ 1 (a HH ) J 1 (a LH )J 2 (a LH ) +[]b 2 LH2a LHJ 0 (a HH )J 1 (a HH ) ∓J 1±1 (a LH ) ± 1a LHJ 1 (a LH ) J 2 (a LH ) + (3.65)1a LH a HH(J21 (a LH )J 0 (a HH ) J 2 (a HH ) + J 2 1 (a HH)J 0 (a LH ) J 2 (a LH ) )} k 2 z.The final step is to specify the odd solution further by choosing eitherJ 1 (a LH ) = 0 or J 1 (a HH ) = 0. Here the discussion is restricted to a HH =j 1,n , which corresponds to the lowest, heavy hole like energy bands. 4 Afterchoosing the convenient signs in (3.65), the determinant equation in thiscase reduces to an expression for the expansion factor b HH of the heavy holelateral wavenumber:22βb 2 aHH= − LHJ 1 (a LH )13 J 0(a LH ) − J 2 (a LH ) = − j 1,nJ 1 ( j 1,nβ )13 J 0( j 1,nβ ) − J 2( j (3.66)1,nβ),4 Note again that for the odd solutions there is no heavy -, light hole mixing any more.57

Chapter 3.Electronic propertieswhere the second expression follows from a LH = 1 β a HH, see (3.48).Expanding also the energy equation E U h= −(γ 1 − 2γ 2 )(kHH(k 2 z ) + kz)2(equation (3.44)) using (3.59)-(3.61), the odd |f z | = 1 2heavy hole bands areapproximately given byEU h= −(γ 1 − 2γ 2 )j 1,n − (γ 1 − 2γ 2 )(1 + b 2 HH)(k z R) 2 , (3.67)where b HH is given in (3.66) and the dependence on the wire radius is explicitlyshown by defining an energy unit U h :U h≡ 22m 0 R 2 . (3.68)This results in an expression for the effective mass of a heavy hole in theodd |f z | = 1 2energy bands of an infinite wire:m ∗ HH, z = m 0 (γ 1 − 2γ 2 ) −1 (1 + b 2 HH) −1 , (3.69)where z is the only direction in which the hole is still free to move.58

3.5. Results3.5 ResultsAs an illustration of the above theory, in this section numerical results aregiven on the basis of specific examples. In particular it is insightful to compareIII-V materials with different properties, in this case different Kohn-Luttinger parameters. Hence, the III-V compounds InP and InAs are investigated,their Kohn-Luttinger parameters given in Table 3.1 are takenfrom [20]. Note that the values of γ 3 are not needed here: the theoreticalγ 1 γ 2InP 5.08 1.60InAs 20.0 8.5Table 3.1: Kohn-Lutinger parameters for InP and InAsframework rests on the assumption γ 3 = γ 2 .3.5.1 Hole energy bands of III-V material nanowiresThe first seven hole energy bands of InP and InAs nanowires are shown infigure 3.1. They are calculated from equations (3.43) and (3.44). The blackline in the graphs corresponds to the reference band −(γ 1 + 2γ 2 )kz2 withk LH = 0, where all bands end because there are no solutions if k LH ≤ 0. TheR dependence is absorbed in the units along the axes, k z R and E R 2 , withR in nm, k z R dimensionless and E R 2 in eV nm 2 .E R 2 eV nm 2 0-1-2-3-4-5-6InPf z ⩵ 12, 1f z ⩵ 12, 1f z ⩵ 32, 1f z ⩵ 32, 1f z ⩵ 32, 2f z ⩵ 12, 2f z ⩵ 12, 2E R 2 eV nm 2 0-1-2-3-4-5-6InAsf z ⩵ 12, 1f z ⩵ 12, 1f z ⩵ 32, 1f z ⩵ 32, 1f z ⩵ 32, 2f z ⩵ 12, 2f z ⩵ 12, 20 1 2 3 4 5 6k z R0 0.5 1 1.5 2 2.5 3 3.5k z RFigure 3.1: Hole energy bands for InP and InAs. The bands are labeledby absolute total angular momentum in the z-direction, |f z |, and parity,denoted with (+) n th (nth even solution) and (−) n th (odd). The blackline in the graphs corresponds to the reference band (γ 1 + 2γ 2 )kz 2 withk LH = 0. The R dependence is absorbed in the units along the axes,with wire radius R in nm.Using the notation given in (3.54) and (3.47), the representation of thefirst seven hole subbands of InP and InAs nanowires are shown in Table 3.2.59

Chapter 3.Electronic propertiesThe subbands v i are ordered with respect to their zone center offset, seeFigure 3.1. For convenience, also the representation of first two electronsubbands c 1 and c 2 are given, within the framework of Subsection 3.3.2.Subband InP InAsv 1 E (+)1E (−)2 ,1 12 ,1v 2E (−)1 E (+)2 ,1 12 ,1v 3 E (+)3E (+)2 ,1 32 ,1v 4E (−)3 E (+)2 ,1 12 ,2v 5 E (+)3v 6 E (+)1E (−)2 ,2 32 ,1E (−)2 ,2 12 ,2v 7E (−)1 E (+)2 ,2 32 ,2c 1 C (+)0, 1 C (+)0, 1c 2C (−)1, 1 C (−)1, 1Table 3.2: Irreducible representation of the first seven hole subbands v iand first two electron subbands c j for InP and InAs nanowires. Thecharacterization is also valid for conduction subbands calculated in afinite potential well.Around k z = 0 the hole band dispersion can be approximated by thequadratic expressions given in Table 3.3. In general, for |f z | = 1 2and oddparity these numerical results are in good agreement with the analyticalexpansion given in (3.67). For instance, for InP the effective mass in the zdirection m ∗ HH, z corresponding to the values in Table 3.3 are 3.45 m 0 and11.49 m 0 for the first even and first odd subband, while the analytical expression(3.69) gives 3.39 m 0 and 10.20 m 0 , respectively.Comparing the two materials, the following remarks are supported byFigure 3.1, Table 3.2 and Table 3.3:60

3.5. Resultshole state InP InAs|f z | = 1 2 , (+) 1 −0.75 − 0.14(k zR) 2 −2.14 − 1.87(k z R) 2|f z | = 1 2 , (−) 1 −1.05 − 0.29(k zR) 2 −1.68 + 0.77(k z R) 2|f z | = 1 2 , (+) 2 −2.16 − 0.07(k zR) 2 −3.91 − 0.72(k z R) 2|f z | = 1 2 , (−) 2 −3.53 − 0.087(k zR) 2 −5.62 − 0.35(k z R) 2|f z | = 3 2 , (+) 1 −1.38 + 0.31(k zR) 2 −2.95 + 0.44(k z R) 2|f z | = 3 2 , (−) 1 −1.79 − 0.59(k zR) 2 −4.07 − 0.45(k z R) 2|f z | = 3 2 , (+) 2 −2.08 − 0.16(k zR) 2 −6.52 + 1.29(k z R) 2Table 3.3: Numerical results for the hole energy E R 2 (eV nm 2 ), fittedto k 2 zR 2 around the wire zone center• The shape of a particular band, including its zone center energy, ismaterial dependent. It depends on the magnitude of the gamma’s byγ 1 − 2γ 2 , but also their ratio γ 1γ 2is a deciding quantity. Consequently,the corrections to the band gap E g caused by the confinement arematerial dependent. For the present two examples the zone centerband gaps of InAs are more shifted by the infinite wire configuration.• Next to the shape of the individual bands, also their mutual orderingis material dependent. This means that the parity of the lowest lyingband (and the others) can differ depending on the material. Forexample, the lowest lying band is even for InP and odd for InAs.Furthermore, it should be noted that the results are valid for all R, withthe only requirement that R should be sufficient large in order to consider theconfinement potential (3.41) as a slowly varying function with respect to theunit cell dimensions. This means that in the limit R → ∞ the confinementcorrection on the band gap has to disappear, which is indeed the case as canbe concluded from the 1/R 2 dependence.3.5.2 Hole wave functions of III-V material nanowiresAs pointed out in section 3.3, the wavenumber along the cylinder axis k z isnot independent of the lateral part of the hole wavefunction. As a consequence,the total wavefunction for a particular band depends in a non trivialway on k z and should be calculated from (3.42) for every value of k z separately.Here the results of this procedure are summarized by focussing, next tothe k z dependence, on three other subjects: the parity of the wavefunctions,the invariance under total z-angular momentum reversion and the influenceof material properties (Kohn-Luttinger parameters).Starting with parity and the invariance under f z → −f z , Figure 3.2shows the φ = 0 radial part of the envelope wavefunction, decomposedinto the different j z components at the same value of k z and for the same61

Chapter 3.Electronic propertiesmaterial. The graphs in the first row correspond to the two E (+)1 solutions,,2those in the second row to the solutions with representation E (−)3 . The total,1 2envelope function is normalized using equation (3.40) and the wire radius Ris absorbed in the dimensionless unit ρ R .2Χ j zΧ j z1.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵2.66667 , 2j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 32 , k z R⩵2.66667 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-1f z ⩵12 , k z R⩵2.66667 , 2j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵32 , k z R⩵2.66667 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRFigure 3.2: Radial (φ = 0, e ikzz omitted) part of the normalized holeenvelope functions, decomposed in the different j z components: the j z =32 components are given in yellow, j z = 1 2 in green, j z = − 1 2in blue andj z = − 3 2 in red. The first row gives the solutions for |f z| = 1 2 , + (2),the second row for |f z | = 3 2, −(1). The Kohn-Luttinger parameters aretaken from InP as given in Table 3.1 and k z R = 2.67 is fixed.Figure 3.2 illustrates that for a given subband E (±)|f the solution at f z|,n zfor a particular j z is the same (apart from minus sign) as at −f z for −j z .Moreover, as expected the two total wavefunctions turn into each otherunder time reversal, because under f z = l z + j z → −f z = −l z − j z anycomponent J l (kρ) |j, j z 〉 → J −l (kρ) |j, −j z 〉, so besides j z → −j z the oddsolutions reverse sign, as shown in Figure 3.2.The wavefunctions in Figure 3.2 are calculated away from the wire zonecenter, at k z R = 1.5 10 −9 with R in nm. As a consequence, next to the j zcomponents with the parity of the zone center, also other j z componentsappear which have the opposite parity. For example, in the first graph thedominant j z = 1 2 component (green curve) and the j z = − 3 2component (redcurve) are the evolved even components which are present at the zone center,while the j z = 3 2 (orange) and j z = − 1 2(blue) curves are odd in ρ → −ρ.62

3.5. ResultsΧ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0. , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵0.25 , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵0.75 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.25 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRΧ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0.125 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵0.5 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1. , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.5 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRFigure 3.3: Radial part of the |f z | = 1 2, + (1) hole envelope wavefunctionsfor InAs. The value of k z R changes from 0 in the first picture to themaximum value 1.5 (at the end of the band) in the last graph.This is a general property: at the wire zone center the total wavefunctionconsist only of either even or odd components, while away from k z = 0 alsosignificant contributions with the other parity arise.63

Chapter 3.Electronic propertiesThe variation of the wavefunction as a function of k z is illustrated in moredetail in Figure 3.3. It shows the radial part of the E (+)1 hole wavefunctions,1 2for InAs for different values of k z , given at the top of each graph.In Appendix A also the first odd solution, E (−)1 , for InAs is shown in,1 2this way; the illustrations can be compared to the results in case of InP,Figure 10 and Figure 11.The following remarks are revealed by Figure 3.3 and the figures inAppendix A:• As noted before, at k z = 0 the total hole wavefunction is either evenor odd under inversion ρ → −ρ. Remarkable is that the shape of thefunctions depends only a little on the choice of material.• Increasing k z slowly, the shapes of the different j z components changein a continuous way: an extra graph between the first and the secondwould give a result in between.• Comparing the results for InP and InAs, it can be seen that for agiven band the altering of the hole wavefunctions by increasing k z ismaterial dependent. Actually, the amount of change depends on theshape of the corresponding band: a smaller effective mass correspondswith a faster change in the hole wavefunctions around k z = 0, whichcan be checked for the present examples with the help of Table 3.3.• At the end of a particular band, one of the j z components disappears,so in general there are three or fewer j z components which contributeto the total wavefunction at the end of a band.The above results have some important consequences, in particular concerningthe calculation of the absorption matrix elements over the entireband, see next chapter. Since the different j z components of the hole wavefunctionchange just slightly over a particular band, it suffices to choosea suitable small number of k z points by which the hole wavefunctions atneighboring points are approximated.3.5.3 Band gap in III-V material nanowiresFinally, it is illustrative to estimate the effect of the infinite confinement inthe present model by comparing the bulk band gap E g with the confinementenergy Ev conf1 →c 1of the fundamental transition v 1 → c 1 between the highestlying hole state v 1 and lowest electron state c 1 at the wire zone center.The values of effective mass of the Γ 6 conduction band and the bulkband gap E g for InP and InAs are given in Table 3.4. Using the values64

3.5. ResultsInP InAsm ∗ c/m 0 0.0795 0.026E g (eV ) 1.4236 0.417V 0 (eV ) 4.28 4.93Table 3.4: The band gap E g , potential well V 0 and effective masses ofthe Γ 6 bulk conduction band for InP and InAsof the effective mass, the first row in Table 3.5 gives the energy of thelowest conduction subband c 1 at the wire zone center. The second rowInPInAsE c1 (eV ) 2.77 R −2 8.48 R −2Ev conf1 →e 1(eV ) 3.52 R −2 10.16 R −2Table 3.5: The energy in eV at k z = 0 of the lowest conduction subbandc 1 and the confinement energy Ev conf1→c 1of the fundamental transitionv 1 → c 1 for InP and InAs in an infinite wire confinement, as derivedwith the model described in this paper. R denotes the wire radius.in Table 3.5 shows the confinement energy Ev conf1 →c 1(R) of the fundamentaltransition v 1 → c 1 . Note that Ev conf1 →c 1(R) does not include the bulk E g , it isdefined asE transv i →c j(R) = E g + E ci (R) − E vi (R) ≡ E g + E confv i →c j(R). (3.70)However, the assumption of an infinite potential well is too strong. Thedifference between the vacuum level and the conduction band edge (electronaffinity) is in the order of electron volts for III-V materials and taking thisfiniteness into account leads to significant corrections, in particular for theconduction subbands[18][21].Here the discussion will be restricted to correcting the conduction subbandc 1 for InP and InAs, given in Table 3.5, with a reduction factor dueto the finite potential well. For the valence subbands it is expected that thecorrection is less crucial, in the first place because the correction is smallerfor bulk bands with a higher effective mass. Another reason is that, apartfrom the extra energy difference by the band gap, the difference with thevacuum level becomes larger for deeper lying subbands, in contrast to theconduction band states.In the finite potential well model, the dependence on the wire radiusbecomes more complicated then the simple R −2 dependence in the infinitecase. Actually, the electronic properties depend on the dimensionless65

Chapter 3.Electronic propertiesquantity R l V, where l Vis defined asl V =√ 22m ∗ V . (3.71)For the conduction band, the potential V equals the vacuum level offset V 0which is given for InP and InAs in Table 3.4.InPInAsE c1 (eV ) 0.07 0.13Ev conf1 →c 1(eV ) 0.11 0.20Table 3.6: The energy in (eV ) at k z = 0 of the lowest conduction subbandc 1 and the confinement energy Ev conf1→c 1of the fundamental transitionv 1 → c 1 for InP and InAs at R = 4.83 nm and R = 4.85 nm, respectively.These are the corrected results of Table 3.5: the conductionsubbands are calculated in the finite potential wells given in Table 3.4.The corresponding energies of the lowest conduction subband c 1 in thefinite wire configuration are shown in Table 3.6. These values of E c1 can becompared to those for the infinite potential well case at R = 4.8 nm: 0.2 eVand 0.36 eV for InP and InAs, respectively. Similarly, for the confinementenergy Ev conf1 →c 1, the values in the infinite potential well case are 0.15 eV and0.43 eV for InP and InAs, respectively. The difference between the twomodels is larger for InAs due to the smaller effective mass of the conductionband.66

Chapter 4EM transition matrixIn this chapter the EM matrix element for band-to-band transitions betweenthe top Γ 8 valence bands and the lowest lying Γ 6 conduction band in III-Vsemiconductor nanowires will be derived. Section 4.1 contains some generaltheory, in section 4.2 the matrix element is developed further in the Blochrepresentation, section 4.3 gives explicit expressions for the band-band matrixelements and section 4.4 treats the selection rules on the intersubbandtransitions. Explicit results for InP and InAs are shown in section 4.5.4.1 General theory4.1.1 Radiation matter interactionIn this paper the interaction between the external EM field and the electronswithin the semiconductor system is described using a macroscopic, semiclassicalapproach. In this method the EM field is treated classically, whilethe semiconductor material is described quantum mechanically in the spiritof the previous chapter. Next to the assumption that this semi-classical pictureapproximates the more realistic QED model, it is also assumed that thesemiconductor heterostructure can be described using macroscopic Maxwellequations, i.e. where the different parts in the system are characterized bymacroscopically averaged quantities, such as the dielectric function.As will be discussed in more detail in Chapter 5, it is unclear if thismacroscopic framework still holds if the size of the system is reduced to nanoscale,when the dimensions of the system are not large any more comparedto the microscopic (atomic) distances. Then a microscopic semiclassical theorywould be a more realistic approach [22].However, proceeding with the macroscopic semiclassical approach, thegauge freedom in choosing the scalar potential φ and vector field A is used67

Chapter 4.EM transition matrixby taking the Coulomb gauge,∇ · A = 0. (4.1)Then the transverse part of the electric field equals − 1 c ∂t. Note that thelongitudinal part −∇φ is absorbed in the matter part of the semiconductorHamiltonian: the averaged potential V (r) in equation (3.1) contains the fullCoulomb interaction between the particles. Further details are found in [22].In this scheme the Hamiltonian representing the radiation-matter interactionis given byH r−m = − em 0 c∂A∑A(r i ) · p i , (4.2)iwhere i labels the N electrons in the material and m 0 the free electronmass. Taking m 0 instead of an effective mass m ∗ i is justified for interbandtransitions [15]. Furthermore, since the EM field is typically is small, theterm of order |A| 2 is neglected in (4.2).4.1.2 EM transition matrixTreating the time dependent EM interaction (4.2) as a small perturbation,it induces a transition between initial state |Ψ i 〉 and final state |Ψ f 〉, whichare eigenstates of the semiconductor system in the absence of the EM field.The probability that the unperturbed system |Ψ i 〉 transforms under theabsorption/emission of light to |Ψ f 〉 is proportional to the transition matrixelementM fi = 〈Ψ f |H r−m |Ψ i 〉. (4.3)In case of a degenerate initial and/or final state, with |Ψ im 〉 and |Ψ fn 〉 them- and n-fold degenerate initial and final states, this degeneracy is takeninto account by|M fi | 2 = ∑ |〈Ψ fn |H r−m |Ψ im 〉| 2 , (4.4)m,ni.e. the different possible transitions at the same energy are summed asprobabilities. If the degeneracy is lifted by a perturbation which breaks aparticular symmetry, say an external magnetic field, then |Ψ im 〉 and |Ψ im ′ 〉are separated in energy and their matrix elements can be distinguished asbelonging to different transitions |M fi | and |M fi ′|.In this paper the absorption process is considered, in the specific caseof direct band-to-band transitions in nanowires. As initial state the many68

4.1. General theorybody groundstate Ψ 0 of the undoped semiconductor nanowire is taken, withall valence bands completely filled and the conduction bands empty:( )∏Ψ 0 (r 1 , r 2 , .., r N ) = A Ψ λi k zi(r i ), , (4.5)iwhere the product of one particle states is anti-symmetrized by the operatorA and λ i denote all valence bands.As final state an excited state is constructed by taking an electron outof one of the top most valence bands,⎛⎞Ψ excited (r 1 , r 2 , .., r N ) = A ⎝Ψ λj = c k zj(r j ) ∏ Ψ λi k zi(r i ) ⎠ , (4.6)i ≠jwhere j = c indicates that the electron is placed into one of the empty conductionbands. In the remainder this notation will be used to distinguishwavefunctions belonging to conduction bands from valence band wavefunctionsΨ λi k zi(r i ).One can proceed further by forming the final exciton state from a linearcombination of the excited states (4.6). However, in this paper only theband-to-band transitions from the groundstate () to a particular excitedstate (4.6) are considered. By writing down the antisymmetrization operatorexplicitly in terms of the permutation operator P , A = √ 1N!∑P (−1)P P , thetransition matrix (4.3) for these specific initial and final states becomesM c v = − em 0 c 〈Ψ excited | ∑ A(r i ) · p i |Ψ 0 〉 = − e 1 ∑∫(−1) P +P ′dr 1 . . . dr N ×mi0 c N!P P(′P Ψ ∗ λ l = c k zl(r l ) ∏ ∑ ( ∏ )k ≠l Ψ∗ λ k k zk(r k ))i A(r i) · p i P ′ j Ψ λ j k zj(r j ) . (4.7)In order to simplify this expression, we recall the orthonormality relations∫dr Ψ ∗ λ i = c k zi(r i ) Ψ λj k zj(r j ) = 0, (4.8)∫dr Ψ ∗ λ i k zi(r i ) Ψ λj k zj(r j ) = δ λi ,λ jδ kzi ,k zj, (4.9)where (4.8) is due to the orthogonality of the zone center atomic functions,s-like and p-like for conduction and valence band wavefunctions, respectively.Equation (4.9) is explained by the orthogonality of (3.27) and thenormalization (3.40) of the total wavefunction.It can be shown [15] that, utilizing (4.8) and (4.9), the transition matrixelement (4.7) simplifies toM c v = − e ∫dr Ψ ∗ λm 0 cc k zc(r) A(r) · p Ψ λv k zv(r), (4.10)69

Chapter 4.EM transition matrixin other words, M c v is the transition matrix for an electron in one particularstate Ψ λvkzvin a valence band λ v which is excited to a conduction bandstate Ψ λc k zcby the EM radiation.It is easy to check the simplification (4.10) of equation (4.7) if N = 2:from the 8 terms only two identical terms are nonzero and they cancel the2! in the denominator.4.2 Bloch representationThe transition matrix (4.10) is developed further by making use of either theWannier or the Bloch representation for the wavefunctions. Here the Blochscheme is chosen, for the practical reason that zone center Bloch functionsare much better known. An other disadvantage of expanding in Wannierfunctions lies in the fact that Wannier functions are extended over a regionlarger then a unit cell, which makes it less defensible to approximate theEM field as constant over the relevant intervals of integration [18].4.2.1 Total wavefunction in Bloch functionsInstead of expanding the wavefunction in zone center Bloch functions by applyingansatz (3.25), the Bloch representation is generalized by expandingthe wavefunction in k space using the Fourier transform of the Wannierfunctions, equation (3.20).Defining the Fourier transform of the envelope function ˆF λkz ,j (k ′ ) asˆF λkz ,j (k ′ ) ≡ N − 1 ∑2R F λk z ,j (R) e −ik′·R , the total wavefunction Ψ λ kz (r)in this way ecomesΨ λ kz (r) = ∑ ∑ ∑F λkz,j (R) a jR (r) = N − 1 2 F λkz,j (R) e −ik′·R ψ nk′(r)jRjR k ′}∑ ∑= N − 1 2 N − 1 2 F λkz ,j (R) e −ik′·R e ik′·r u jk′(r)jk ′ {∑= N − 1 2Rjk ′ ˆFλkz,j (k ′ ) e ik′·r u jk′(r), (4.11)where in the second line the normalized Bloch functions (3.3) are used.This expression is specified further in the infinite wire configuration bydecomposing the envelope F λ kz , j in its radial part and the plane wave alongthe cylinder axis,F λ kz , j(R) = χ λ kz , j (R ⊥ ) eikzZ√M, (4.12)70

4.2. Bloch representationwhere the envelope is normalized in the Z direction by denoting the numberof atoms in this direction with M. Note that the not normalized lateral partχ λ kz , j (R ⊥ ) in general depends on k z , as concluded in the previous section.Utilizing equation (4.12), the Fourier transform of the envelope functionˆF λkz ,j (k ′ ) is decomposed aswhere∑∑ˆF λkz,j (k ′ ) = N − 1 2 χ λ kz, j (R ⊥ ) e −ik′ ⊥·R ⊥R ⊥ Zξ λkz ,j (k ′ ⊥ ) ≡ ( NMe i(k z−k ′ z )Z√M= ξ λkz ,j (k ′ ⊥ ) δ k z,k ′ z , (4.13)) −12 ∑R ⊥χ λ kz , j (R ⊥ ) e −ik′ ⊥·R ⊥(4.14)is the Fourier transform of the radial part χ λ kz, j (R ⊥ ).Inserting (4.13) in equation (4.11), the total wavefunction expanded inBloch functions u j (k′⊥ ,k z) reads∑Ψ λ kz (r) = N − 1 2jk ′ ⊥ξ λkz ,j (k⊥ ′ ) ⊥·r ⊥+k z z) ei(k′ u j (k′⊥ ,k z) (r). (4.15)Within the framework developed in the previous chapter, this expressionis valid around the zone center of any bulk band. This general form thusalso describes the total wavefunction in a confined conduction band, wherethe sum over j usually disappears because this band is non degenerate inmost of the III-V materials. Furthermore, equation (4.15) is also valid foran arbitrary strength of the confinement V 0 (as in equation (3.37)).4.2.2 EM transition matrix in Bloch functionsUtilizing the general expression (4.15) both for the valence and conductionband wavefunction, the transition matrix (4.10) is expanded in Blochfunctions byM c v = − em 0 c= − em 0 c1N∫∫dr Ψλ ∗ c k zc(r) A(r) · p Ψ λv k zv(r)∑ ∑ξλ ∗ c k zc ,j c(k ′ ⊥c ) ξ λ v k zv ,j v(k ′ ⊥v ) × (4.16)j c j vk ′ ⊥c k′ ⊥vdr e −i(k′ ⊥c·r ⊥+k zc z) u jc (k ′ ⊥c ,k zc)(r) A(r) · p e i(k′ ⊥v·r ⊥+k zv z) u jv (k ′ ⊥v ,k zv)(r),where the labeling ⊥ indicates that the corresponding vector lies in the planeperpendicular to the wire axis, as in section 4.2.1.71

Chapter 4.EM transition matrixFor the moment, we narrow the focus to the last line in equation (4.16).The integral over the entire space r can be split up into N integrals over aunit cell, ∫ drf(r) = ∑ ∫R Ω 0dr ′ f(R + r ′ ). Using the commutation relation[p, e ik·r ] = −k e ik·r , the third line in (4.16) becomes1 ∑e −i(k′ ⊥c −k′ ⊥v )·R ⊥e −i(kzc−kzv)Z × (4.17)NR∫dr e −i(k′ ⊥c −k′ ⊥v )·r ⊥e −i(kzc−kzv)z u jc(k ′ ⊥cΩ ,kzc) (r) A(R + r) · (p + k) u jv(k ′ ⊥v ,kzv) (r).0Further progress is made by assuming the variation of the EM waves to besmall over a unit cell, i.e.A(R + r) ≃ A(R), (4.18)This approximation is justified by the fact that the wavelength λ 0 of theEM radiation typically is much larger then the interatomic distances a 0 . Asin the case of the effective mass approximation, where the restriction on δV(slowly varying over one unit cell) leads to a separation between atomic wavefunctionsand ”macroscopic” envelope functions, the envelope and Blochparts of the transition matrix elements will factor into separate integrals dueto assumption (4.18).In order to show this explicitly, reconsider expression (4.17) and includethe following:• Assuming the EM field constant over a unit cell leads to the eliminationof the k term, because in this case the integral contains just twoorthogonal Bloch functions.• In the case of an infinite cylinder the EM field is of the formA(R) = A(R ⊥ )e iq zZ(4.19)To be more precise, in Section 1.3 it was derived that q z = −k 0 sin θ,with θ the angle of incidence compared to a plane perpendicular tothe wire axis.• It is more convenient to rewrite k c and k v in terms of the total momentumK of the system by definingK ≡ k c − k v ; (4.20)k ≡ k v . (4.21)72

4.2. Bloch representationNow (4.17) is simplified by1 ∑∑A(R ⊥ )e −iK′ ⊥·R ⊥e −i(Kz−qz)Z · (4.22)NR ⊥ Z∫dr e −iK′ ⊥·r ⊥e −iKzz u jc (K ′ ⊥Ω +k′ ⊥ ,K z+k z )(r) p u jv (k ′ ⊥ ,k z)(r),0With the notion that under assumption (4.18) q z is much smaller then areciprocal lattice vector G iz and that K z lies in the first Brillouin zone,the summation over M lattice points results in a momentum conservationrelation in the Z direction:∑e −i(K z−q z )Z = Mδ qz ,K z. (4.23)ZThis simplifies (4.22) further intoδ qz ,K z Â(K ′ ⊥ ) · ∫Ω 0dr e −iK′ ⊥·r ⊥e −iK zz u jc (K ′ ⊥ +k′ ⊥ ,K z+k z )(r) p u jv (k ′ ⊥ ,k z)(r),where Â(K ⊥) is defined as the Fourier transform of A(R ⊥ ),Â(K ⊥ ) ≡ M ∑A(R ⊥ )e −iK ⊥·R ⊥. (4.24)NR ⊥Furthermore, since the envelope functions are also assumed to be slowly varyingover a unit cell, now it is possible to factor the envelope and Blochparts of the transition matrix elements into separate integrals. Any wavevectorκ in (4.16) is much smaller then a reciprocal lattice vector G i becauseit is assumed that R ≫ a 0 and κ ∼ 1 R and G i ∼ 1 a 0. Under this conditionu jκ ∼ u j0 and e −iK′ ⊥·r ⊥e −iK zz ∼ 1. Inserting (4.24) into (4.16) and using∑k ′ ⊥ K′ ⊥ ξ∗ λ ck zc,j c(k ′ ⊥ + K′ ⊥ )Â(K′ ⊥ )ξ λ vk zv,j v(k ′ ⊥ ) = ∑ R ⊥χ ∗ λ ck zc,j c(R ⊥ )A(R ⊥ )χ ∗ λ vk zv,j v(R ⊥ )the final form of the transition matrix element in Bloch functions is obtained:M c v = − em 0 c δ ∑ ∑k zc ,k zvχ ∗ λ ck zc,j c(R ⊥ )A(R ⊥ )χ ∗ λ vk zv,j v(R ⊥ ) ·j cj v R∫⊥dr u jc0(r) p u jv0(r). (4.25)Ω 0This result is also obtained by using ansatz (3.25), a simplified expansionof the wavefunction in Bloch states which is justified if the wire radius Ris sufficiently large, i.e. the Fourier related k values are so small that thecorresponding energy difference ɛ j (k) − ɛ j remains much smaller then theband edge differences ɛ j − ɛ n ′.It should be noted that the degeneracy of the hole and electron energybands is not taken into account in (4.25). It can be included in the sameway as by replacing (4.3) with (4.4), as will be done further on when thematrix elements are calculated explicitly.73

Chapter 4.EM transition matrix4.3 Reformulation of transition matrix elementIn this section the transition matrix element (4.25) is reconsidered by includingthe degeneracy of the conduction and valence subbands in case oftransitions between the top Γ 8 valence bands and the lowest lying Γ 6 conductionband in III-V semiconductors. Apart from the trivial degeneracyin ± k z , the transitions are degenerate in the quantum numbers ± l z c and± σ of the conduction subbands and in ± f z v of the valence subbands, seeChapter 3.The EM field is specified further by considering two approximations withrespect to the wire radius as derived in Part I. First, the spatial variationof the EM field across the wire diameter is entirely neglected, i.e. the transitionmatrix element is formulated in the dipole limit, where λ 0 ≫ R soA(R ⊥ ) ≃ A. Secondly, a spatial variation of the EM field is taken intoaccount by applying the scattering fields (2.8), (2.9) and (2.10), which areexpansions up to second order in 2πλ 0R for normal incident light.Recall that the Coulomb gauge is chosen by deriving the transition matrixelement, so the vector potential is related to the transverse electric fieldby E = − 1 ∂Ac ∂t. In case of absorption this yieldsE = − iω cA, (4.26)with ω the frequency of the EM field.4.3.1 EM field in dipole approximationIn Part I it was shown that the strength of the EM field inside the wire dependson the polarization of the incident light. In the dipole approximationthis resulted in (2.6) and (2.7) for polarization parallel and perpendicular tothe wire axis, respectively. In order to separate this polarization anisotropyfrom the dielectric mismatch, a matrix element T c v is defined by|M c v (k z )| 2 e≡ (m 0 ω )2 | E 2 |2 |T c v (k z )| 2 , (4.27)where k z = k zc = k zv and E is the strength of the electric field inside thewire in the dipole approximation, E ≡ E ˆε.Including the degeneracy, from equation (4.4) and (4.25) one obtains forthe transition matrix T c v between valence and conduction subband v and c:|T c v (k z )| 2 = ∑∑2〈χ lzc |χ fzv;jz (k z ) 〉〈Sσ| ˆε · p | 3 2∣j z 〉, (4.28)j z∣74σ l d zc f d zv

4.3. Reformulation of transition matrix elementwhere lzc d denotes l zc = {|l zc | , −|l zc |}, the degeneracy at a given |l zc |. A similardefinition applies to fzv. d The notation of the atomic part is explainedin (4.34). Furthermore, 〈χ lzc |χ fzv ;j z(k z ) 〉 = ∫ dR ⊥ χ ∗ l zc(R ⊥ )χ fzv k z ;j z(R ⊥ ),where the replacement of the summation ∑ R ⊥by an integral is justifiedsince in the effective mass approximation the dimensions of the wire are assumedto be much larger then interatomic distances. The additional step sizeby going from summation to integration is eliminated by the normalizationof the wavefunctions.Note that indeed the strength of the internal field is separated fromT cv . This classical penetration effect is absorbed in E, as shown in equation(4.28).4.3.2 EM field including Mie scatteringIn case of a spatially varying EM field the separation in (4.27) is not possibleany more. Instead, the EM field has to be integrated between the conductionand valence subband envelope functions over the wire cross section areaπR 2 , as shown in (4.25). However, from equations (2.8), (2.9) and (2.10)one finds that the reduction factor caused by the penetration into the wirestill can be divided out, just as in the dipole limit. Thus, by separatingout the factor E = E 0 in case of polarization parallel to the wire axis andE = 21+ε E 0 at perpendicular polarization, the matrix element T c v containsonly the scattering (optical focusing) and expansion terms due to the wavebehavior of the EM field inside the wire. Denoting E ′ (R ⊥ ) as the electricfield without the penetration strength, E E ′ (R ⊥ ) ≡ E(R ⊥ ), the right handside of (4.28) now is replaced with|T c v (k z ; ε, R)| 2 = ∑2∑〈χ lzc ,n|E ′ |χ fzv ,n;j z(k z ) 〉 · 〈Sσ| p | 3 2∣j z 〉(4.29) .j z∣σ l d zc,f d zvNote that, contrary to the dipole limit, the direction of the internal EMfield is in general different from that of the incident field . Furthermore,due to the scattering field the matrix elements now depend on the dielectricfunction and the wire radius.4.3.3 Polarization anisotropy of the transition matrixAt this point it is instructive to define a polarization anisotropy purely originatingfrom the matrix elements. As demonstrated by equation (4.28), inthe dipole limit it is possible to separate the anisotropy caused by the dielectricmismatch from that which is due to the polarization in the transitionmatrix. In other words, in principle one is able to determine the polarizationanisotropy caused by the transition matrix elements alone once itis (experimentally) possible to eliminate the polarization anisotropy of the75

Chapter 4.EM transition matrixdielectric mismatch, for instance by a sufficient increase of the intensity ofthe incident field in the perpendicular case or by changing the surrounding(for instance if the nanowire is covered by an oxide). In analogy with (2.21),the polarization anisosotropy ρ Tcv of the matrix element alone is defined asρ Tcv ≡ |T cv, ‖ | 2 − |T cv, ⊥ | 2|T cv, ‖ | 2 + |T cv, ⊥ | 2 , (4.30)where T cv, ‖ = T cv, z is the matrix element corresponding to a polarizationparallel to the wire and T cv, ⊥ denotes the perpendicular case.4.4 Selection rulesThe interband transition matrix (4.25) is investigated in more detail byconsidering the different kind of selection rules it imposes. A selection ruleoriginates from an underlying symmetry of the system under considerationand generally disappears if the symmetry on which it relies is broken. Bya selection rule some transitions are ”selected” to be allowed, while othersare said to be forbidden.As stated in section 4.3, a spatially varying EM field has to be treateddifferently then the more common dipole approximation. The dipole approximationis crucial for the selection rules originating from the envelopepart of the transition matrix element (4.25). Away from this limit the variationof the EM field starts to break the symmetry of the matrix elementbetween the envelope parts of the electron and hole wavefunctions.On the other hand, the selection rules originating from the atomic likematrix element of the momentum operator p are independent of E, providedthat the field can be considered as constant over a unit cell, an approximationwhich was made earlier.Leaving the discussion of a spatially varying EM field to paragraph 4.5.2,the selection rules in the dipole approximation fall into three different classes:• Polarization selection rules• Selection rules on the l z -angular momentum of the envelope wavefunction• Parity selection rules.The polarization rules are caused by the atomic like matrix element of themomentum operator p, while the selection rules on the l z -angular momentumof the envelope wavefunction and the related parity selection rules are76

4.4. Selection rulesdue to the envelope part of the transition matrix (4.25).Explicit investigation of the transition matrix requires a restriction to amore specific case. The selection rules depend on which particular system isconsidered and subsequently which symmetry properties are valid. Thereforethe focus will be on the band-to-band transitions between the top mostvalence bands and the lowest lying conduction band in III-V materials. Inthis case the matrix element is given by (4.28), or by (4.29) in case of aspatially varying EM field.4.4.1 Polarization selection rulesIn this paragraph the polarization selection rules in case of transitions betweenthe top Γ 8 valence bands and the lowest lying Γ 6 conduction band inIII-V semiconductors are derived. However, since the theory strongly relieson the results of atomic physics, it is instructive to summarize these shortlyin advance.Suppose an atomic system which is built from the orthonormal base{|η j m〉}, where the quantum numbers j and m come from an angular momentumoperator J and η refers to other possible quantum numbers whichcomplete the basis of the system in consideration. Then for any vectoroperator V applies〈η j m ′ |V + |η j m〉 = 0 if m ′ − m ≠ 1,〈η j m ′ |V − |η j m〉 = 0 if m ′ − m ≠ −1,〈η j m ′ |V z |η j m〉 = 0 if m ′ − m ≠ 0, (4.31)where V + ≡ V x + iV y and V − ≡ V x − iV y , as usual.In case of the absorption of light, where p is the vector operator of interest,this result gets its physical interpretation if one realizes that a photoncarries spin 1, so m = {1, 0, −1}: in this case (4.31) is just a consequenceof the conservation of angular momentum.Turning to the optical transitions in III-V semiconductors, rememberthat an analogy was made between the band edge Bloch states and atomicfunctions, see paragraph 3.1.1. Now it becomes more clear what is actuallymeant with ”atomic-like”: in the k · p method the optical matrix elementsare used as input. Without specifying the precise form of a particular bandedge Bloch function, it can be argued that its symmetry properties are thesame as a particular atomic function. All what remains is to determine(experimentally) the optical matrix elements.Concerning transitions between the Γ 8 valence bands and the Γ 6 con-77

Chapter 4.EM transition matrixduction band, the only nonzero matrix elements are given by− im 0〈S|p x |X〉 = − im 0〈S|p y |Y 〉 = − im 0〈S|p z |Z〉 = P, (4.32)where |S〉 denotes the band edge Bloch function of the conduction bandwhich is s-like. The magnitude P of the matrix elements in (4.32) is relatedto the Kane matrix element E p byE P = 2m 0 P 2 , (4.33)which can be determined experimentally for each particular III-V bulk semiconductormaterial.As an upshot, the polarization selection rules in semiconductor materialsresult from restrictions imposed on the matrix element of the momentumoperator by the symmetry properties of the atomic-like Bloch states. Generally,from group theory it is known that a matrix element 〈ψ 1 |Ô|ψ 2〉 isonly nonzero if the symmetry S 1 of ψ 1 is the same as one of the irreduciblerepresentations of the direct product O ⊗ S 2 , where O denotes the symmetryof the operator Ô. Indeed, analyzing the symmetry properties of themomentum operator, one finds that p-like and s-like states lead to the onlynonzero matrix elements given in (4.32)[14].Narrowing the focus to band-to-band transitions between the top Γ 8 valencebands and the lowest lying Γ 6 conduction band in III-V semiconductors,the the atomic part of the transition matrix (4.25) is specified furtherby∫Ω 0dr u jc0(r) p u jv0(r) = 〈S σ | p | 3 2 j z 〉, (4.34)again with σ = ↑, ↓ and j z ɛ { 3 2 , 1 2 , − 1 2 , − 3 2}. This matrix can be calculated interms of the only nonzero matrix elements (4.32) by using the decompositionof | 3 2 j z 〉 in the states |X〉,|Y 〉 and |Z〉, as given in (3.13).Table 4.1 shows the result for the unpolarized interband matrix element〈S σ | p x + p y + p z | 3 2 j z 〉.Here the matrix elements P u are introduced for convenience,〈S|p u |U〉 ≡ P u , u = {x, y, z}, U = {X, Y, Z}. (4.35)In terms of the Kane matrix elements, equations (4.32) and (4.33), Table4.2 gives the quantitative results of the polarization dependence of the1matrix element 〈S σ | p(m 0 E p) 1 u | 3 2 j z 〉. This result plays a dominant role2by determining the polarization anisotropy in the EM transition matrix elements,see paragraph 4.3.3.78

4.4. Selection rulesp x + p y + p z | 3 232 〉 | 3 212 〉 | 3 2 − 1 2 〉 | 3 2 − 3 2 〉〈S ↑ | − 1 √2P x − i √2P y√2 √3P z1 √6 P x − i √6P y 0〈S ↓ | 0 − 1 √6P x − i √6P y√2 √3P z1 √2 P x − i √2P yTable 4.1: Result for the unpolarized interband matrix elements〈S σ | p x + p y + p z | 3 2 j z 〉 in terms of P u ≡ 〈S|p u |U〉. For a particulartransition, the spin σ of the conduction band electron is shown inthe left column and the Bloch angular momentum j z belonging to thevalence band in the first row.For instance, Table 4.2 clearly demonstrates that the ratio of ∑ σ |〈S σ | p z | 3 2 j z =± 1 2 〉|2 and ∑ σ |〈S σ | p x | 3 2 j z = ± 1 2 〉|2 equals 4. The polarization selectionrule has an even stronger effect on the valence subband states which aredominated by terms with j z = ± 3 2 : in this case the matrix element of p zis strictly zero. A strictly zero matrix element of a particular transitionand direction of the momentum operator is said to be polarization forbidden,or polF in short notation. This qualitative result is summarized inTable 4.3, which shows the allowed polarizations. Here x, y denotes the allowedpolarizations perpendicular to the wire axis and z the allowed parallel1(m 0 E p) 1 2p u | 3 232 〉 | 3 212 〉 | 3 2 − 1 2 〉 | 3 2 − 3 2 〉p x − i 20i2 √ 3〈S ↑ | p y12012 √ 300p z 0i √30 0p x 0 − i2 √ 3〈S ↓ | p y 012 √ 30i2012p z 0 0i √30Table 4.2: Selection rules on the atomic-like interband matrix elements1〈S σ | p u | 3 2 j z 〉.(m 0 E p ) 1 279

Chapter 4.EM transition matrixpolarization. Another feature which will be extracted from Table 4.2 furtherˆε · p | 3 232 〉 | 3 212 〉 | 3 2 − 1 2 〉 | 3 2 − 3 2 〉〈S ↑ | x, y z x, y polF〈S ↓ | polF x, y z x, yTable 4.3: Selection rules on the atomic-like interband matrix elements〈S σ | ε · p | 3 2 j z 〉. The allowed polarizations are denoted with x, y andz, where x, y are perpendicular to the wire axis and z gives the parallelpolarization. Polarization forbidden transitions are denoted with polF .on is that the matrix elements corresponding to polarizations perpendicularto the wire axis are the same, i.e. |T cv, x | 2 = |T cv, y | 2 , which is a consequenceof the rotational symmetry around the wire axis in the dipole approximation.Finally, it is instructive to come back to the selection rules in the atomiccase. In fact, the results in Table 4.2 lead to the same selection rules as givenin (4.31). For this purpose, first note that Table 4.2 can be formulated inan algabraic way by〈Sσ|p x | 3 2 j z 〉 = i(m 0 E p ) 1 2√16 |j z| (δ σ−jz,1 − δ σ−jz,−1),〈Sσ|p y | 3 2 j z 〉 = (m 0 E p ) 1 2√16 |j z| (δ σ−jz ,1 + δ σ−jz ,−1),〈Sσ|p z | 3 2 j z 〉 = i(m 0 E p ) 1 2√23 |j z| δ σ,jz . (4.36)In terms of p + ≡ p x + ip y and p − ≡ p x − ip y the matrix elements of p x andp y yield〈Sσ|p + | 3 2 j z 〉 = i(m 0 E p ) 1 2√23 |j z| δ σ−jz,1,〈Sσ|p − | 3 2 j z 〉 = (m 0 E p ) 1 2√23 |j z| δ σ−jz ,−1. (4.37)Together with the matrix element of p z in (4.36), these are the same selectionrules as in the atomic case, now originating from the conservation of angularmomentum on the Bloch part of the transition matrix.4.4.2 Selection rules on the envelope wavefunctionsThe selection rules on the envelope part of the transition matrix are not asgeneral as the polarization selection rules derived in paragraph 4.4.1. While80

4.4. Selection rulesthe polarization selection rules originate from the bulk, atomic like matrixelement of the momentum operator, the selection rules on the envelope partare dependent on the configuration of the system (wire radius, length) andalso depend on whether the EM field can be considered in the dipole limit.T ransition P olarization Class |j z |C 0, 1 → E 12 ,n 1x, y, z MP2C 0, 1 → E 32 ,n 3x, y SP2C 0, 1 → E 52 ,n − lF −C 1, 1 → E 12 ,n 1x, y, z MP2C 1, 1 → E 32 ,n 1x, y, z MP2C 1, 1 → E 52 ,n 3x, y SP2Table 4.4: Summary of the polarization and class for the lowest band-tobandtransitions in C ∞ nanowires with a constant EM field. Polarizationperpendicular to the wire axis is denoted with x, y, parallel polarizationwith z. The envelope angular momentum forbidden transitions are denotedwith lF while SP and MP denote the polarization class, singleand mixed polarization respectively. The last column gives the allowedvalues of |j z |.Proceeding with the transitions between the Γ 8 valence bands and theΓ 6 conduction band in III-V semiconductors in the dipole limit, equation(4.28), the φ part of 〈χ lzc ,n|χ fzv ,n;j z(k z ) 〉 gives∫1 2πdφ e −i(l zc−(f zv −j z ))φ2π 0= δ lzc ,f zv −j z, (4.38)which can be considered as a selection rule on the envelope angular momentum,since l zv = f zv − j z . As a direct consequence, transitions for whichl zc − f zv ≠ {− 3 2 , − 1 2 , 1 2 , 3 2} are l-angular momentum forbidden (lF ). Thisresult can be found in Table 4.4, which shows a combination of the polarizationand l-angular momentum selection rules. The last row gives theallowed values of |j z |.81

Chapter 4.EM transition matrixThe constraint (4.38) also leads to a selection rule which is a consequenceof the parity of the subbands at the wire zone center. An even (odd) wavefunctioncorresponds to an even (odd) Bessel function J lz and by (4.38) l zcand l zv = f zv −j z has to be the same, i.e. the parity of the valence subbandwavefunction has to match with the parity of the conduction subband. Inother words, transitionsE (±) → C (∓) (4.39)are parity forbidden (pF) at k z = 0. Away from the zone center this selectionrule generally is broken: the valence subband wavefunctions are notcharacterized by parity any more.82

4.5. Results4.5 ResultsIn this section the above theoretical framework is applied to specific examples.Again InP and InAs are chosen since those III-V materials are two kind ofextremes regarding their electronic properties. As a matter of fact, althoughthis is also the case for the Kane matrix element E p , the effect of thisdifference will be small since E p is hardly material dependent, see Table 4.5.InP InAsE p (eV ) 20.7 21.5Table 4.5: The Kane matrix element E p for InP and InAsSince a spatially varying EM field requires a different approach comparedto the theory derived in the dipole limit, this case is treated separately inparagraph 4.3.2.4.5.1 Dipole approximationThe topmost illustration in Figure 4.1 (a) and Table 4.6 show the nume-E Trans eV, R⩵4.85 nmΡ T v c T v c 2 arb. units0.140.120.100.080.060.040.021.000.800.600.400.2000.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.82a : k z R⩵0. b : k z R⩵0.450.140.120.100.08zpol.0.06ypol.0.040.02v2 c1v3 c1v4 c1Ρ0Ρ0v7 c1v1 c11.000.800.600.400.200.49 0.5 0.5 0.51 0.52 0.53 0.48 0.49 0.5 0.5 0.51 0.52 0.53 0v2 c1v3 c1v4 c1v5 c1v6 c1v7 c1E Trans eV, R⩵9.96 nmFigure 4.1: Matrix elements |T cv,‖ | 2 (first row, white bars) and |T cv,⊥ | 2(black) in arbitrary units and corresponding polarization anisotropy ρ Tcv(second row) of the first 7 transitions v i → c 1 for InAs at a) : k z R = 0and b) : k z R = 0.45. The colors are the same as in Figure 3.1; for therepresentations see Table 4.6. The energy scale for all graphs is given attwo values of R, indicated at the top and bottom of the figure.83

Chapter 4.EM transition matrixrical results at k z = 0 for InAs of the matrix elements perpendicular andparallel to the wire axis, |T cv, y | 2 and |T cv, z | 2 respectively. The upper graphFigure 4.1 (b) gives the same results away from the wire zone center, atk z R = 0.45. The matrix elements are calculated for wavefunctions validin an infinite potential well both for the valence and the conduction subbandsand so the values are independent of R. Contrary to the energies, thecorrection on the subband wavefunctions by taking a finite potential intoaccount is expected to be negligible, provided the dimensionless quantity l Vin (3.71) is not too small.The last two columns in Table 4.6 give the transition energy and correspondingwavelength. In this case the conduction subbands are calculated inthe finite potential wells given in Table 3.4, at R = 4.85 nm.T ransition Representation Class |T cv, y | 2 |T cv, z | 2 E trans (eV ) λ trans (nm)v 1 → c 1E (−)12(+)→ C,1 0, 1 pF − − 0.616 2011v 2 → c 1 E (+) (+)1 → C,1 0, 1 MP 2.94 10 −2 11.77 10 −2 0.636 19492v 3 → c 1 E (+) (+)3 → C,1 0, 1 SP 1.23 10 −2 − 0.670 18502v 4 → c 1 E (+) (+)1 → C,2 0, 1 MP 6.15 10 −3 24.6 10 −3 0.712 17542v 5 → c 1v 6 → c 1E (−)32E (−)12(+)→ C,1 0, 1 pF − − 0.718 1726(+)→ C,2 0, 1 pF − − 0.784 1581v 7 → c 1 E (+) (+)3 → C,2 0, 1 SP 8.39 10 −2 − 0.818 15152Table 4.6: Interband matrix elements |T cv, y | 2 and |T cv, z | 2 of the firstseven transitions v i → c 1 for InAs at k z = 0. The parity forbiddentransitions are denoted with pF , SP and MP refer to single and mixedpolarization, respectively. The last two columns show the transitionenergy and corresponding wavelength calculated at R = 4.85 nm withthe conduction subband in the finite potential well model.Furthermore, Table 4.6 summarizes the representation and class of thefirst seven transitions v i → c 1 as derived in the previous sections. In Figure4.1 the transitions are indicated in the same colors as used in Figure 3.1.The second row in Figure 4.1 gives the polarization anisotropy ρ Tcv correspondingto the matrix elements alone, as defined by equation (4.30).84

4.5. ResultsA closer investigation and comparison with the theory leads to the followingconclusions:• At the zone center only transitions between subbands with the sameparity are allowed. Away from the zone center, the parity selection isbroken since the hole states are not characterized by parity any more.• The polarization anisotropy is completely determined by the polarizationrules. All E (+)1 → C (+)0 transitions show a polarization of 0.6,2which is in agreement with the analytical result of paragraph 4.4.1,where a polarization contrast |T cv, ‖| 2|T cv, ⊥of 4 was determined for |j| 2 z | = 1 2 .Indeed only these states contribute to the matrix elements, since in thedipole limit it is required that |j z | = |f zv | by the l-angular momentumselection rule on E 12expected: for the |j z | = 3 2(polF).→ C 0 . Also the results of E (+)32→ C (+)0 are asstates a parallel polarization is forbidden• The matrix elements are independent of the wire radius. It should benoted that the finite confinement for the conduction subbands is nottaken into account, but this correction is expected to be small. On theother hand, the transition energies strongly depend on the wire radius.Therefore, the energy scale is given at two values of R in Figure 4.1,R = 4.85 nm and R = 9.96 nm indicated at the top and bottomof the figure, respectively. Since the correction is substantial for thetransition energies, these are calculated with the conduction subbandin the finite potential well model.• The present effective mass approach can be compared successfully withthe results derived in an atomistic approach. In Appendix D, Figure 14the band-to-band matrix elements for an R = 4.8 nm InAs wire areshown based on an atomistic, empirical pseudo-potential plane-wavemethod [4]. It is noted that the C ∞ v representations given in Table IIof the article differ from those derived in the present paper and are inconflict with the basic symmetry considerations of Chapter 4. Withthis in mind, comparing Figure 4.1 with FIG. 3 in [4] it is concludedthat also in the more accurate atomistic approach the E (+)3 → C (+)0, 12transitions are completely y-polarized. Considering the polarizationanisotropy of the E (+)12→ C (+)0, 1transitions, the deviations from 0.6 inFIG. 3 in [4] are explained by the possible corrections of including thesplit-off band, or even diagonalizing the full 8 × 8 Hamiltonian of thethree Γ 8 valence bands and Γ 6 conduction band in the present effectivemass approximation. This also explains that some pF transitions in85

Chapter 4.EM transition matrix[4] still have a small contribution at k z = 0. For the shifts in thetransition energies the additional argument holds that taking also finiteconfinement for the valence subbands into account may change thevalence energies slightly.E Trans eV, R⩵4.83 nmΡ T v c T v c 2 arb. units0.140.120.100.080.060.040.021.000.800.600.400.201.53 1.55 1.58v1 c1v3 c1a : k z R⩵0.zpol.ypol.Ρ0Ρ001.45 1.46 1.46v5 c1v6 c11.55 1.58 1.6 1.62 1.65b : k z R⩵0.93v1 c1v3 c1v2 c1v4 c1v5 c1v6 c11.46 1.46 1.47 1.47 1.48v7 c10.140.120.100.080.060.040.021.000.800.600.400.200E Trans eV, R⩵10. nmFigure 4.2: Matrix elements |T cv,‖ | 2 and |T cv,⊥ | 2 and corresponding polarizationanisotropy ρ Tcv of the first 7 transitions v i → c 1 for InP ata) : k z R = 0 and b) : k z R = 0.93.The colors are the same as in FIgure 3.1, for the representations seeTable 4.7. The energy scale for all graphs is given at two values of R,indicated at the top and bottom of the figure.The above results of InAs can be compared with InP, see Figure 4.2 andTable 4.7. This leads to the following conclusions:86• The large difference in the transition energies, for instance the transitionE (+)1 → C (+)0, 1 corresponds to E trans = 0.636 eV for InAs and2E trans = 1.529 eV for InP, is caused by the difference in the bulk bandgap E g , see Table 3.4. It is slightly reduced by the difference in confinementenergies for the electron subband, see Table 3.6: since InP hasa larger conduction band effective mass m ∗ c than InAs, its confinementenergy of the conduction subbands is lower.• As already stated in Chapter 3, the ordering of the valence subbands ismaterial dependent, due to the differences in the heavy- and light holeeffective masses. For instance the first two transitions E (−)1 → C (+)0, 12

4.5. ResultsT ransition Representation Class |T cv, y | 2 |T cv, z | 2 E trans (eV ) λ trans (nm)v 1 → c 1 E (+) (+)1 → C,1 0, 1 MP 3.4 10 −2 13.7 10 −2 1.529 8112v 2 → c 1E (−)12(+)→ C,1 0, 1 pF − − 1.542 804v 3 → c 1 E (+) (+)3 → C,1 0, 1 SP 9.1 10 −2 − 1.556 7972v 4 → c 1E (−)32(+)→ C,1 0, 1 pF − − 1.574 788v 5 → c 1 E (+) (+)3 → C,2 0, 1 SP 1.2 10 −2 − 1.586 7822v 6 → c 1 E (+) (+)1 → C,2 0, 1 MP 3. 10 −4 12. 10 −4 1.590 7802v 7 → c 1E (−)12(+)→ C,2 0, 1 pF − − 1.648 752Table 4.7: Interband matrix elements |T cv, y | 2 and |T vc, z | 2 of the firstseven transitions v i → c 1 for InP at k z = 0. The parity forbiddentransitions are denoted with pF , SP and MP refer to single and mixedpolarization, respectively. The last two columns show the transitionenergy and corresponding wavelength calculated at R = 4.83 nm withthe conduction subband in the finite potential well model.and E (+)12→ C (+)0, 1are ordered the other way around in case of InP.As an important consequence, the parity selection rule works on thelowest possible transition in case of InAs, while for InP this selectionrule at k z = 0 works on the second possible transition.• Next to the energies, also the transition strengths are material dependent.This is caused by the different heavy- and light hole effectivemasses which result in different valence subband wavefunctions.4.5.2 EM field including Mie scatteringIn Figure 4.3 the numerical results are shown of the matrix elements |T cv,‖ | 2and |T cv,⊥ | 2 and corresponding polarization anisotropy ρ Tcv for InAs, includingthe effects of spatial variation of the EM field up to second order. Thepenetration strength, which is also present in the dipole limit, is not takeninto account and the results are obtained by using equation (4.29). Sincethe matrix elements now depend on the wire radius by the scattering field,87

Chapter 4.EM transition matrixR = 4.85 nm is fixed. Contrary to the expressions for the expanded electricfield at normal incidence in Part I, which were given in cylindrical coordinates,here there components in cartesian coordinates are used. As canbe concluded from (2.9) and (2.10), for polarization perpendicular to thewire axis the internal field does not have the same direction as the incidentfield. However, contributions from the other direction (say x) to the matrixelements are negligible.E Trans eV, R⩵4.85 nmΡ T v c T v c 2 arb. units0.140.120.100.080.060.040.021.000.800.600.400.200.62 0.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.820.14a : k z R⩵0. b : k z R⩵0.450.12zpol.0.10ypol.0.080.060.040.02v1 c1v2 c1v3 c1v4 c1v5 c1Ρ0Ρ0v6 c1v7 c100.62 0.65 0.68 0.72 0.75 0.78 0.82v1 c1v2 c1v3 c1v4 c1v5 c1v6 c1v7 c11.000.800.600.400.62 0.65 0.68 0.72 0.75 0.78 0.82 0 0.20E Trans eV, R⩵4.85 nmFigure 4.3: Matrix elements |T cv,‖ | 2 and |T cv,⊥ | 2 and corresponding polarizationanisotropy ρ Tcv of the first 7 transitions v i → c 1 for InAs, calculatedincluding the scattering terms in the EM field at R = 4.85 nm.The corresponding energy scale is given at the top and bottom of thefigure.Before proceeding further by analyzing the results and comparing themwith the dipole limit, at this point it is of particular importance to note thatthe bulk value of the dielectric function is used by calculating the matrixelements. As will be discussed in more detail in Chapter 5, this can only givea first estimation since a proper calculation of the matrix elements requiresa self-consistent determination of the dielectric function.Nevertheless, starting with a qualitative approach, at first sight Figure4.3 seems to be not very different from the results in the dipole limit,Figure 4.1. The selection rules in the dipole limit still determine almostcompletely the strength of the matrix elements and away from the zone centerthe corresponding polarization anisotropy ρ Tcv . At k z = 0, however,88

4.5. Resultsthe parity selection rule is broken and the resulting transition strengths,regardless of their magnitude, are subject to a selection rule which requiresa strictly zero matrix element |T cv,‖ | 2 in case of the |f zv | = 1 2 subbands.E (−)12The explanation is subtle. First recall from paragraph 3.4.1 that thevalence subbands are heavy hole like. At the wire zone center, forf zv = 1 2 the corresponding lateral part of the envelope wavefunction (the j zcomponent) is given byχ 12 ,j z (ρ, φ) = |HH1〉 j zJ 12 −j z (j 1,nρR )} ei( 1 2 −jz)φ , (4.40)apart from a normalization constant and with j 1,n the n th zero of J 1 (x) = 0.Here |HH1〉 is given in (3.35), with k z = 0.Turning to the band-to-band transitions E (−)1 → C (+)2 ,1 0, 1 , the transitionmatrix for the ∼ ρ cos φ term in the internal field, see (2.9) and (2.10), is ofthe form2∑∑〈χ lzc =0| ρ cos φ |χ 12σ ∣,j 〉〈Sσ|p zy| 3 2 j z〉, (4.41)j z∣again only considering f z = + 1 2and focussing on the perpendicular (y)polarization. The integral over φ in the envelope part of (4.41) is onlynonzero if j z = 3 2 or j z = − 1 2and the integral over ρ gives the same valuein these two cases, apart from a minus sign in case of j z = 3 2(coming fromJ −1 (x) = −J 1 (x)). Absorbing the value of the integral over ρ in a constantc, which also includes the overall normalization of the wavefunctions, andusing (4.40), equation (4.41) equals∑∣∣∣−c 2π|HH1〉 3 〈Sσ|p y | 3 32 2 2 〉 + c 2π|HH1〉 − 1 〈Sσ|p y | 32 2 − 1 ∣∣2 〉 2σ= |c| 2 (2π) ∑ 2 ∣ − 1∣√ 〈Sσ|p y | 3 3∣∣∣2σ 32 2 〉 + 〈Sσ|p y| 3 2 − 1 2 〉∣ ∣∣∣= |c| 2 (2π) 2 −√ 1 13 2 + 12 √ 23∣= 0, (4.42)where in the first equality the |HH1〉 jz are inserted utilizing (3.35) at k z =0, while for the second equality the polarization selection rules, given inTable 4.2, are used.Since the only nonzero contributions for transitions from E (−)1 to the2first conduction subband are coming from the ρ cos φ term (all others areparity forbidden), equation (4.42) clearly demonstrates that for perpendicular(y) polarization the f zv = + 1 2part of this transition is strictly forbidden.89

Chapter 4.EM transition matrixIn a similar way it can be shown that this is also the case for f zv = − 1 2 , sowe conclude that the transitions E (−) (+)1 → C,1 0, 1 at perpendicular incidence2are polarization forbidden, if the envelope angular momentum selection rule4.38 is changed by the ρ cos φ term in the scattering field.Turning to the quantitative aspects of taking the spatial variation intoaccount, it can be concluded that the corresponding corrections are toosmall to overcome the parity selection rule at k z = 0 significantly. Thisis a general result: in Appendix C the effect of the Mie scattering is alsoinvestigated for InP at different R. The maximal contribution at k z = 0 of|T cv,‖ | 2 for the E (−) (+)1 → C,1 0, 1 transitions is of the order 10−6 and a similar2order was found for the matrix elements of the E (−) (+)3 → C,1 0, 1 transitions.2For the transitions already present in the dipole approximation it canbe concluded from Figure 4.2, Figure 12 and Figure 13, that the scatteringreduces the strength slightly, with a comparable amount for both |T cv,‖ | 2and |T cv,⊥ | 2 . While the polarization anisotropy of the matrix elements moreor less remains the same, the strength of the transition thus reduces at largerR. As an example, for ε = 12 and a transition wavelength of 900 nm, thereduction is estimated with the expansion parameter |ε| 1 2 k 0 R to be about1 % at R = 5 nm and 6 % at R = 10 nm, which is indeed confirmed by theresults in Appendix C.90

Chapter 5Dielectric function nanowireFinally, in this chapter the dielectric function and polarization anisotropyof III-V semiconductor nanowires are obtained including the quantum confinementcorrections by the band-to-band transitions between the top Γ 8valence bands and the lowest lying Γ 6 conduction band. Section 5.1.3 considersgeneral theory about the dielectric response of a particular group ofinterband transitions. In section 5.2 a total dielectric function is formulatedincluding the background response of all transitions which are not connectedwith the quantum confinement. Subsequently, in section 5.3 the totalpolarization anisotropy of a nanowire is derived and in section 5.4 explicitresults are given for InP and InAs.5.1 General theoryThe transition matrix elements, derived in the previous chapter for bandto-bandtransitions, are of crucial importance for determining the opticalresponse of a system to an incident EM field. There are mainly two approachesfor determining the macroscopic dielectric function from the quantummechanical oscillator strengths, one based on the atomic polarizability, theother using the absorption transition rate. Both methods rely on the assumptionthat the EM field can be considered in the dipole limit, i.e. E(R ⊥ ) ≃ E.Next to this it is required that the semiconductor heterostructure can be describedusing macroscopic Maxwell equations, i.e. where the different partsin the system are characterized by macroscopically averaged quantities.In the following both approaches are applied to the nanowire case, inparticular concerning the band-to-band transitions.5.1.1 Atomic polarizability approachIn order to derive the dielectric function of the nanowire by using the atomicpolarizability, the transition matrix has to be expressed in the position91

Chapter 5.Dielectric function nanowireoperator x instead of the momentum operator p.Already at this stage the dipole approximation is of crucial importance.If E(R ⊥ ) = E the electric field can be taken out of the integral as in(4.27). Omitting the subband characterization for convenience by denotingthe conduction and valence subband wavefunctions with |Ψ c 〉 and |Ψ v 〉 nowit suffices to find the relation between 〈Ψ c | p |Ψ v 〉 and 〈Ψ c | x |Ψ v 〉. Utilizingthe commutation relationim 0p = [x, H], (5.1)where H is the one-electron crystal Hamiltonian in the wire configurationderived in Chapter 3, one obtainswhere〈Ψ c | p |Ψ v 〉 = m 0i 〈Ψ c|[x, H]|Ψ v 〉= im 0Ecv trans (k z )〈Ψ c | x |Ψ v 〉, (5.2)E transcv (k z ) ≡ ω cv (k z ) ≡ E g + E c (k z ) − E v (k z ), (5.3)is the transition energy. The R dependence is omitted for the moment.Following Ziman[23], the dipole moment is proportional to the local field,〈 −ex(t) 〉 = α ′ (ω)E(t) (5.4)where α ′ denotes the real part of the atomic polarizability. Also this onlymakes sense in the dipole approximation, the proportionality is integratedout in case of a spatially varying field. From equations (5.2), (4.27) and(4.28) the atomic polarizability for the band-to-band transitions betweenthe Γ 8 and Γ 6 subbands considered in this paper is given byα ′ (ω) = 1 Ne 2 ∑m 0cv∑2m 0 ω cv (k z ) |T c v(k z )| 2 1ω 2 k zcv(k z ) − ω 2 , (5.5)where contrary to the one atom model in [23] here a factor 1 Nappears sincealready N atoms are taken into account in (5.5) by the sum over the Ninitial valence subband states.The formal prove of equations (5.4) and (5.5) requires time-dependentperturbation theory and in the present case this means that in the excitedstate (4.6) the time dependence has to be included by solving the correspondingtime-dependent Schrodinger equation. Here we make an analogy withan atomic system, since the basic arguments are the same in the two pictures.Denote a ground-state orbital with |Φ 0 〉 corresponding to a ground-state92

5.1. General theoryenergy ɛ 0 , from which an electron can be excited to higher orbitals |Φ j 〉 withenergy ɛ j . Assuming the electric field in the x direction for simplicity 1 , theelectron wavefunction |Ψ(t)〉 at a particular moment can be written as|Ψ(t)〉 = |Φ 0 〉e −iɛ 0t/ + ∑ jc j (t)|Φ j 〉e −iɛ jt/ , (5.6)where the coefficients c j (t) ∝ eE x 〈Φ j | x |Φ 0 〉 are the solutions of the timedependentSchrödinger equation. The expectation value of the dipole moment−〈Ψ(t)| ex |Ψ(t)〉 consequently becomes proportional to the electricfield as in (5.4), where the atomic polarizability in the present case is givenbyα ′ (ω) = e2 ∑ f jm (ɛj j − ɛ 0 ) 2 − ω 2 , (5.7)with f j = 2m (ɛ 2 j − ɛ 0 ) |〈Φ j |x|Φ 0 〉| 2 the oscillator strength of these atomictransitions.Turning back to the nanowire, using (5.2) it is easy to show that theoscillator strength of a transition v → c at k z corresponding to (5.5) equalsf cv (k z ) =2m 0 ω cv (k z ) |T c v(k z )| 2 , (5.8)Classically, the oscillator strength is the number of oscillators with frequencyω cv (k z ). Indeed, the quantity (5.8) is dimensionless. Quantum mechanicallyit has to satisfy the Thomas-Reiche-Kuhn sum rule, in the present case∑cv∑k zf cv (k z ) = N.Once the real part of a linear response is known, the imaginary part isuniquely determined by the Kramers-Kronig relations which are a consequenceof the causality condition. This leads toα(ω) = 1 Ne 2 ∑ ∑m 0cv(f cv (k z )k z1ωcv(k 2 z)−ω+ iπ 2 2ω δ(ω − ω cv)). (5.9)Apart from the oscillator strength, this is basically the same result as obtainedin the one atom model [4].Since the atomic dipole moment (5.4) described in this way is the samefor all the N electrons involved in transitions between the Γ 8 and Γ 6subbands, the total dipole moment per unit volume P is given byP = N VαE. (5.10)1 An electric field represented by E x e −iωt corresponds to a real electric field with amplitude2E x , so take E(t) = E x (e iωt + e −iωt )93

Chapter 5.Dielectric function nanowireAll what remains is to determine the relation between this macroscopicallyaveraged quantity and the dielectric function. Again assuming a linear response,the dielectric displacement D equalsD = εE = E + 4πP , (5.11)which leads toε − 1 = 4πN α. (5.12)VCombining this with (5.9) the imaginary part of the dielectric function isobtained:ε ′′ (ω) = 2π2 e 2m 0 ωV∑ ∑cvk zf cv (k z )δ(ω − ω cv (k z )), (5.13)where V is the volume πR 2 L of the nanowire.In the above derivation it is assumed that the local field which excites anelectron equals the macroscopic field obtained from the Maxwell equations.It is well known [8][12][23] that a more realistic result requires inclusion ofthe Lorentz correction, which has to be reconsidered when dealing with ananostructure. In the present paper this correction is not taken into account.5.1.2 Transition rate methodIn the second approach the quantum mechanically determined transitionprobability is related to the macroscopic power loss in the wire due to theabsorption process. The method is explained here shortly since it is extendedmore easily to the case of a varying EM field.Starting with the transition matrix M c v , the probability P cv for a transitionbetween subbands c and v is obtained from from Fermi’s Golden Rule,P cv (k z ) = 2π |M c v(k z )| 2 δ(ω − E transcv (k z )). (5.14)The total transition rate P subsequently equals ∑ k z∑cv P cv(k z ) and multiplyingwith the energy ω in each photon this equals the power loss in thewire volume due to absorption. With equations (4.27) and (4.28), in thedipole limit this results inP ower loss = 2π ωe 2m 2 0|E| 2 ∑ cv,k z|T c v (k z )| 2 δ(ω − E transcv (k z )). (5.15)In Part 1 this quantity already was derived macroscopically for an infinitecylinder by94W ext = C ext I 0 = C extc8π |2E 0| 2 = Q extc8π |2E 0| 2 2RL, (5.16)

5.1. General theorysee equations (1.58) and (1.59). In the second step it is taken into accountthat an electric field represented by E 0 e −iωt corresponds to a real electricfield with amplitude 2E 0 .Utilizing the efficiency factors (2.17)-(2.20) in the dipole limit, with thenotion that the scattering contribution can be neglected in that case, thetotal power loss W abs due to absorption equalsW abs = ε ′′ V ω 2π |E|2 , (5.17)where E is defined in the same way as in (4.27).Finally this givesε ′′ (ω) = 1 ( ) 2πe 2 ∑ ∑|T c v (k z )| 2 δ(ω − Ecv trans (k z )), (5.18)V m 0 ωcvk zwhich is the same result as in (5.13), here shown with f cv explicitly writtenout.As in the case of the polarizability approach, the current derivation relieson the dipole approximation, but here it is more easy to see the consequencesof allowing a spatial variation of the EM field compared to the scale ofR. Starting with the classical absorption rate, equation (5.17) in fact is anexpansion in mk 0 R up to first order, with an additional factor R coming fromthe geometrical cross section. Increasing R and subsequently mk 0 R requiresa further expansion than (5.17), here denoted with W abs (ε, R). In case ofthe quantum mechanically determined transition probability, in (5.14) nowequation (4.29) has to be used instead of (4.28).Equalizing the two transition rates yieldsW abs (ε, R) = ω ∑ ∑P cv (k z ; ε, R) (5.19)k z= 2π ωe 2m 2 0cv|E| ∑ 2 |T c v (k z ; ε, R)| 2 δ(ω − Ecvtrans (k z , R)),cv,k zwith the R, ε dependence is indicated, also for Ecvtrans . Together with theKramers-Kronig relations, (5.19) has to be solved self-consistently in orderto find ε ′ (ω) and ε ′′ (ω).As a final remark, in the above expressions (5.13) and (5.18) for ε ′′ (ω)in the dipole approximation and even more (5.19) as a relation for ε(ω) incase of a spatial varying EM field, it is implicitly assumed that the dielectricfunction is a constant, not depending on the wire radius. A correct description,however, requires allowing ε(R), also in the dipole limit: the inclusionof the Lorentz correction has to be reconsidered within a microscopic, semiclassicalapproach, when dealing with nanoscale systems which are not95

Chapter 5.Dielectric function nanowire”microscopically large” any more: the resonant states are extended over thewhole volume by the envelope part of the wavefunctions and consequentlythe induced polarization is position dependent.As a first step, however, in this paper the dielectric function is approximatedwith a constant, which depends on the orientation of the incident fieldbut is calculated in a local approach, not including the spatial variation.5.1.3 Dielectric function expressed in reduced effective massThe dielectric function derived in the previous section is expressed in a moreeasily applicable form by taking advantage of the effective mass approximation.For this purpose, first note that the sum over k z in (5.18) is replacedby an integral over the first Brillouin zone if L → ∞. Using L = Ma andtaking the reciprocal distance 1 Mε ′′ (ω) = 2 R 2 e 2m 2 0 ω2 ∑cv2πa∫ πa− π ainto account, (5.18) is replaced withdk z |T c v (k z )| 2 δ(ω − E transcv (k z )), (5.20)where the integral is taken over the first Brillouin zone.Apart from the correction by the finite potential well on the conductionsubbands, the simple kz 2 dependence of Ecvtrans (k z ) derived in the effectivemass approximation results inwheredk zdE trans cv (k z )1µ ∗ cv, z≡1 m ∗ c=( ) 12µ∗ 2 cv z1 2 2 √ , (5.21)Ecvtrans (k z ) − Ecvtrans+ 1m ∗ v, z(5.22)is the reduced effective mass in the z-direction of the nanowire obtainedfrom the results in Chapter 3 and Ecvtrans ≡ Ecv trans (0).With the density of states expression (5.21) the integral part in (5.20)equals∫1dE √ |T c v (E)| 2 δ(ω − E) (5.23)E − E transcvand after evaluating the δ distribution finally one obtainsε ′′ (ω) = 2 ( ) 1 ( )2µ ∗cv z2 2e∑R 2 2 m 0 ω cv |T c v(ω)| 2 1where |T c v (ω)| 2 is given by (4.28).√ω−E transcv, (5.24)96

5.2. Dielectric function for finite group transitionsIn the last step of the above derivation it is more realistic to add abroadening term, for instance by replacing the δ function with a Gaussiandistribution, since in practice never an infinite sharp line is seen. The quantumefficiency is always reduced by the radiative relaxation of the levels, orby impurities for instance at the surface of the nanowire.5.2 Dielectric function for finite group transitionsAlthough the results in the previous sections apply to band-to-band transitionsbetween the Γ 8 valence and Γ 6 conduction subbands in a cylindricalnanowire, up till now we did not specify precisely how to consider the sumover v and c. In principle, the real part of the dielectric function at a frequencyω away from any absorption peak is built up from all responses withfrequencies ω ′ > ω, i.e. from all atomic oscillators which are able to followthe oscillation of the EM field. In practice, however, it is hardly feasible tocalculate all contributions, for example those of bulk bands lying deep in aparticular system, or even all relevant symmetry points of the highest lyingbands.Nevertheless, focussing again on the nanowire, there is a way to takeonly a particular group of transitions into account explicitly without neglectingthe others entirely, provided the bulk dielectric function ε bulk is knownreasonably well. Denoting this group with c and v, referring to the notationabove, a background dielectric function ε bg can be defined which is basicallythe bulk dielectric function, from which the c, v group is projected out. Inother words, denoting ε w cv as the dielectric response of one particular transitionc, v in the nanowire and ε bulkcv as its contribution at the same frequencyin the bulk material, the total dielectric function in the wire configurationis given byε w (ω) = ε bg (ω) + ∑ cvε w cv(ω), (5.25)withε bg = ε bulk − ∑ cvε bulkcv . (5.26)As a first approximation, away from a bulk absorption peak ε bg is approximatedwith ε bulk , the bulk dielectric function. Contrary to common literature[4][5][14][15], where ε bg usually is not taken into account at all, this approachwill be retained carefully in the remaining of this paper.97

Chapter 5.Dielectric function nanowire5.3 Polarization anisotropy nanowireClosely related to the discussion above, the common approach [4][5][14][15]to derive the polarization anisotropy of an infinite cylinder in the dipoleapproximation is insufficient when the quantum confinement becomes essential.In the usual procedure the polarization anisotropy is estimated byconsidering the absorption coefficient of the nanowire. In contrast, by usingthe efficiency factors, here an approach is followed in which the correct dielectricfunction ε w (ω) appears in a natural way. Also the dipole limit canbe taken more precisely.By defining the relative difference δ ⊥ between the macroscopically determinedinternal field at parallel and perpendicular polarization byδ 2 ⊥ (ω, x) ≡ ∣ ∣∣∣ 21 + ε ⊥ w (ω, x)∣2, (5.27)the efficiency factors derived in Part 1 are summarized for a nanowire atnormal incidence byQ ext‖ (ω, x) = ε ′′ πx‖ w(ω, x)2 + O(x3 ), (5.28)Q ext⊥ (ω, x) = ε ′′ πx⊥ w (ω, x)2 δ2 ⊥ (ω, x) + O(x3 ), (5.29)where the expansion parameter x is defined asx ≡ k 0 R, (5.30)while ε ′′‖ w (ω, x) and ε′′ ⊥ w(ω, x) are the dielectric functions of the nanowireat parallel and perpendicular incidence, respectively. Consequently, thepolarization anisotropy due to the extinction is given byρ ext (ω, x) = Q ext‖ (ω, x) − Q ext⊥ (ω, x)Q ext‖ (ω, x) + Q ext⊥ (ω, x)= ε′′ ‖ w (ω, x) − ε′′ ⊥ w (ω, x) δ2 ⊥ (ω, x) + O(x2 )ε ′′‖ w (ω, x) + ε′′ ⊥ w (ω, x) δ2 ⊥ (ω, x) + O(x2 ) . (5.31)Taking the wire dipole limit by neglecting terms of higher order in x, thisleads toρ dip (ω, R) = ε′′ ‖ w (ω, R) − ε′′ ⊥ w (ω, R) δ2 ⊥(ω, R)ε ′′‖ w (ω, R) + ε′′ ⊥ w (ω, R) (5.32)δ2 ⊥(ω, R),where the R dependence is explicitly shown. Note that the scattering processin the wire dipole limit is negligible compared to absorption. It should be98

5.3. Polarization anisotropy nanowirestressed again that for practical purpose it is more convenient to use apolarization contrast, in the present case denoted withC dip (ω, R) =ε ′′‖ w(ω, R)ε ′′ ⊥ w (ω, R) (5.33)δ2 ⊥(ω, R).Utilizing the framework of section 5.2 it is of particular interest to considerthese results in two special cases.Starting with the one which leads in a natural way to the result in theusual procedure, if the transitions are investigated at frequencies ω whereε ′′bulk ∼ 0 and ε′ bulkis large compared to both the real and imaginary partof ∑ cv εw cv(ω) , then the background contribution to ε ′′ w(ω) is negligible andδ ⊥ is approximated with the bulk value. In this caseρ dip ∼ ∑ cv|T cv,‖ | 2 − |T cv,⊥ | 2 δ⊥2|T cv,‖ | 2 − |T cv,⊥ | 2 δ⊥2 , (5.34)which indeed equals the usual expression [14][4][5].Secondly, take a macroscopically small, but microscopically large R whichallows to neglect the quantum corrections, but still satisfies the dipole limitλ 0 ≫ R. Considering (5.25) and (5.26), we conclude that in this casethe quantum correction ∑ cv εw cv(ω)− ∑ cv εbulk cv (ω) becomes negligible, whichleads back to the classical result (2.22), as required.99

Chapter 5.Dielectric function nanowire5.4 ResultsUsing the results of the previous chapters, in this section the above theoreticalframework is applied to the specific examples InP and InAs. Initiallythe focus will be on the dielectric response purely due to the band-to-bandtransitions between the Γ 8 valence and Γ 6 conduction subbands in nanowiresby neglecting the imaginary part ε ′′bgof the background dielectric functionin (5.25). Hereby the following aspects will be discussed in more detail:• The effect of the k z dependence of the transition matrix on the imaginarypart ε ′′ w of the nanowire dielectric function .• The polarization anisotropy (4.30) of the matrix elements alone and includingthe polarization due to the dielectric mismatch only by takingε ′ bginto account, as in (5.34).• The R dependence of ε ′′ w and corresponding polarization anisotropy(5.34).• Material dependence and comparison with literature [4][5].Finally, in paragraph 5.4.2, ε ′′bgis taken into account by calculating thecorrect expression for the polarization contrast, corresponding to equation(5.33) and equivalent to the polarization anisotropy given in (5.34). Allresults are obtained in the dipole approximation.5.4.1 Estimation k z dependence of |T cv | 2In Chapter 4 it was shown that the transition matrix elements depend onk z in a nontrivial way.4zpolypol.a: kz⩵0.4v2zpolypol.b: kz⩵0.45Ε'' wire321v2v3v4v7Ε'' wire321v1v3v4v70.65 0.7 0.75 0.8 0.85EnergyeV0.65 0.7 0.75 0.8 0.85EnergyeVFigure 5.1: Complex part of the dielectric function ε w at parallel (z) andperpendicular (y) polarization for InAs and R = 4.85 nm, fixing |T cv | 2at a) : k z R = 0 and b) : k z R = 0.45. Only the first 7 transitions v i → c 1are taken into account, the corresponding peaks are labeled in the samecolors as in Figure 4.1. The background response ε ′′bgis neglected.100

5.4. ResultsAs a first simple trial, Figure 5.1 shows the imaginary part of the dielectricfunction ε w for InAs and R = 4.85 nm , obtained by fixing |T cv | 2 atits value at a) : k z R = 0 and b) : k z R = 0.45 in equation (5.24). The singularitiesat ω − Ecvtrans = 0 are broadened in a qualitative way by adding adisplacement of 0.004 eV to ω in the denominator of (5.24). Although thisis not the common way to include the broadening, it qualitatively gives thethe same results as in the usual procedure, where the Dirac distribution isreplaced for instance with a Gaussian. Since the non-radiative decay whichcauses the broadening is not estimated yet it makes little practical difference.Focussing on the first peaks in Figure 5.1, it is concluded that simplytaking |T cv (0)| 2 in (5.24) not only quantitatively, but also qualitatively failssince it does not include the contribution of the first peak, corresponding tothe transition E (−) (+)1 → C,1 0, 1 which is parity forbidden at k zR = 0, but not2at finite k z R.In Figure 5.2 the k z dependence of |T cv | 2 is taken into account properly.At each fixed value of ω equation (5.24) is calculated using the correctvalue of |T cv | 2 , which means that for every point in the figure the correctvalence subband wavefunction is used in (4.28). Only the first two peaksare shown.43zpolypol.Ε'' wire21v1v200.62 0.64 0.66 0.68 0.7 0.72EnergyeVFigure 5.2: Parallel (dots) and perpendicular (line) contributions to ε wof the first two transitions v i → c 1 for InAs and R = 4.85 nm. At eachfixed value of ω equation (5.24) is calculated using the correct value of|T cv | 2 , which means that for every point in the figure the correct valencesubband wavefunction is included in (4.28).Since it is desirable to avoid such an extensive calculation, from nowon the simpler approach of taking |T cv | 2 constant is used, but at a finitek z value such as in Figure 5.1 b). In this way the results of Figure 5.2are reproduced qualitatively: the peaks which are parity forbidden (pF)101

Chapter 5.Dielectric function nanowireat k z = 0 are included. Quantitatively, the slope corresponding to oneparticular transition is underestimated in most of the cases, since except forthe pF transitions, the matrix elements become smaller for larger k z . Theheight of the peaks will be corrected with respect to the strength of thetransitions which are already present at k z = 0. As a consequence, the pFtransition peaks are slightly overestimated in the following paragraphs.Furthermore, it is important to note that the tails of the different transitioncontributions to ε ′′ w only represent a first rough qualitative estimation.In the effective mass approach the bulk bands are assumed to be quadraticand as stated in Chapter 3 this rests on the assumption that k is sufficientlysmall. In fact, a particular transition stops contributing when it reaches theboundary of the Brillouin zone, but in the present procedure this point isnot reached at all since the hole subbands derived in Chapter 3 have a finiteextent.5.4.2 Polarization anisotropy and R dependenceUtilizing the above mentioned procedure, Figure 5.3 a) again illustrates theimaginary part of the dielectric function ε w for InAs and R = 4.85 nm ,now obtained by fixing |T cv | 2 at k z R = 0.45 and correcting the height ofthe peaks with respect to the strength of the transitions which are alreadypresent at k z = 0. As stated before, the background response ε ′′bgwill beneglected up till paragraph 5.4.4. The remarks about broadening remainthe same.Ε'' wire4321v1v2v3zpolypol.v4v70.65 0.7 0.75 0.8 0.85EnergyeVΡ10.750.50.250-0.25-0.5Ρ bulkΡ T v cΡ wire0.65 0.7 0.75 0.8 0.85EnergyeVFigure 5.3: a): Complex part of the dielectric function ε w at parallel(z) and perpendicular (y) polarization for InAs and R = 4.85 nm, fixing|T cv | 2 at k z R = 0.45 and correcting the height of the peaks with respectto the strength of the transitions which are already present at k z =0. Only the first 7 transitions v i → c 1 are taken into account, thecorresponding peaks are labeled in the same colors as in Figure 4.1. b):Polarization anisotropy, calculated from a) alone (dotted), only frombulk mismatch (line) and both together (red line).In Chapter 4 it already was shown that v 1 → c 1 is the only pF transitionwhich contributes significantly. Indeed this is confirmed by Figure 5.3.102

5.4. ResultsSecondly, the influence of the valence subband effective mass on thereduced effective mass µ ∗ cv z (5.22) is negligible: in all relevant situationsconsidered here, the effective mass of the conduction subband c 1 , also in thefinite potential well model, is much smaller then m ∗ v, z.As in Chapter 4, the transition energies include the correction by the finitepotential well model on the conduction subbands. The correction on thematrix elements is not taken into account, since it can be argued to be small.Figure 5.3 b) gives the polarization anisotropy ρ Tcv due to the matrixelements alone (dotted line), ρ bulk only from bulk mismatch (black line) andboth together (red line). Starting from the left, up till the appearance ofthe peak v 3 , ρ Tcv has a constant value of 0.6, as expected (compare Figure4.1 b)): for all k z both transitions are subject in the same way to theenvelope angular momentum and polarization selection rules described inthe previous chapter. The peak v 3 changes ρ Tcv drastically: in the caseof the E 32 ,n → C 0, 1 transitions the perpendicular component is polarizationforbidden. This is also the case for the v 7 peak, here the polarizationanisotropy of the matrix elements alone becomes even negative.Compared to the bulk value ρ bulk due to the dielectric mismatch, whichis almost a constant in the region of interest, it is concluded that ρ Tcv causesgiant changes in the total polarization anisotropy of the wire. This willbecome even more visible in paragraph 5.4.4 considering the polarizationcontrast.Ε'' wire4321v1v2v3zpolypol.v4v70.52 0.54 0.56 0.58 0.6 0.62EnergyeVΡ10.80.60.40.20Ρ bulkΡ T v c-0.2 Ρ wire-0.40.52 0.54 0.56 0.58 0.6 0.62EnergyeVFigure 5.4: a): Complex part of the dielectric function ε w at parallel(z) and perpendicular (y) polarization for InAs and R = 7.5 nm. b):Polarization anisotropy, calculated from a) alone (dotted), only frombulk mismatch (line) and from both together (red line).The statements about the R dependence of ε ′′ w and corresponding polarizationanisotropy (5.34) are summarized if one compares Figure 5.3 with thesame results at R = 7.5 nm, Figure 5.4. Without including the backgrounddielectric response ε bg , the dielectric function of the wire behaves formallyas ε ′′ w ∝ 1 R 2 while the polarization anisotropy ρ Tcv remains the same. Howe-103

Chapter 5.Dielectric function nanowirever, the different transitions come closer to each other for increasing R, withtheir mutual distances also behaving like 1 in the infinite-well approximation.Moreover, more and more higher transitions which are not taken intoR 2account here enter the energy region of interest. Apart from the remarksabout the tails corresponding to the different transitions one should thuskeep in mind that a correct picture of the higher energy part includes morepeaks, for instance coming from v i → c 2 transitions.5.4.3 Material dependenceFor InP the present effective mass approach can be compared with resultsobtained from an tight-binding approach [5].Ε'' wire0.80.60.40.2v1v2v3v4v5v6zpolypol.Ρ10.80.60.40.20-0.2Ρ bulkΡ T v cΡ wire1.54 1.56 1.58 1.6 1.62 1.64EnergyeV-0.41.54 1.56 1.58 1.6 1.62 1.64EnergyeVFigure 5.5: a): Complex part of the dielectric function ε w at parallel(z) and perpendicular (y) polarization for InP and R = 4.83 nm, fixing|T cv | 2 at k z R = 0.6. Only the first 7 transitions v i → c 1 are taken intoaccount. b): Polarization anisotropy, calculated from a) alone (dotted),only from bulk mismatch (line) and from both together (red line).Comparing Figure 5.5 with Figure 15 in Appendix D and noting thatthe present results are obtained with the parameters at T = 0 K only forthe first seven transitions v i → c 1 , we conclude that the effective massapproximation, without including the split off Γ 8 band and assuming aninfinite potential well in case of the valence subbands, successfully describesthe overall features of ε ′′ w . Remarkable is the difference in the lowest twotransition(s): in FIG 3 of [5] the band edge optical transition is fully z-polarized. This polarization selection cannot be explained from an effectivemass approach at all considering the E (+)1 → C (+)0, 1 transitions, but we note2that also the more accurate atomistic approach [4] contradicts this strictselection found in [5], see paragraph 4.5.1.With the effective mass approach it is relatively easy to change the materialparameters and wire radius. In Figure 5.6 the imaginary part ofthe dielectric function ε w is shown at R = 10 nm both for InAs and InP.Comparing the two materials it is concluded that a particular band-to-bandtransition has a significantly larger contribution in the dielectric function104

5.4. ResultsΕ'' wire1.751.51.2510.750.50.25v2v3a: InAsv4zpolypol.v70.48 0.49 0.5 0.51 0.52 0.53 0.54EnergyeVΕ'' wire1.751.51.2510.750.50.25v1v3b: InPv4zpolypol.1.46 1.47 1.48 1.49EnergyeVFigure 5.6: Complex part of the dielectric function ε w at parallel (z) andperpendicular (y) polarization for a): InAs and b): InP at R = 10 nm.The pF transitions are neglected by fixing |T cv | 2 at k z R = 0.of InAs. This is directly caused by the larger band gap of InP since thetransition probability ∝ 1ω 2 .5.4.4 Effect of the dielectric backgroundFinally, in this paragraph the effect of the background response ε bg is estimatedby looking at the polarization contrast. For this purpose, comparethe results shown in Figure 5.7 and Figure 5.8.C200160175a: InAs 140b: InP150120125100100C bulkC bulk75C80wire C wire605040250.52 0.54 0.56 0.58 0.6 0.621.48 1.5 1.52 1.54EnergyeVEnergyeVCFigure 5.7: Polarization contrast C bulk (black line), only due to thedielectric mismatch, and C wire (red line), including the polarization anisotropycaused by quantum confinement, for a:) InAs and b:) InP, atR = 7.5 nm. The dielectric background is not taken into account properly:ε ′′bgis set to zero.First of all, we recover the result that the effect of the quantum confinementis huge if ε bg is not taken into account: both for InP and InAs Figure5.7 shows a maximum enhancement by a factor 4 due to the quantumconfinement, as already predicted in paragraph 4.4.1. However, in generalthis overestimates the polarization anisotropy substantially.After including ε bg in a proper way, see Figure 5.8, the effect of thequantum confinement is still large for InAs (maximum enhancement ≃ 2),105

Chapter 5.Dielectric function nanowireC100C bulk a: InAs 48C bulk b: InP90C wire C wire804670446050424040300.52 0.54 0.56 0.58 0.6 0.621.48 1.5 1.52 1.54EnergyeVEnergyeVCFigure 5.8: Polarization contrast C wire compared to C bulk for a): InAsand b): InP, at R = 7.5 nm using the correct dielectric function of thewire: ε w (ω) = ε bg (ω) + ∑ cv εw cv(ω). It is assumed that ε bg ≃ ε bulk .but considerably reduced for InP (maximum enhancement ≃ 1.2). Theseresults are far more conform reality and reflect the difference between InPand InAs: as shown in the previous section the confinement correction toε w is much smaller in the case of InP.Furthermore, for both materials the effect of the quantum confinementdisappears if R is increased sufficiently. This is only achieved if ε wire isconsidered in the right way: ε w (ω) = ε bg (ω) + ∑ cv εw cv(ω). To be moreprecise, taking all transitions c, v corresponding to one particular bulk transitioninto account, ∑ cv εw cv(ω) ≃ ∑ cv εbulk cv (ω) for sufficiently large R andconsequently ε w (ω) ≃ ε bulk (ω). So for R → ∞ the quantum confinementcorrection ∑ cv εw cv(ω) equals the contribution of the same group of transitionsprojected in the bulk system.106

Summary and ConclusionsIn this paper we analyzed the optical absorption of III-V semiconductor cylindricalnanowires with the aim to get a theory which describes the opticalproperties for arbitrary wire thickness and for a wide range of semiconductormaterials.In Part I we started with a classical theory describing the scattering oflight by an infinite cylindrical structure. At arbitrary angle of incidence, inChapter 1 general expressions were found for the cross sections and correspondingefficiency factors, which are measurable quantities in the region farfrom the cylindrical wire.Subsequently, in Chapter 2 we focussed on the case of cylindrical wiressmall compared to the wavelength of the incident light and derived analyticsolutions explicitly as a function of the material properties (dielectricconstant, wire radius R), geometric configuration (angle of incidence) andwavenumber (k 0 ) of the incident light. Next to reproducing the well knownresults in the dipole limit, we extended the theory by Mie expansion of theEM field inside cylinder up to second order in the dimensionless parameterk 0 R. We concluded that for increasing cylinder radius, besides the wavebehavior of the EM field inside the wire, the effect of optical focussing getsa more important role.Furthermore, numerical results of the efficiencies and corresponding polarizationanisotropy are given for InP in the region between 350 and 600 nmand wire radii up till 5 nm. Hereby the nanowire is treated classically bytaking the bulk value of the dielectric function.In Part II we have included the effects of quantum confinement by meansof a corrected description of the dielectric function of cylindrical nanowires.For this purpose, in Chapter 3 we first derived the electronic structureusing effective mass theory. This method utilizes the already well knownbulk energy gaps and optical matrix elements at the band extreme. The resultingbulk dispersion obtained from the crystal Hamiltonian H 0 is treatedas a kinetic energy term, which after including the cylindrical confinementpotential and assuming the wire radius R large compared to interatomicdistances, results in a one-particle Schrodinger equation acting on the enve-107

lope of the nanowire wavefunction. Neglecting the small anisotropic termsin H 0 by applying the spherical approximation, the total Hamiltonian of thenanowire becomes diagonal with respect to the total angular momentumalong the z axis F z = J z + L z , with L z and J z the z-projection of the envelopeangular momentum and the total angular momentum of the atomicstates, respectively. Consequently the eigenvalue f z of F z is a good quantumnumber.In case of the Γ point valence band in III-V semiconductors we neglectedthe split-off band and diagonalized the 4 × 4 Hamiltonian H Γ 8F zof the heavyand light hole band in the basis j z v = { 3 2 , 1 2 , − 1 2 , − 3 2 }. At k z = 0, the wirezone center, H Γ 8F zdecouples into two 2 × 2 blocks with solutions which arecharacterized by parity: the envelopes are even or odd under ρ → −ρ.Assuming an infinite confinement potential, the complete set of solutionsfor the Γ 8 valence band in a cylindrical nanowire is thus characterized with|f z |, the parity (±) and n, denoting the n th solution at this parity. Awayfrom the zone center, the valence subband wavefunctions depend in a nontrivial way on k z by the the lateral part of the envelope functionFor the Γ 6 conduction band the Hamiltonian is already diagonal in L zand consequently the irreducible representation of the conduction subbandsis given by the eigenvalue l z of L z and again (±), n. We have taken thefiniteness of the potential well into account by including a reduction factorto the conduction subband energies calculated in an infinite confinementmodel, which results in a large correction since the bulk conduction bandhas a relative small effective mass. Contrary to the energies, no correctionis made to the subband wavefunctions since in this case the difference withthe finite potential well model is expected to be negligible.In Chapter 4 we analyzed the radiation-matter interaction − em 0 c A · pbetween the external electromagnetic (EM) field and the electrons withinthe semiconductor system using a macroscopic, semiclassical approach: wetreated the EM field classically, while the semiconductor nanostructure isdescribed in the spirit of Chapter 3. Similar to the separation of the nanowirewavefunction into an atomic part and ”macroscopic” envelope functions, theEM transition matrix factors into separate integrals: a bulk, atomic likematrix element of the momentum operator and the integral of the EM fieldbetween the envelopes. This separation relies on the natural assumptionthat the EM field varies slowly compared to atomic distances.Both the envelope and the atomic part of the transition matrix are subjectto selection rules. The last ones are the semiconductor variant of thewell known selection rules on p in atomic systems. In case of the p-likeΓ 8 valence states and the s-like Γ 6 conduction states we concluded for the|j z | = 3 2valence states that the atomic-like matrix elements correspondingto components of p parallel to the wire axis are polarization forbidden. Con-108

sidering the |j z | = 1 2states we found a ratio of 4 between the parallel andperpendicular p components, respectively. Physically, these polarization selectionrules are based on the conservation of angular momentum on theatomic part of the transition matrix.While the polarization selection rules originate from the bulk, atomiclike matrix element of the momentum operator, the selection rules on theenvelope part depend on the configuration of the system (wire dimensions,wavelength of incident EM field). Starting with the envelope integral overthe coordinate z parallel to the wire axis, in Part I we found for infinitecylinders that the EM field always is of the form E(r ⊥ )e iqzz and togetherwith the ∝ e ikzz dependence of the envelope functions this results in theconservation of momentum equation along the z-direction.Considering the lateral envelope part of the transition matrix, a spatiallyvarying EM field requires a different approach than the dipole limit.In the common dipole approximation, the valence and conduction subbandwavefunctions (∝ e ilzφ ) must have the same envelope angular momentum.In addition, this envelope angular momentum conservation naturallyleads to parity selection at the wire zone center: transitions between stateswith different envelope parity are not allowed. Away from the zone centerthis is the only selection rule which is broken since the valence subbandwavefunctions are not characterized by parity any more. We calculated theband-to-band transitions between the first 7 valence subbands and the lowestΓ 6 conduction subband for InP and InAs at different wire radii R. Firstof all, while the transition energies strongly depend on the wire radius, thematrix elements are independent of R, approximatively even if the finiteconfinement for the conduction subband wavefunctions would be taken intoaccount. Secondly, comparing the results of InAs with InP it can be concludedthat next to the energy of a particular transition, also its strengthis material dependent. Finally we conclude that the polarization anisotropyof the transition matrix is completely determined by the polarization selectionrules. At all k z , the transitions from |f z | = 3 2valence subbands to thelowest l z = 0 conduction subband are strictly forbidden for parallel polarizationand for |f z | = 1 2 the transitions to the l z = 0 subband have a fixedpolarization anisotropy of 0.6. These observations compare favorably withthe results derived in an atomistic, empirical pseudo-potential plane-wavemethod[4]. The deviations can be explained by the possible corrections ofincluding the split-off band, or even diagonalizing the full 8× 8 Hamiltonianof the three bulk Γ 8 valence bands and Γ 6 conduction band in the presenteffective mass approximation.One of the purposes of this paper was to estimate the effect of classicalscattering for wire dimensions in the quantum confinement regime. Generally,a spatially varying EM field breaks the symmetry in the envelope partof the transition matrix. We analyzed this by including the Mie scatteringterms found in Part I into the envelope integral. Although the qualitative109

features indeed are different, since new peaks arize which are parity forbiddenin the wire dipole limit, we conclude that the order of magnitude ofthese kind of corrections is too small to overcome the parity selection ruleat k z = 0 significantly. The transitions that become weakly allowed areweaker by a factor ∼ 10 −5 for InP and InAs and wire radii up till 15 nm.A second effect is that the strength of transitions which are already allowedin the wire dipole limit changes, but also this effect is not large. For a InPnanowire with a radius of 10 nm the Mie correction on these already allowedstates is about 5 %. Consequently, the effect of classical Mie scattering canbe neglected in the quantum regime.Finally, in Chapter 5 we analyzed the full optical absorption process incylindrical nanowires by deriving an expression for the nanowire dielectricfunction including the quantum confinement effects. Hereby we made a firstorder approximation by assuming that the local field acting on a particularelectron at a lattice site is the same as the averaged field obtained from themacroscopic Maxwell equations. Thus, we neglected the additional internalfield due to the induced polarization of the neighboring atoms. It is notedthat the inclusion of such a Lorentz correction has to be reconsidered withina microscopic, semiclassical approach, when dealing with nanoscale systemswhich are not ”microscopically large” any more: the resonant states areextended over the whole volume by the envelope part of the wavefunctionsand consequently the induced polarization is position dependent.As a first step, however, the dielectric function is approximated by aconstant and, based on the results in Chapter 4, the EM field can be consideredin the dipole limit. In this framework we used the advantages of theeffective mass approach by deriving a simplified expression for the imaginarypart of the nanowire dielectric function, in which the integral over k z isreplaced by an integral over the energy utilizing an explicit expression forthe 1D density of states.In this paper we focussed on the band-to-band transitions close to theband gap of III-V materials. In order to include also the dielectric responseof all other transitions, a background dielectric function ε bg is definedwhich is basically the bulk dielectric function ε bulk , in which the group ofband-to-band transitions is projected out. As a first approximation we tookε bg ≃ ε bulk and neglected the quantum confinement contribution of the groupband-to-band transitions to the real part of the total dielectric function.By making some rough simplifications, for instance neglecting the k zdependence of the transition matrix and inserting line broadening of theabsorption peaks by hand, we obtained successfully the qualitative behaviorof the nanowire dielectric function. In agreement with recent literature[4][5], we find that the dielectric response of the band-to-band transitionsis strongly polarization dependent, which completely relies on the resultsof Chapter 4. Furthermore, comparing InP with InAs it is concluded that110

the confinement has a significantly smaller effect on the dielectric functionof InP, which is explained by the larger bulk band gap of InP resulting insmaller band-to-band transition probabilities.Compared to the bulk polarization anisotropy due to the classical dielectricmismatch, which is almost a constant in the region of interest, itis concluded that the polarization anisotropy due to quantum confinementcauses giant changes in the total polarization anisotropy of the wire. Inaddition, by including the dielectric background the effect of quantum confinementdisappears in a natural way if R is increased sufficiently.After all, we have thus established that the effective mass approach providesa fast and flexible tool to analyze the diameter dependent propertiesof nanowires for a wide range of semiconductor materials. Possible improvementsof the current framework are achieved if the Γ 8 spin-off band isincluded, or eventually diagonalizing the full 8 × 8 Hamiltonian of the threebulk valence bands and Γ 6 conduction band in the present effective massapproximation.111

A Hole wavefunctions fordifferent k z112

Χ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0. , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵0.25 , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵0.5 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1. , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRΧ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0.125 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵0.375 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵0.75 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.125 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRFigure 9: Radial part of the |f z | = 1 2, − (1) hole envelope wavefunctionsfor InAs. The value of k z R changes from 0 in the first picture to themaximum value 1.125 (at the end of the band) in the last graph.113

Χ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0. , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵0.333333 , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵1. , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.66667 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRΧ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0.166667 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵0.666667 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.33333 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵2.03333 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRFigure 10: Radial part of the |f z | = 1 2, + (1) hole envelope wavefunctionsfor InP. The value of k z R changes from 0 in the first picture to themaximum value 2.033 (at the end of the band) in the last graph.114

Χ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0. , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵0.333333 , 1j z ⩵ 32j z ⩵ 12f z ⩵ 12 , k z R⩵1. , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.66667 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRj z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRΧ j zΧ j zΧ j zΧ j z1.510.50-0.5-11.510.50-0.5-11.510.50-0.5-11.510.50-0.5-1f z ⩵ 12 , k z R⩵0.166667 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵0.666667 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵1.33333 , 1j z ⩵ 32j z ⩵ 120 0.2 0.4 0.6 0.8 1ΡRf z ⩵ 12 , k z R⩵2.76667 , 1j z ⩵ 32j z ⩵ 12j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵32j z ⩵12j z ⩵320 0.2 0.4 0.6 0.8 1ΡRFigure 11: Radial part of the |f z | = 1 2, − (1) hole envelope wavefunctionsfor InP. The value of k z R changes from 0 in the first picture to themaximum value 2.767 (at the end of the band) in the last graph.115

B Polarization selection rules116

1m 0〈S ↑ | ˆε · p | 3 232 〉 ˆε x ˆε y ˆε zpropagation ‖ to ê x impossibleΠ √2 polFpropagation ‖ to ê y −i Π √2impossible polFpropagation ‖ to ê z −i Π √2Π √2 impossibleTable 1: Selection rules on the atomic-like interband matrix elements1m 0〈S ↑ | ε · p | 3 32 2〉. The propagation direction of the EM-wave isdenoted in the left column. The unit directions of E are denoted with ε x ,ε y and ε z ; Π is related to the Kane matrix element E p by E p = 2m 0 Π 2 .The polarization forbidden transitions are denoted with polF .1m 0〈S ↑ | ˆε · p | 3 212 〉 ˆε x ˆε y ˆε zpropagation ‖ to ê x impossible polF i 2Π √6propagation ‖ to ê y polF impossible i 2Π √6propagation ‖ to ê z polF polF impossibleTable 2: Selection rules on the atomic-like interband matrix elements1m 0〈S ↑ | ˆε · p | 3 12 2 〉.1m 0〈S ↑ | ˆε · p | 3 2 − 1 2 〉 ˆε x ˆε y ˆε zpropagation ‖ to ê x impossibleΠ √6 polFpropagation ‖ to ê y i Π √6impossible polFpropagation ‖ to ê z i Π √6Π √6 impossibleTable 3: Selection rules on the atomic-like interband matrix elements1m 0〈S ↑ | ˆε · p | 3 2 − 1 2 〉. 117

1m 0〈S ↑ | ˆε · p | 3 2 − 3 2 〉 ˆε x ˆε y ˆε zpropagation ‖ to ê x impossible polF polFpropagation ‖ to ê y polF impossible polFpropagation ‖ to ê z polF polF impossibleTable 4: Selection rules on the atomic-like interband matrix elements1m 0〈S ↑ | ˆε · p | 3 2− 3 2〉.The propagation direction of the EM-wave isdenoted in the left column. The unit directions of E are denoted with ε x ,ε y and ε z ; Π is related to the Kane matrix element E p by E p = 2m 0 Π 2 .The polarization forbidden transitions are denoted with polF .118

C Interband matrix elements119

E Trans eV, R⩵4.83 nmΡ T v c T v c 2 arb. units0.140.120.100.080.060.040.021.000.800.600.400.201.53 1.55 1.58 1.6 1.62 1.65v1 c1v2 c1v3 c1v4 c1v5 c1v6 c1a : k z R⩵0.zpol.ypol.Ρ0Ρ0v7 c101.53 1.55 1.58 1.6 1.62 1.651.55 1.58 1.6 1.62 1.65b : k z R⩵0.93v1 c1v3 c1v2 c1v4 c1v5 c1v6 c11.55 1.58 1.6 1.62 1.65v7 c10.140.120.100.080.060.040.021.000.800.600.400.200E Trans eV, R⩵4.83 nmFigure 12: Matrix elements |T cv,‖ | 2 and |T cv,⊥ | 2 and corresponding polarizationanisotropy ρ Tcv of the first 7 transitions v i → c 1 for InP, calculatedincluding the scattering terms in the EM field at R = 4.85 nm.The corresponding energy scale is given at the top and bottom of thefigure.120

E Trans eV, R⩵10. nmΡ T v c T v c 2 arb. units0.140.120.100.080.060.040.021.000.800.600.400.201.45 1.46 1.46 1.47 1.47 1.48v1 c1v2 c1v3 c1v4 c1v5 c1v6 c1a : k z R⩵0.zpol.ypol.Ρ0Ρ0v7 c101.45 1.46 1.46 1.47 1.47 1.481.46 1.46 1.47 1.47 1.48b : k z R⩵0.93v1 c1v3 c1v2 c1v4 c1v5 c1v6 c11.46 1.46 1.47 1.47 1.48v7 c10.140.120.100.080.060.040.021.000.800.600.400.200E Trans eV, R⩵10. nmFigure 13: Matrix elements |T cv,‖ | 2 and |T cv,⊥ | 2 and corresponding polarizationanisotropy ρ Tcv of the first 7 transitions v i → c 1 for InP,calculated including the scattering terms in the EM field at R = 10 nm.121

D Reference articles122

Figure 14: Califano and Zunger [4], pg. 6. In FIG. 3 the interbandmatrix elements for a R = 4.8 nm InAs wire are shown based on anatomistic, empirical pseudopotential plane-wave method. The C ∞ v representationsgiven in Table II differ from those derived in the presentpaper.123

124Figure 15: Persson and Xu [5], pg. 3. In FIG. 3 ε ′′ w is shown for aR = 2.9 nm InAs wire based on an atomistic tight-binding approach.

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