Overview of spectroscopies I (MC) - TDDFT.org

Manuel Cardona, Max-Planck- Institut für FestkörperforschungStuttgart, GermanyIn honor of Max Planck‘s 150 th birthday (b. April 23, 1858, † Oct. 1947)Lectures 1 and 21979Dr. Rer.Nat.Dielectric function of semiconductorsDependence of electronic structure 1901onEinstein lookTemperature, including zero-point vibrations1930(isotope effects)193019301910 Einstein look1979Dr. Rer. NatBernasque, Sept.2008~1946, withwife Marga1

GermaniumPseudopotential band structure ofgerrmanium including spin-orbitinteraction (0.2 eV for EDielectric function of germanium1, E 1+ Δ 1)obtained from reflectivity spectrum Fundamentals of Semiconductors,by Kramers-Kronig analysis, at 300K Yu+CardonaPhilipp and Taft Phys Rev 113 1001 (1959)E 1transitions should be spin-orbit split butNot enough resolutionExperimental discoveryof spin-orbit splitting ofE 1gapTauc and AntončikPRL 5 253 1960Spin-orbit splittings have played a very important role in the interpretation of themeasured optical spectra in the visible and eV. Most state of the art codesto perform such calculations do not include it. They also assume that the atomsare at fixed lattice positions and neglect thermal vibrations, including zero-pointqm vibrational effects.2

Benedict, Shirley, Bohn, PRB 57, R9385 (1998): Electron-hole quasiparticals plusdirect (screened) and exchange (unscreened) exciton interactionSolid line:Measured,Unfortunately 300KCalculation lower thanExper. because noexciton interactionE 1 E 2Calculations:No temperature,No zero-point vibrationsNo spin-orbit couplingE 1E 2E 1,E 1+Δ 1E 23

Shift 0.2eVIn part 300K E 1siliconDiamond, ellipsometry 300KEllipsometryAspnes+Studna300KExp. at 300 KHahn, Seino, Schmidt, FurtmüllerBechstedt, p.stat. sol.(b) 242 2720 (2005)R.Ramírez et al.,Phys Rev B77,045210(2008); Path Integral Molecular Dynamicscalculation1.5 eVThe lowest direct gap of diamondIs lowered by 0.7 eV by e-p interact.(right). The shift in the lower flank ofIm ε (ω) may be due, in part to e-pZero point0.7 eVDiamond4

Albrecht et al. PRL 80.4510(1998)Cardona, Lastras et al. PRL 83.3970(1999)Albrecht et al.PRL80,4150(1998)Cardona et al.PRL83,3970(1999)State of the art calculation, no phononsZero point?90meVNo excitonsEllipsometry at 300KEllipsometry at 20K, Lautenschlageret al. PRB 36.4821(1987)5

The electronic properties of crystals are affected by the lattice vibrations throughelectron-phonon interaction. These effects depend strongly on the temperature Tbut do not disappear even at T=0 because of the zero-point vibrations. The T=0effects are quantum effects which decrease with increasing isotopic massBelow is shown as an example the temperature dependence of the spectra of theimaginary part ot the dielectric function of Gallium Arsenide for several T ‘s,Measured by ellipsometry.P.Lautenschlager et al., PRB 35, 9174 (1987)Photon Energy6

Comparison of calculated with measured imaginary part of dielectric functionof Gallium Arsenide. Calculations performed by averaging 2 for various staticphonon displacements „frozen phonons“ whose amplitude depends ontemperature. Lots of computation. No SO splitting, zero-point „by hand“See Ibrahim, Shkrebtii et al, PRB 77, 125218 (2008), theory and experiment.Probably first such calculation! Phonon displacements determined bymolecular dynamics. Use supercells.It is possible to calculate the shift with temperature and zero-point vibs. of specificcritical point energies. This will be discussed next7

Recent work by A. Marini, PRL in press (2008)ab initio calculation of imaginary part of dielectricfunction of silicon including electron-phononinteraction and effects of this interaction onexciton effects. Blue curves are experimental dataof Lautenschlager et al., PRB 36, 4821 (1897)8

Three electron-phonon interaction terms must be consideredto lowest order in T :1. Thermal expansion: can be included as a change of the lattice constants with Tor with isotopic mass at T=0 : Quasistatic approximation (somewhat trivial) dEE ln VdlnV ln V q [2 ( ) 1], jq , jnB q , jBV2. First and second order electron phonon interaction: renormalization ofelectron propagators:Fan termsAntončik termsu u u 2 uq,j {2[exp( / kT) 1] 1}4M2 1 fM;4MM4 f1212 Complications at low T discussed laterZero pointAt high T, kT , independent of M (classical effect)At low T, M -½ , zero point vibrations (quantum)9

Information on electron-phonon effects can be obtained eitherfrom the temperature dependence or from isotope effects at T=0Natural Abundance of some Stable IsotopesCarbon12 C 99%; 13 C 1%; 14 C unstable (dating) ; Nitrogen 14 N 99.6% ; 15 N 0.4%Silicon28 Si 92%; 29 Si 5%; 30 Si 3%Germanium 70 Ge 22%; 72 Ge 28%; 73 Ge 8%, has nuclear magnetic moment I = 9/2; 74 Ge 35%; 76 Ge6%Tin (gray)112 Sn 1%; 114 Sn 0.6%; 115 Sn 0.4%;116 Sn 14.3%; 117 Sn 7.6%; 118 Sn 24.2%119 Sn 24.2%; 120 Sn 32.6%; 122 Sn 4.7%; 124 Sn 5.8%CopperChlorineZincSulfur63 Cu 69.2%; 65 Cu 30.8% nuclear magnetic moment35 Cl 75.8%; 37 Cl 24.2% “ “ “64 Zn 48.6%; 66 Zn 27.9%; 67 Zn 4.1%; 68 Zn 18.8%32 S 95.2%; 33 S 0.75%; 34 S 4.21%; 36 S 0.02%10

A single crystal of 29 Si (5% natural isotopic abundance) grown byVITCON (Germany, Dr.Pohl) from 29 Si separated in Russia byEFFECTRON (Moscow, Zelenogorsk)Because of its nuclear spin, 29 Si has been suggested for the realizationof quantum bits [T.D. Ladd et al., PRL 89, 017901 (2002)]1 cm11

CdS Phonon dispersion relations:In. Neutron ScatteringA.Debernardi et al. Solid State Comm. 103, 297(1997)Natural Cd has 113 Cd,the strongest neutronabsorber. Hence, till thiswork no dispersion relationsof CdS, the canonicalwurtzite-type materialavailable. These INS measurementswere performedwith 114 CdS which does not absorb neutrons.13

2.9meVParks et al. PRB49,14244(1994)Renormalization of the direct gap of Geby electron-phonon interaction0.95Linear extrapolationE 0= 60meVThe renormalization E 0can be determined either by thelinear extrapolation of E 0(T) or from the dependence ofthe gap on isotopic mass which is proportional to M -1/2The extrapolation gives 60meVThe isotope shift 2.9x2x(73/6)=70meVThe most recent semi-ab-initio calculation[Olguin et al.SSC 122 575 (2002)]gives 64meVCan be fitted withsingle oscillator:Einstein modelThis renormalization should be taken into account whenstate of the art ab initio calculations of the gap are comparedwith experimental results.ΔE(T)= ΔE o [ 1+2(1-exp(ω E /kT) -1 ]14

Dependence of the indirect gap of diamond on temperature and isotopic massFit of E ind(T) with a single Bose-EinsteinLinearextrapolationterm of average frequency1080 cm -1 . The Ramanfrequency of diamond is1300 cm -1 . = 370meVΔ(M) = AM -1/2 = 370 meVΔ(M 13 )- Δ(M 12 )=-(1/2)AM -3/2 =-(1/2)• Δ(M)•M= 14.8 meVIsotope shift=15meV12CThe T=0 renormalization calculated from the isotope shift usingthe M -1/2 law is = 15meVx2x(12/1)=360meV,VERY LARGE! (= 60meV for Ge, 65 meV for Si )13CElectron-phonon interaction calculations byZollner et al.[ PRB 45, 3376 (1992)] usingsemiempirical pseudopotentials givean even larger value: = 600meV!!! STRONGELECTRON-PHONON INTERACTION:SUPERCONDUCTIVITY?A.T.Collins et al., PRL 65,891(1990)Computational band structure theorist beware!!Tiago, Ismail-Beigi, Louie PRB 69 125212 (2004) 15Si, Ge and GaAs, GW approximation, no e-p interact.

7.37.27.4ΔE = 0.7 eVClassical molecular dynamicsLowest direct gapof diamond (E ‘ o) ,Γ 25‘ Γ 157.1Measured,Logothetidis et al.PRB 46, 4483 (1992)1000Ramírez et al.PRB 73,2452022006Squares ,Quantum (path integral)MD calculations.For diamond the volume effectis negligible:0.03eV at T=0compared with 0.7 eV total16

Because of strong electron-phonon interaction, superconductivitypossibleSuperconductivity in boron-doped (~2x10 21 cm -3 )diamond films produced by CVDMore recent T c =11.4 KY. Takano et al.Diamond and rel.mat.16, 911 (2007)Y.Takano et al., APL 85, 2851 (2004);Ekimov E.A. et al. Nature 424, 542(2004)17

Although no isotope effect measurements available,all seems to indicate that we are dealing with BCS-type superconductivity. Scanning tunneling microscopyhas yielded a gap Δ=0.87 meV for T=0.47K,Hence 2Δ /kT c = 3.7 (BCS).The points are experimentalFor B-doped diamond, the solidlines are guides to the eye, Thedashed line a fit with McMillan‘sequation.From M.Cardona, Sci. Tech. of advancedmats. 7, S60 (2006)Exp. Points: J. Kačmarčik et al.,Phys. Stat. Sol.(a) 202,2160 (2005)H. Mukuda et al., PRB 75, 033301 (2007)18

Estimate of the critical temperature T cAlthough no measurements of isotope effect on T c available yet,because of strong electron-phonon interaction we conjecture that thesuperconductivity mechanism is BCS-like, Cooper pairs glued byphonons.●McMillan equation:●1 12 2TcTDexp{ [ (1 ) *] ; =NdD / MDWhere is the hole-phonon coupling constant and * represents thehole-hole Coulomb repulsion. N d is the density of hole at the FermiLevel, T d and d the “Debye” temperature and frequency, resp. and Dan average hole-phonon deformation potential. Taking standard valuesof these parameters from the literature and assuming * 0we obtained (M.Cardona,Solid State Com. 133,(2005),3) T c 5.2 K.For hole concentration N h = 2.2x10 21 cm -3 . .19

Fermi Surface of diamond and electron-phonon processes relevant forDSUPERCONDUCTIVITY●2e-pDe-pHeavyholesN DMd ; 1/2 =(0.55d 0 /a 0 )eV /Ǻ0Diamond : d 0= 85eV; Silicon d 0= 35eVSpin split holesLight holes85 2 370meV( ) 5.9; ( ) 6.135 60meVGap RenormalizationsWith these values of d 0one obtains T c=5.2K for diamond with 2x10 21 holes/cm 31.2% Boron; Basically T c≈ 0 for silicon (for μ* = 0) Cardona SSC 133 (2005) 3.Boeri et al. PRL 93,237002(2004): T c= 0.2K (for μ* = 0.1) for 3% BLee and Pickett PRL 92, 237003(2004): T c= 7K (for μ* = 0.15) for 2.5% BXiang et al., cond:mat/0406446 v3 T c= 4.4 K (for μ* = 0.15) for 2.8% B20

Recent calculations:Giustino, Yates, Souza,Cohen, Louie U.C. BerkeleyPRL 98, 047005 (2007)Use millions of points in k-space and either C-B virtual crystalor a supercell with boron in the center. Include boron vibrationsFind for N h ~ 2x10 21 cm -3 T c = 4K21

Why is the electron-phonon interaction so muchstronger in diamond than in germanium or silicon?The large e-p interaction with the top of the valence bandseems to be a property of the elements of the second row ofthe periodic table ( carbon, nitrogen, oxygen…). We shall seenext that that the e-p interaction, as obtained from thedependence of the gap on T and on M, is also large in GaN ain ZnO and, as will be seen later, in SiC.Handwaving argument:C, N, O.. have no p-electrons in the core and the p valence electrons, as the atomsvibrate, can get much closer to the core than in cases where p-electronsare present in the core: germanium, silicon, GaAs….22

Another possible heuristic explanation is that for materialswith atoms of the second row the gaps are large. In pert.theory intraband and interband terms contribute to theself-energy. Usually (Ge, Si) they compensate each other.For large gaps intraband terms increase, interband onesdecrease and the sum can become larger.IntrabandInterbandThe two heuristic explanations may beequivalent in some way !interband23

2.9meVParks et al. PRB49,14244(1994)Renormalization of the direct gap of Geby electron-phonon interaction0.95Linear extrapolationE 0= 60meVThe renormalization E 0can be determined either by thelinear extrapolation of E 0(T) or from the dependence ofthe gap on isotopic mass which is proportional to M -1/2The extrapolation gives 60meVThe isotope shift 2.9x2x(73/6)=70meVThe most recent semi-ab-initio calculation[Olguin et al.SSC 122 575 (2002)]gives 64meVCan be fitted withsingle oscillator:Einstein modelThis renormalization should be taken into account whenstate of the art ab initio calculations of the gap are comparedwith experimental results.ΔE(T)= ΔE o [ 1+2(1-exp(ω E /kT) -1 ]24

Temp. Dep. of gap: SiC, path integral molecular dynamicsTotalRenorm.Si contrib.Carbon contributionHernández, Herrero, Ramírez, Cardona, PRB 200825

Superconductivity in boron-doped siliconE. Bustarret et al., Nature 444,465 (2006)1. (5x10 21 holes/cm 3 )2. (2.8x10 21 „)26

Recent paper: B-doped SiC, Zhi-an Ren, J.Akimitsu et al., J.Phys Soc.Japan 76, 103710 (2007)27

Specific heat of diamond vs.T: Einstein, Nernst-Lindemann, DebyeNernst-LindemannZ.Elektrochemie 817 (1911) experimental dataDebye, T D =1830 KAnn. Phys. 39,789 (1912)Einstein, T E =1450 Kaccording to DebyeEinstein, T E =1300 KAnn.Phys. 22, 180 (1907)So far we used fits with 1 Einstein oscillator, Sometimes two Einstein oscillators betterFrequency,ThzNernst-LindemannReduced wavevector28

Walter NernstF.A. Lindemann,Viscount Cherwell29

Luis Viňa (Stuttgart, 1981)Patrici Molinàs (Stuttgart, 1990)Javier Tejeda (Stuttgart,1976);Gerardo Contreras (Stuttgart,1986)Jorge Serrano (Stuttgart, 2003)Manuel Cardona (Harvard, 1959)William Paul (Aberdeen, 1951)R.V. Jones (Oxford, 1934)F.A. Lindemann (Berlin, 1911)W.H. Nernst (Wurzburg, 1887)Friedrich Kohlrausch (Gottingen, 1863)Wilhelm Weber (Halle, 1826)Johann Schweigger (Erlangen, 1800)Georg Hildebrandt (MD, Gottingen, 1783)Johann Gmelin (MD, Tubingen, 1769)Philipp Gmelin (MD, Tubingen, 1742)Burchard Mauchart (Lic. Med., Tubingen, 1722)Elias Camerarius Jr., (MD, Tubingen, 1691)Elias Camerarius Sr., (MD, Tubingen, 1663)Those Germans sure know how to keep records!30

Temperature and isotope effects in binary compounds: GaNThe zero-point vibrations of different atoms contribute differently to T=0 renormalizations:Different atoms show different isotope effects, especially if their M’s are very different:GaNSo far we have used single oscillator B-E fits (Einstein model of C v ). For binarieswith very different M’s two oscillators are needed (Nernst-Lindemann C v ).The Debye model may be necessary at very low temperatures, as it is for C vdEET ( ) E0 2 MGa [2 nB ( Ga/ kT) 1]dMGadE2 MN[2 nB( N/ kT) 1]dMNThe temperature shift gives directly the sumof anion and cation contributions. The isotopeeffects are more informative since they givethe separate contributionsManjon et al.Europ.phys.j. B40,4553(2004)170meVGaNdE/dM N =4.2meVdE/dM Ga =0.4meVFor diamonddE/dM C =7meV/atom!,why C and Nso large? No p-electrons in core? N =790cm -1 Ga =350cm -131

Contribution of zinc and oxygen to the zero-point renormalization of the ZnO gapExcitonic band gap vs. M zn+ M o(M Zn+M o) (amu)Oxygen 135 meVZinc 29meVFor GaN it is:Gallium 56 meV, nitrogen 203 meVFor diamond it is 360meV= 180 +180 meVRB Capaz… and SG Louie PRB 94, 036801 (2005) use twooscillators to fit T-dependence of gaps of nanotubes32

Anion and cation contributions to the zero-point renormalization andthe temperature dependence of gaps can be very different. For CuCl,CuBr, and CuI they even have opposite signs! Unusual effects appear.A.Göbel, PRB 57, 15183 (1998)65CuCl65CuCl63CuCldE 0/dM Cu= -0.076meV/amuBranchCu-likeCu 35 ClCu 37 CldE 0/dM Cl=+0.036meV/amubranchCl-likeThe isotope data have been crucial for understanding 33the strange behavior of E 0(T) in copper halides. CuGaS 2 ?

Periodic Table ( Section)Mg Al Si P S ClCu Zn Ga Ge As Se Br (3d-core electrons)Ag Cd In Sn Sb Te I (4d-core electrons)Core d-electrons-10 eV-20 eV-30 eV-40 eV-50 eVCu and Ag d-core electrons hybridize withvalence p-electrons34

Low temperature behavior of E 0 (T): Debye ModelThe average energy of the phonons is:.Since < u 2 > (M ) -1/2 we find for the gap renormalization at T0:0Cep1[ n( / kT) ] Nd( )d2This equation differs from that above by a factor of -2 in the integrand. However:C e-p must vanish for T 0 ( 0 phonons are a uniform translation) and C e-pmust be proportional to 2 . Hence in the low T limit behaves like T 435

Sharp indirect edge luminescence in isotopically pure SiThe T 4 dependence of the gap of a semiconductor at low T( T

Very high resolution(< 0.2 eV) excitation spectroscopy measurementswere performed last year by M.Thewalt et al. on pure 28 Si. Theyconfirmed, for the first time, the T 4 dependence of the gap shift.(F in GHz)T 40.15GHz=0.6eV0.08GHz=0.3eV( T in Kelvin)Cardona,Meier,Thewalt,PRL 92,196403(2004)3729

DIAMONDT. Sato et al. PRB 65, 092102 (2002)300 KT 4 (10 9 K 4 )For diamond the average γ isNearly temperature independentV n A Tln q, jq, j[2 B( q,j) 1] 0.017BV q,j4If constant move out of the sumthe sum is then proportional to T 4 plus zero point effect38

Ultrahigh resolution of photoluminescence excitation spectrum of excitons bound to neutralphosphorus in isotopically pure Si (99.991% 28 Si). Width of exciting laser: 0.3neV. Observedwidths: 150neV(inhomogeneous still, should be ~ 5 neV).These lines are split by half thehyperfine splitting = 486/2 = 243neVExciton boundto neutralPNeutral donorM.L.W. Thewalt et al. JAP 101 081724 (2007)39

90 meV, confirmed by isotope measurements, 28 Si vs. 30 SiP. Lautenschlager et al., P.R B31, 2163 (1985)L.F. Lastras Martínez et al. PRB 6112946 (2000)E 1, E 1+ Δ 1(eV)GermaniumRönnow, Lastras et al. Eur. Phys. J.5, 29 (1998)Germanium40

CONCLUSIONS1. State of the art calculations of dielectric function ofsemiconductors is in good shape but does not includeelectron-phonon interaction (Temperature dependence)nor spin-orbit coupling2. Measurements of dependence on temperature andisotopic mass are beginning to be compared withstate of the art ab initio calculations3. Particularly interesting are the isotope effects in binaryor multi-nary materials ( ZnO, CuGaSe 2 )4. Dependence on mass at T~ 0 is strong for atoms of thesecond row of the Periodic Table ( C, N, O ). It seems relatedto the appearance of supercunductivity in B-doped diamond5. Superconductivity has been seen in heavy B-doped Si, SiCIt should be observable in heavy p-type GaN, ZnO…p-type InN:R.E.Jones,E.E. Haller et al., PRL 96,12505 (2006) ?6. Ultra- sharp edge emission spectra for isotopically pure materials:hyperfine splittings41