Terahertz transmission characteristics of high-mobility GaAs and InAs twodimensional-electron-gas systems N. A. Kabir, Y. Yoon, J. R. Knab, J.-Y. Chen, A. G. Markelz, J. L. Reno, Y. Sadofyev, S. Johnson, Y.-H. Zhang, and J. P. Bird Citation: Applied Physics Letters 89, 132109 (2006); doi: 10.1063/1.2357605 View online: http://dx.doi.org/10.1063/1.2357605 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/89/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The influence of charged InAs quantum dots on the conductance of a two-dimensional electron gas: Mobility vs. carrier concentration Appl. Phys. Lett. 99, 223510 (2011); 10.1063/1.3665070 High-mobility window for two-dimensional electron gases at ultrathin Al N ∕ Ga N heterojunctions Appl. Phys. Lett. 90, 182112 (2007); 10.1063/1.2736207 High-mobility two-dimensional electron gas in In Al As ∕ In As heterostructures grown on virtual InAs substrates by molecular-beam epitaxy Appl. 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Markelza兲 Department of Physics, University at Buffalo, The State University of New York, Buffalo, New York 14260-1920 J. L. Reno Nanostructure & Semiconductor Physics Department, Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87185-1415 Y. Sadofyev, S. Johnson, and Y.-H. Zhang Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287-5706 and Center for Solid State Electronics Research, Arizona State University, Tempe, Arizona 85287-5706 J. P. Birdb兲 Department of Electrical Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-1920 共Received 16 May 2006; accepted 2 August 2006; published online 26 September 2006兲 Frequency-dependent complex conductivity of high-mobility GaAs and InAs twodimensional-electron-gas 共2DEG兲 systems is studied by terahertz time domain spectroscopy. Determining the momentum relaxation time from a Drude model, the authors find a lower value than that from dc measurements, particularly at high frequencies/low temperatures. These deviations are consistent with the ratio ␶t / ␶q, where ␶q is the full scattering time. This suggests that small-angle scattering leads to weaker heating of 2DEGs at low temperatures than expected from dc mobility. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2357605兴 There is significant interest in developing terahertz sources and detectors that utilize high-mobility twodimensional-electron-gas 共2DEG兲 systems.1–6 In this letter, we therefore use terahertz time domain spectroscopy 共TDS兲 to measure the ac conductivity of GaAs and InAs 2DEGs. Terahertz TDS allows for contactless measurement of the frequency-dependent 共⬃5 – 85 cm−1兲 complex conductivity by coherently detecting the transmitted amplitude and phase shift of a laser-generated electric-field pulse. In such studies, it is often assumed that free-carrier absorption is described by a Drude model, using the relaxation time obtained from dc transport. The optical conductivity of metal films has been found to be best described by a “modified Drude model,” however, that incorporates a frequency-dependent scattering time.7 A similar model has also been used in studies of terahertz conductivity in transient photoconductivity of bulk GaAs.8 In this letter, we show that a modified Drude model fits the measured terahertz complex conductivity of GaAs and InAs 2DEGs, with relaxation times 共␶THz兲 shorter than those from dc mobility ␮, which yields the mobility lifetime ␶t. This discrepancy is increasingly pronounced with decreasing temperature 共T兲. Near 4.2 K, our analysis suggests that ␶THz is actually intermediate between ␶t and the quantum lifetime ␶q, which defines the total-scattering rate, extracted from analysis of the Shubnikov–de Haas 共SdH兲 effect. We suggest that, due to the role of small-angle scattering, Drude heating a兲 Author to whom correspondence should be addressed. Electronic mail: jbird@buffalo.edu b兲 of high-mobility 2DEGs by terahertz radiation can be significantly weakened at low T, compared to estimates based on dc transport. These results are therefore relevant for terahertz photonic-device development since they reveal that one cannot use dc transport to infer free-carrier loss. In terahertz TDS, the sample transmission is referenced to that of a substrate 共we use a piece of the same heterostructure with its 2DEG layer removed by etching兲. Its complex conductivity is then related to the transmission characteristics by the thin-film approximation,9 t= 1 + nsub Efilmei␸film tfilm = = , Esubei␸sub tsub 1 + nsub + Z0␴s 共1兲 where ␴s = ␴s共1兲 + i␴s共2兲, 冋册 Im 冋册 Re Z0␴s共1兲 1 =1+ , t 1 + nsub Z0␴s共2兲 1 = . t 1 + nsub Here, Efilm 共Esub兲 is the magnitude of the electric field transmitted by the film 共substrate兲, tfilm 共tsub兲 is the complex terahertz transmission of the film 共substrate兲, Z0 is the characteristic impedance of free space 共⬃377 ⍀兲, and nsub 关=3.3 for GaAs 共Ref. 10兲兴 is the refractive index of the substrate. In the modified Drude model, the complex ac conductivity at frequency ␻,11 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 0003-6951/2006/89共13兲/132109/3/$23.00 89, 132109-1 © 2006 American Institute of Physics 209.147.144.20 On: Sun, 08 Feb 2015 00:25:55 132109-2 Appl. Phys. Lett. 89, 132109 共2006兲 Kabir et al. TABLE I. Summary of the key parameters of the GaAs and InAs 2DEG samples. Hall ns at 300 K 共1015 m−2兲 Hall ns at 4.2 K 共1015 m−2兲 8.1 84 2.3 79 GaAs InAs SdH ns at 4.2 K 共1015 m−2兲 2.3 36a, 26b ␮ at 300 K 共cm2 / V s兲 ␮ at 4.2 K 共cm2 / V s兲 ␶t at 4.2 K 共ps兲 ␶q at 4.2 K 共ps兲 ␶t / ␶q at 4.2 K 7000 21 000 2 000 000 450 000 76 1.4 0.14 0.10 540 14 a Density due to first 2DEG subband. Density due to second 2DEG subband. b ␴s = 1 n se 2␶ , m *共 ␻ 兲 1 − i ␻ ␶ 共 ␻ 兲 共2兲 where ns is the 2DEG carrier density, m*共␻兲 is the frequencydependent effective mass, and ␶共␻兲 is the frequencydependent relaxation time. A strength of terahertz TDS is that one can directly extract ␶共␻兲 共which we will refer to hereafter as ␶THz兲, independent of m*共␻兲 and ns, ␶THz = ␴s共2兲 ␻␴s共1兲 = 1 Im共1/t兲 . ␻ 关Re共1/t兲 − 1兴 共3兲 oscillations and the low-temperature value of ␶q. This does not change the quantitative conclusions of our work, however. Terahertz TDS 共Fig. 1兲 was performed in a cryostat system cooled to 10 K. The system uses transient current generation from an antenna on semi-insulating GaAs and electro-optic detection. A 65 fs, 350 mW, and 82 MHz Ti:sapphire laser was used to generate/detect the terahertz pulse. The cryostat has two sets of terahertz-transparent polyethylene windows in the path of the terahertz beam. The scan resolution was 0.02 THz and the time-domain data were Fourier transformed to give the magnitude/phase of the transmitted field for the sample and reference substrate. The bandwidth is 2.5 THz, but since the absorbance falls off dramatically above 0.5 THz we emphasize here the range over which the frequency dependence is apparent. Transmission 共兩t 兩 兲 and phase 关tan−1共Im共t兲 / Re共t兲兲兴 of the InAs and GaAs 2DEGs are shown in Fig. 1. These data are plotted as discrete points, and are averaged values calculated over a 0.1 THz window. The high ns of the InAs 2DEG is responsible for its stronger absorption 共particularly below 1 THz兲 compared to the GaAs system. Figure 2 shows the frequency dependence of ␶THz 关from Eq. 共3兲兴 for the 2DEGs at several temperatures. At 300 K, ␶THz appears to be independent of frequency. At lower tem- GaAs and InAs 2DEGs were investigated in this study. The GaAs 2DEG is comprised of a 10 nm undoped GaAs cap layer, 98 nm of undoped AlGaAs, 2 nm of undoped AlAs, 2.3 nm of undoped GaAs, a Si delta-doped layer with an areal density of 1012 cm−2, 0.6 nm of undoped GaAs, 2 nm of undoped AlAs, a 75 nm AlGaAs spacer layer, a 30 nm GaAs quantum well, a 95 nm AlGaAs lower spacer layer, 2 nm of undoped AlAs, 2.3 nm of undoped GaAs, a lower Si delta-doped layer with an areal density of 1012 cm−2, 0.6 nm of undoped GaAs, 2 nm of undoped AlAs, 98 nm of undoped AlGaAs, 300 periods of a GaAs/ AlGaAs smoothing superlattice, and a 100-nm-thick undoped GaAs buffer. The InAs 2DEG 共Ref. 12兲 was grown on GaAs and was comprised of a 6 nm GaSb cap layer, 25 nm of undoped AlSb, a Te delta-doped layer, a 15 nm undoped AlSb spacer layer, a 15 nm InAs quantum well, a 15 nm AlSb spacer layer, a Te delta-doped layer, a 20 nm undoped AlSb layer, ten periods of a 2.5 nm AlSb/ 2.5 nm GaSb smoothing superlattice, and undoped buffer layers of 2.5 mm AlSb, 100 nm AlAs, and 200 nm GaAs. For transport measurements, GaAs Hall bars with alloyed contacts were formed by optical lithography and lift-off. The InAs 2DEG was cleaved into small pieces and alloyed contacts were formed by In soldering. SdH and Hall effects were measured in the GaAs and InAs 2DEGs at 1.4– 40 K. These measurements were made by standard lock-in techniques and yield information on the T dependence of ␶t. An analysis of the SdH oscillations also allows ␶q to be determined.13 Table I lists the parameters extracted for the 2DEGs. The different ns extracted for InAs from the SdH and Hall effects indicate partial filling of a second subband. Due to nonparabolicity of the InAs conduction band, the dc effective mass was extracted from the temperature dependence of the SdH oscillations.13 This gave m* = 0.060m0 ± 0.005m0, in good agreement with the predicFIG. 1. 共Color online兲 Field transmission magnitude 共upper panel兲 and tions of the Kane model for this density.14 The main uncerphase 共lower panel兲 of the GaAs 共red data兲 and InAs 共black data兲 2DEGs. tainty in our analysis is in our estimate of ␶q. The error bars The solid symbols are the original experimental data. The dotted lines are in Fig. 3 show the range of ␶q obtained from our analysis, but fits to Eq. 共2兲 using the variation of ␶THz inferred from the thin-film approxieven after considering this uncertainty there is still some 共a mation 关Eq.terms 共3兲兴. at: Circles: 300 K, triangles: 125 K, and squares: Downloaded 20 K. Inset: to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the http://scitation.aip.org/termsconditions. factor of 2兲 discrepancy between the turn-on field of the SdH schematic of the key features of the terahertz TDS system. 209.147.144.20 On: Sun, 08 Feb 2015 00:25:55 132109-3 Appl. Phys. Lett. 89, 132109 共2006兲 Kabir et al. FIG. 2. Frequency dependence of ␶THz inferred from the thin-film approximation 关Eq. 共3兲兴. Circles: 300 K, open squares: 200 K, triangles: 125 K, and filled squares: 20 K. The solid lines are interpolated between data points. peratures, however, ␶THz shows a significant decrease with increasing frequency. Similar behavior has been reported in midinfrared measurements of metal films.7 In Fig. 1, we have substituted the frequency-dependent values of ␶THz into Eq. 共2兲 to calculate the transmission amplitude/phase and we plot * these results as the dotted lines. We have used mInAs * = 0.06m0 and mGaAs = 0.067m0 in these calculations, along with the total 2DEG densities 共Table I兲. The fits show good agreement with experiment, with small deviations arising from the frequency dependence in the effective mass. Figure 3 shows the various lifetimes obtained from our measurements. Near 300 K, the dc mobility is limited predominantly by electron-phonon scattering.15 As the temperature is lowered below this, the values of ␶t and ␶THz differ significantly. This is clearest for the GaAs 2DEG in which, at the lowest temperatures, ␶THz is several orders of magnitude smaller than ␶t but close to ␶q. The difference between ␶q and ␶t in such 2DEGs is known16 to result from the predominance of small-angle scattering at low-temperatures, which does not significantly limit the mobility. The large ratio of ␶t / ␶q in the GaAs 2DEG is due to its thick spacer layers. The suggestion therefore is that at low temperatures, small-angle scattering results in reduced terahertz absorption, relative to the value expected from the dc mobility. Previously, the modified Drude model was shown to agree well with terahertz measurements of bulk GaAs.8 Our analysis suggests a different picture to describe the interaction of terahertz radiation with a high-mobility 2DEG. Near 300 K, where electron-phonon scattering should dominate the mobility,15 the results of our experiment approach the predictions of the dc Drude model. At low T, however, where impurity scattering dominates, large-angle scattering predominantly limits the dc mobility while the terahertz transmission seems to be sensitive to the total-scattering 共␶q兲 rate, particularly above ⬃0.5 THz. Since ␶t / ␶q can be very large 共⬎100 for GaAs兲, low-temperature Drude heating of 2DEGs by terahertz radiation can be significantly weaker than simple estimates based on the 2DEG mobility. Our measurements of the InAs 2DEG are consistent with this idea. The ratio ␶t / ␶q for this 2DEG is nearly 40 times smaller than that FIG. 3. Temperature dependence of ␶THz, ␶t, and ␶q in the GaAs 共upper兲 and InAs 共lower兲 2DEGs. The solid lines are guides for the eyes. in GaAs 共see Table I兲, and the discrepancy between ␶THz and ␶t is also less pronounced 共Fig. 3兲. The smaller value of ␶t / ␶q reflects the fact that the InAs 2DEG has no spacer layers and is unintentionally doped by donors that are believed to be located close to the interfaces of the quantum well. The work at Buffalo is supported by the Department of Energy and NYSTAR. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DEAC04-94AL85000. 1 X. G. Peralta, S. J. Allen, M. C. Wanke, N. E. Harff, J. A. Simmons, M. P. Lilly, J. L. Reno, P. J. Burke, and J. P. Eisenstein, Appl. Phys. Lett. 81, 1627 共2002兲. 2 E. A. Shaner, M. Lee, M. C. Wanke, A. D. Grine, J. L. Reno, and S. J. Allen, Appl. Phys. Lett. 87, 193507 共2005兲. 3 W. Knap, Y. Deng, S. Rumyantsev, J.-Q. Lu, M. S. Shur, C. A. Saylor, and L. C. Brunel, Appl. Phys. Lett. 80, 3433 共2002兲. 4 W. Knap, Y. Deng, S. Rumyantsev, and M. S. Shur, Appl. Phys. Lett. 81, 4637 共2002兲. 5 I. V. Kukushkin, S. A. Mikhailov, J. H. Smet, and K. von Klitzing, Appl. Phys. Lett. 86, 044101 共2005兲. 6 P. S. Dorozhkin, S. V. Tovstonog, S. A. Mikhailov, I. V. Kukushkin, J. H. Smet, and K. von Klitzing, Appl. Phys. Lett. 87, 092107 共2005兲. 7 J. Cerne, D. C. Schmadel, M. Grayson, G. S. Jenkins, J. R. Simpson, and H. D. Drew, Phys. Rev. B 61, 8133 共2000兲. 8 M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, Phys. Rev. B 62, 15764 共2000兲. 9 S. W. McKnight, K. P. Stewart, H. D. Drew, and K. Moorjani, Infrared Phys. 27, 327 共1987兲. 10 O. Madelung, Semiconductors: Basic Data, 2nd ed. 共Springer, Berlin, 1996兲, pp. 109–117. 11 N. W. Ashcroft and N. D. Mermin, Solid State Physics 共Saunders, Philadelphia, 1976兲, pp. 1–27. 12 Yu. G. Sadofyev, A. Ramamoorthy, B. Naser, J. P. Bird, S. R. Johnson, and Y.-H. Zhang, Appl. Phys. Lett. 81, 1833 共2002兲. 13 T. Ando, in High Magnetic Fields in Semiconductor Physics II, edited by G. Landwehr 共Springer, Berlin, 1989兲, pp. 164–173. 14 W. Zawadzki, Adv. Phys. 23, 435 共1974兲. 15 T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 共1982兲. 16 P. T. Coleridge, Phys. Rev. B 44, 3793 共1991兲. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 209.147.144.20 On: Sun, 08 Feb 2015 00:25:55