Negative differential conductance in two-dimensional electron grids

APPLIED PHYSICS LETTERS 92, 052104 2008Negative differential conductance in two-dimensional electron gridsW. Pan, 1,a S. K. Lyo, 1 J. L. Reno, 1 J. A. Simmons, 1 D. Li, 2 and S. R. J. Brueck 21 Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87185, USA2 Center for High Technology Materials, University of New Mexico, Albuquerque, New Mexico 87108, USAReceived 8 July 2007; accepted 16 January 2008; published online 5 February 2008Negative differential conductance has been observed in grid-shaped surface superlattices, realized ina high mobility two-dimensional electron system. The current-voltage characteristics vary with themodulation strength, indicating that the two-dimensional electronic transport properties can bemanipulated in a controllable way. Theoretical modeling yields reasonable agreement with theexperimental data. © 2008 American Institute of Physics. DOI: 10.1063/1.2840996a Electronic mail: wpan@sandia.gov.In a periodic structure of electron potential, under anexternal electric field E, if an electron can reach the boundaryof the Brillouin zone BZ without being scattered, itundergoes Bragg reflection, passing back into the BZ on theopposite side. This results in a high frequency f oscillationof electron, i.e., Bloch oscillation BO, atf =eEd/h, whered is the period of the potential, e is the electron charge, andh is Plank’s constant. When the magnitude of electric fieldand the period of potential are properly chose, the Blochfrequency can fall in the terahertz region. Proposed about80 years ago, 1 recently, BO has gained a renewed interest, asa Bloch oscillator can be utilized as a solid-state, electricallybiased, frequency-tunable terahertz source and detector.So far, work on BO has mainly been carried out in verticalquantum well superlattice structures. 2–10 On the otherhand, a surface superlattice patterned in a two-dimensionalelectron system 2DES has long been proposed as an alternativedevice structure to generate BO. 11–16 A surface superlatticehas three-dimensional quantization, as opposite to theconventional vertical quantum well superlattice. 13 As a result,gaps in the energy spectrum exist in all three dimensions.This provides the possible existence of BO at relativelymoderate electric field. It has also been shown that, 13,16in surface superlattices, the electron-optical phonon scattering,one of the limiting mechanisms in achieving BO in theterahertz range in vertical superlattice, can be completelysuppressed, and electrically generated BO can become possible.Furthermore, compared to a vertical superlattice, a surfacesuperlattice offers other various advantages, for example,easy device fabrication and multiple fundamentalfrequencies. It can also fully utilize the benefit of long coherenttransport time achieved in high mobility 2DES. Todate, however, very few studies of BO in surface superlatticeshave been reported.In this paper, we report the experimental observation ofnegative differential conductance NDC in surface superlattices,one of the signatures of BO. Two devices with varyingcoupling strengths are studied and, in both samples, NDC isobserved. Our theoretical simulation, based on an energyrelaxation-time approximation and a microscopic short-rangeelastic scattering model for the momentum relaxation, 17yields reasonable agreement with the experimental data.To minimize electron scattering by residual impuritiesand to achieve long coherent transport time, the starting materialis a high mobility 2DES realized in a GaAs quantumwell QW of width 300 Å. The low temperature density andmobility of 2DES are n2–310 11 cm −2 and 110 6 cm 2 /V s, respectively, before patterning. After the patterning,the electron mobility was reduced somewhat. Twospecimens samples A and B, cut from two wafers, werestudied. The growth structure for the two wafers is basicallythe same, except that in wafer A, the quantum well is 200 nmbelow the surface, while in wafer B is 150 nm below. A lowfrequency lock-in technique was used to measure the lowtemperaturemagnetoresistance R xx and a Keithley K236source meter was used to supply the source-drain dc biasV dc and to measure the source-drain current I ds .In our experiments, a surface superlattice was realized ina two-dimensional 2D electron grid device. The 2D electrongrid was fabricated by depositing a periodic 2D metalgrid on the surface of QW samples. The metal consists ofTi/Au, about 80 nm thick. A scanning electron microscopySEM picture of metal grid is shown in the inset of Fig.1a. The pitch of the metal grid is 350 nm and the diameterof the holes is 150 nm. To achieve this large-area nanometerscale patterning, the interferometric lithographytechnique was employed. 18 In experiments, the grid deviceswere cooled in darkness to 4 K and a low temperature illuminationby a red-light-emitting diode was used to realizethe low temperature density and mobility. During eachcooldown, the sample, including the metal grid, wasgrounded. Due to the different thermal expansion coefficientsFIG. 1. a I-V trace in sample A. b I-V trace in sample B. Gray dash linerepresents the theoretical data. c Zoom-in plot of I-V trace in a differentcooldown for sample B. Jumps in current in the negative differential conductanceregime are apparent.0003-6951/2008/925/052104/3/$23.0092, 052104-1© 2008 American Institute of PhysicsDownloaded 05 Feb 2008 to 64.106.37.205. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

052104-2 Pan et al. Appl. Phys. Lett. 92, 052104 2008of the metal and the GaAs/AlGaAs semiconductor, the mechanicalstress induces a two-dimensional electron potentialmodulation of the 2DES. To characterize this modulation, wehave adopted well-documented methods 19 by measuring R xx .Similar to previous results, a positive magnetoresistance nearzero magnetic B field and the commensurate oscillationswere observed. From the saturation B field of the positivemagnetoresistance, 20 we estimated that the modulation amplitudeis 3% of the Fermi energy E F or 0.3 meV forsample A, and 15% 1 meV for sample B. To compare, ina reference sample without patterning neither positive magnetoresistancenor commensurate oscillation was observed inR xx .Current-voltage I-V measurement was carried out at4 K, using a Keithley K236 source meter. In Fig. 1, the I-Vcurves of samples A and B are shown. At small V dc , bothdevices show Ohmic behavior, and the corresponding resistanceis 1500 for sample A and 3000 for sample B.These resistances include wire and contact resistance in additionto the sample resistance. In sample A Fig. 1a, I dsreaches a maximal value of 0.92 mA at V dc 2.7 V, andthen decreases as V dc continues to increase, reaching a localminimum of 0.89 mA around 6.5 V. After this, I ds increasesagain with increasing V dc . In other words, a negativedifferential conductance is observed between 2.7 and6.5 V. In sample B, where a stronger modulation isachieved, again, a NDC is observed. In Figs. 1b and 1c,we show the I-V curves from two different cooldowns. Twofeatures are worthwhile emphasizing: 1 the maximal currentis significantly higher than that in sample A, 2.5 mAat V dc 3.8 V; 2 more complex structures are observed inthe NDC regime. In particular, in Fig. 1c, three currentjumps are observed at V dc 3.8, 4.0, and 4.6 V.The observation of NDC is exciting. Recall that NDCwas predicted and taken as the evidence of Bloch oscillationsin vertical quantum well superlattices by Esaki and Tsu intheir original paper. 2 In surface superlattices, it wasshown 11–15 that the bend over in the I-V curve could also bedue to the onset of Bloch oscillations, where electrons areable to cycle many times through the reduced Brillouin zonebefore a scattering event can happen. 13 Thus, the observedNDC may indeed represent the evidence of electron selfoscillationsin our 2D electron grids. Furthermore, the I-Vcurves show different characteristics in the two samples, indicatingthat the I-V characteristics and the 2D electronictransport properties can be manipulated by adjusting thestructure. Finally, we note the observation of current jumpsin the NDC region in our GaAs surface superlattice. Similarcurrent jumps have been observed in InGaAs/InAlAs verticalsuperlattices 21 and in InAs/AlSb resonant-tunnelingdiodes, 22 and were attributed to self-rectification of the electronoscillation. 23Of course, other mechanisms such as the formation ofhigh electric-field domains and the so-called thermal runaway,can also induce an apparent NDC. It has been shownthat stationary or propagating domains can form in 2Dstructures. 24 In this regard, it is possible that current jumpsmight be produced as the domain boundary moves throughthe interface fluctuations induced by surface superlattice. 25On the other hand, the two-dimensional nature of conductingchannels should help to stabilize the electrical field in surfacesuperlattices and prevent the formation of high electric-fielddomains. It is interesting to study in future experimentsFIG. 2. Color online Differential measurement for verifying the thermalrunaway origin of the observed NDC. The inset shows the experimentalsetup. R 0 is a thin-film resistor.whether the current jump observed in our device is related tothe domain formation.To experimentally eliminate the possibility of thermalrunaway origin, we employ a “differential” measurementsetup as shown in the inset of Fig. 2, where a thin filmresistor of constant value R 0 =1500 is connected in serieswith the two-dimensional electron grid sample B. A smallalternating current bias V ac is added to V dc . The voltagemeasured by a lock-in amplifier is given by V=V ac R 0 /R 0+r, where r=dV/dI is the differential resistance of the twodimensionalelectron grid. Here, we have omitted the wireand contact resistance, since they are relatively small comparedto r in the NDC region. If the observed NDC is ofthermal runaway origin, r is always positive. Consequently,V is expected to be positive, at any V dc . On the other hand, ifr is caused by dynamic localization through BO, then r0.In the case of r R 0 , V is expected to be negative in theNDC region. In Fig. 2, we show V as a function of V dc .Atsmall V dc , V is nearly constant, consistent with the observationin Fig. 1 that our electron grid is Ohmic at small V dc .Starting from V dc 8V, V begins to decrease. At V dc8.9 V, V becomes negative. 26 The observation of negativeV shows that, indeed, the observed NDC is not a result ofthermal effects.To help understand the physical origin of NDC, a theoreticalstudy was carried out for sample B with 15% modulation.We first calculate the energy band structures. In ourdevice with a periodic in-plane potential modulation, theoriginal modulation-free single conduction band is foldedinto many Bloch minibands with small miniband gaps. Surprisingly,each miniband can be represented by a fewcomponentcosine function similar to Esaki’s band withnegligible contributions from higher harmonics. To calculatethe I-V characteristic, we use a model similar to that byGerhardt, 17 except that our model treats a degenerate 2Delectron gas and assumes that the matrix element for elasticscattering is independent of the initial and final momentumrelevant to short-range impurity potentials, thereby treating2D interband scattering with an equal weight. Also, in ourcalculation, a simplified model, where the electron potentialmodulation is one-dimensional in the field direction, is used.We believe that the physics for a qualitative understandingDownloaded 05 Feb 2008 to 64.106.37.205. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

052104-3 Pan et al. Appl. Phys. Lett. 92, 052104 2008should essentially be the same as in the two-dimensionalelectron grid. Details of our theoretical calculations will bepublished elsewhere. The theoretical result is shown in Fig.1b by the gray dash line. Overall, the theoretical result is ina reasonable agreement with the experimental data, exceptthat the decrease of current in the NDC regime is steeperthan that of the experimental data. We also note here that theeffective current cross section employed in theoretical calculationsis smaller than the geometric cross section of realdevice by a factor of a few, and the electron scattering rate islarger than that deduced from the mobility. Those discrepanciesprobably are related to the assumptions of 1 a onedimensionalmodel and 2 field-independent inelastic scatteringrate we made in the calculation. Finally, for a selfconsistentcheck, for a parabolic band in the absence ofperiodic potential modulation, our model yields linear fielddependence of current and therefore no NDC.In conclusion, we have observed a NDC in twodimensionalelectron grid devices, and in one sample, severalcurrent jumps in the NDC region. Our theoretical modelingyields reasonable agreement with the experimental data. Theobserved NDC may represent the evidence of Bloch oscillationin our two-dimensional electron grids.We are grateful to Mark Lee for suggesting to us themeasurement in Fig. 2, and Mike Wanke for bringing ourattention to Ref. 24. We thank Denise Tibbetts for her excellenttechnical assistance. This project was supported, in part,by LDRD and DOE/BES at Sandia National Laboratories.Sandia is a multiprogram laboratory operated by Sandia Corporation,a Lockheed Martin company, for the United StatesDepartment of Energy’s National Nuclear Security Administrationunder Contract No. DE-AC04-94AL85000. The facilitiesof the NSF-sponsored NNIN node at the Universityof New Mexico were used for the fabrication.1 F. Bloch, Z. Phys. 52, 555 1928.2 L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 611970.3 K. K. Choi, B. F. Levine, R. J. Malik, J. Walker, and C. G. Betha, Phys.Rev. B 35, 4172 1987.4 J. Bleuse, G. Bastard, and P. Voisin, Phys. Rev. Lett. 60, 220 1988.5 E. E. Mendez, F. Agullo-Rueda, and J. M. Hong, Phys. Rev. Lett. 60,2426 1988.6 A. Sibille, J. F. Palmier, H. Wang, and F. Mollot, Phys. Rev. Lett. 64, 521990.7 F. Beltram, F. Capasso, D. L. Sivco, A. L. Hutchinson, S. N. G. Chu, andA. Y. Cho, Phys. Rev. Lett. 64, 3167 1990.8 K. Utnterrainer, B. J. Keay, M. C. Wanke, S. J. Allen, D. Leonard, G.Medeiros-Ribeiro, U. Bhattacharya, and M. J. W. Rodwell, Phys. Rev.Lett. 76, 29731996.9 Y. Shimada, N. Sekine, and K. Hirakawa, Appl. Phys. Lett. 84, 49262004.10 T. Feil, H.-P. Tranitz, M. Reinwald, and W. Wegscheider, Appl. Phys. Lett.87, 212112 2005.11 H. Sakaki, K. Wagatsuma, J. Hamasaki, and S. Saito, Thin Solid Films 36,497 1976.12 G. J. Iafrate, D. K. Ferry, and R. K. Reich, Surf. Sci. 113, 4851982.13 R. Reich, Ph.D. thesis, Colorado State University at Fort Collins, 1982.14 G. Bernstein and D. K. Ferry, J. Vac. Sci. Technol. B 5, 9641987.15 K. Ismail, W. Chu, D. A. Antoniadis, and H. I. Smith, Appl. Phys. Lett.54, 460 1989.16 I. A. Dmitriev and R. A. Suris, Semiconductors 35, 212 2001.17 R. F. Gerhardt, Phys. Rev. B 48, 9178 1993.18 S. C. Lee and S. R. J. Brueck, J. Vac. Sci. Technol. B 22, 1949 2004.19 D. Weiss, M. L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, andG. Weimann, Phys. Rev. Lett. 66, 27901991.20 P. H. Benton, E. S. Alves, P. C. Main, L. Eaves, M. W. Dellow, M. Henini,O. H. Hughes, S. P. Beaumont, and C. D. W. Wilkinson, Phys. Rev. B 42,9229 1990.21 E. Schomburg, R. Scheuerer, S. Brandl, K. F. Renk, D. G. Pavel’ev, Yu.Koschurinov, V. Ustinov, A. Zhukov, A. Kovsh, and P. S. Kop’ev, Electron.Lett. 35, 1491 1999.22 E. R. Brown, J. R. Soderstrom, C. D. Parker, L. J. Mahoney, K. M.Molvar, and T. C. McGrill, Appl. Phys. Lett. 58, 2291 1991.23 H. C. Liu, Appl. Phys. Lett. 53, 4851988.24 B. K. Ridley, Semicond. Sci. Technol. 3, 5421988.25 K. J. Luo, K.-J. Friedland, H. T. Grahn, and K. H. Ploog, Phys. Rev. B 61,4477 2000.26 Due to the load resistance R 0 , the NDC region is pushed to higher V dc ,compared to Fig. 1b.Downloaded 05 Feb 2008 to 64.106.37.205. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp