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Spin effects in InSb quantum wells
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Physica E 20 (2004) 386 – 391 www.elsevier.com/locate/physe
Spin e!ects in InSb quantum wells G.A. Khodaparasta;∗ , R.C. Meyera , X.H. Zhanga , T. Kasturiarachchia , R.E. Doezemaa , S.J. Chunga , N. Goela , M.B. Santosa , Y.J. Wangb a Department
of Physics and Astronomy, Center for Semiconductor Physics in Nanostructures, The University of Oklahoma, Norman, OK 73019, USA b National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA
Abstract Among the III–V semiconductors, InSb has the smallest electron e!ective mass and the largest g-factor. We make use of these properties to explore some aspects of electron spin in InSb quantum wells with far-infrared magneto-spectroscopy. We observe the clear signature of spin-resolved cyclotron resonance caused by the non-parabolicity of the conduction band. We observe avoided-level crossings at magnetic ?elds where Landau levels of the same spin are predicted to intersect. We also study electron spin resonance in the far infrared over a wide range of magnetic ?eld. In samples with symmetrically designed quantum wells we ?nd cyclotron masses and observed g-factors in good agreement with a Pidgeon–Brown analysis adapted to the two-dimensional band structure. However, the spin splitting approaches ∼3 meV as the magnetic ?eld approaches zero in samples intentionally asymmetrically doped. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.70.Ej; 78.67.De; 78.20.Ls Keywords: Heterostructures; Magneto-optics; Spin e!ects; Cyclotron resonance; Spin resonance
1. Introduction Recently, there has been growing interest in the study of spin phenomena in semiconductor heterostructures. A new ?eld, “spintronics” has emerged for the purpose of developing devices combining both the charge and spin degrees of freedom. In particular, spin splitting in heterostructures caused by bulk inversion asymmetry and structural inversion asymmetry (SIA—often called Rashba splitting), has attracted much attention. Understanding the spin–orbit ∗ Corresponding author. Department of Electrical and Computer Engineering, Rice Quantum Institute, Rice University, Houston, TX 77005, USA. E-mail address: [email protected] (G.A. Khodaparast).
interaction caused by SIA in heterostructures, which leads to zero ?eld spin splitting, is important for developing spin-based devices. It has been theoretically shown that lack of inversion symmetry and a small energy gap are the two mechanisms responsible for electric-dipole spin resonance. In narrow gap semiconductors with large spin–orbit coupling, the strong interaction between the conduction and valence bands leads to mixing of the electronic energy states. As a result, the wave function for a nominal spin direction and Landau quantum number n depends on both spin quantum numbers as well as n and n ± 1 [1,2]. In this case, the matrix elements for electric-dipole transitions allow the spin to change from up to down or both spin and orbital states can change by ±1. Electric dipole
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.08.042
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excited electron-spin resonance was ?rst observed in the far-infrared (FIR) spectral range in Hg1−x Cd x Te [3] and later in bulk InSb [4]. For many years, InSb has been considered as an ideal narrow-gap semiconductor. It was ?rst recognized by Kane that the small band gap in this material resulted in strong non-parabolicity in the dispersion relation. There have been numerous experimental and theoretical works to explore the characteristics of this narrow gap system. Because of its large effective g-factor and large predicted Rashba splitting [5], InSb is potentially well-suited for use in spintronic devices. Furthermore, the spin and cyclotron resonances in this material occur at FIR frequencies at an accessible range of magnetic ?elds, making InSb a good material for investigating certain spin phenomena. In this paper, we focus on magneto-optical experiments that are made possible by the large spin-splitting in InSb quantum wells: spin-resolved cyclotron resonance (SRCR), spin-resolved avoided-level crossings, and electron spin resonance (ESR). 2. Experiment The samples studied in this work are InSb single-quantum wells with widths of 20 –30 nm, grown on GaAs (001) substrates [6]. The Alx In1−x Sb (x = 0:09 and 0.15) barrier layers are -doped with Si. The Si -layers are located either on one side of the quantum well (asymmetric samples) or equidistant on both sides of the quantum well (symmetric samples). The -doped layers within the barrier layers are typically located 70 nm from the well center. We expect the shape and symmetry of the wells to be determined only by the well/barrier mismatch (symmetric in all samples) and the mismatch in doping layers. Recently, the inQuence of barrier materials on spin splitting due to SIA has been investigated which demonstrates the importance of the spin-dependent boundary conditions at the interface in narrow-gap materials [7]. The electron concentrations in the wells are in the range (1–4) × 1011 cm−2 , where only the ground-state subband is occupied. The mobility of the samples studied range from 70 000 cm2 =V s to 130 000 cm2 =V s. The energy-gap discontinuity [8] as well as the band o!sets [9] in this strained-layer system has been determined earlier.
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The magneto-transmission experiments in this work were performed using two measurement systems. The SRCR measurements were carried out at the National High Magnetic Field Laboratory, using a Fourier transform infrared spectrometer (100 –700 cm−1 ) and a superconducting magnet with magnetic ?eld up to B = 17:5 T. FIR radiation was emitted and detected by a glowbar and a Si bolometer, respectively. The sample normal was tilted at small angles (0 –4◦ ) with respect to B. The propagation direction of the incident FIR radiation is the same as the direction of B. The ESR measurements were carried out using a gas laser with wavelengths of 47–458 m and B up to 7 T. The sample was tilted between 30◦ and 50◦ . The laser beam was chopped mechanically at 150 Hz and a Ge:Ga photoconductor was used to detect the transmission signal. The sample temperature was 4:2 K in both measurement systems. In addition to observing ESR through transmission measurements, we observed ESR through a change in the photoconductivity due to resonant absorption of the laser light. In this case, the sample itself functioned as a detector. An AC current of 1 A (∼130 Hz) is applied to the sample while the incident laser light is chopped at a frequency that is suRciently small to ensure an adequate response time (∼13 Hz). Features in the photoconductivity can be attributed to non-resonant phenomena or to resonant mechanisms such as CR and ESR. As in the case of InSb inversion layers [10], the ESR signal is strongest when the sample is tilted. We have been unable to observe the ESR signal with no tilt. Detailed experimental [11,12] and theoretical studies [13] on bulk InSb demonstrated the e!ect of magnetic ?eld orientations on the g-factor anisotropy and magneto-optical selections rules in n-type zinc-blende semiconductors. In our experiment, we vary the tilt angle to extend the range of B where ESR is observable for a given electron density. In this case, the spins (and ESR) respond to the total B but the two-dimensional Landau quantization (and thus the Shubnikov–de Haas (SdH) oscillations and CR) depends only on the component of B along the sample normal. 3. Spin resolved cyclotron resonance SRCR ?rst was observed in InAs quantum wells with AlSb barrier layers [14,15]. The experimental
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Fig. 1. Calculation of Landau level energies for the lowest two subbands, as functions of applied magnetic ?eld. The solid lines (dashed lines) represent spin up (down) states. The three dash–dot lines indicate values of B where the Fermi energy changes discontinuously at low temperature. The three circles indicate values of k and B where Landau levels of the same spin intersect.
observations were explained by using a modi?ed Pidgeon and Brown bulk band structure model [16] that takes into account the non-parabolicity. In the modi?ed model, the bulk energy gap is replaced by an effective energy gap in the two-dimensional band structure equal to the energy spacing between the ground state valence- and conduction-subbands. Fig. 1 shows the calculated Landau level energies for a 30 nm-wide InSb quantum well with Al0:09 In0:91 Sb barriers at zero tilt Em;n = Em + (n + 12 )˝!c ± gB B=2; where Em is the subband energy (m = 0; 1), n = 0; 1; 2; : : : is the Landau level index, and B is the Bohr magneton. The calculated Fermi energy is shown for an electron density of ns =2:2×1011 cm−2 . In general, three spin-conserving transitions with Tn = 1 are possible at a given B, except for Landau level ?lling factors of 1 ¡ ¡ 2 where only two transitions are possible and ¡ 1 where only one transition is possible. The non-parabolicity of the conduction band makes the energies di!erent for each transition at a given B. Fig. 2 shows normalized transmission spectra for sample S499 (ns ∼ 2:2 × 1011 cm−2 ) at several values of B and a tilt angle of 0◦ . The arrows indicate SRCR features. The many features at 180 cm−1 6 k 6 190 cm−1 are artifacts of the strong InSb-like phonon absorption in the Alx In1−x Sb and
Fig. 2. Normalized transmission spectra for sample S499 (ns ∼2:2 × 1011 cm−2 ) at several values of B. The arrows indicate SRCR features.
InSb layers. Two CR peaks are resolved for most values of B. The energies of the CR peaks are plotted as ?lled circles in Fig. 3. The solid lines indicate the calculated positions for the E0; 0; up → E0; 1; up (line at highest k), E0; 0; down → E0; 1; down ; E0; 1; up → E0; 2; up and E0; 1; down → E0; 2; down (line at lowest k) transitions. The crosshatched area indicates the region of InSb-like phonon absorption. It is evident that the experimental data are in agreement with the calculation. Within experimental error, transitions disappear and reappear as predicted at integer values of . The Landau level fan diagram shown in Fig. 1 shows that Landau levels with the same spin intersect at several values of B where the transition energy is ∼(E1; n − E0; n ) or ∼(E1; n − E0; n )=2. These states should interact in a manner similar to Landau levels in GaAs-based two-dimensional systems that are not spin resolved [17]. Since the interaction should be stronger with increased parallel B, an avoided-level crossing is expected when the sample is tilted. For 30 nm-wide InSb quantum well with Al0:09 In0:91 Sb barriers the
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Fig. 4. Normalized transmission spectra for sample S716 (ns ∼ 1:5 × 1011 cm−2 ) between 2.4 and 3 T in steps of 0:1 T. An avoided level crossing for the E0; 1; up → E1; 0; up transition is indicated by the arrows. Points on the lower (upper) branch are indicated by the downward (upward) pointing arrows. The large unlabeled peaks are the E0; 0; down → E0; 1; down transition.
Fig. 3. The ?lled circles indicate the positions of the SRCR features for sample S499. The solid lines indicate the calculated positions for the four highest-energy spin-conserving transitions with n = 1. The crosshatched area indicates the region of InSb-like phonon absorption.
transition energies for the two highest-energy crossings (E0; 0; up → E1; 0; up and E0; 0; down → E1; 0; down ) are calculated to be in GaAs restrahlung band (∼250– 350 cm−1 ), and therefore would not be observable. However, as shown in Fig. 4, an avoided-level crossing is observed (between 2.4 and 3 T) for the third highest-energy (E0; 1; up → E1; 0; up ) crossing in sample S716 (ns ∼ 1:5 × 1011 cm−2 ). The splitting is 6 cm−1 with a 4◦ tilt. We observe the two highest-energy avoided-level crossings in other samples with narrower wells and higher barriers (that increase the subband spacing above GaAs restrahlung band), and higher densities (to populate the second Landau level at the required B). The large Zeeman e!ect and strong non-parabolicity in InSb enables the study of spin-resolved subband/cyclotron combined resonances. Investigation of the avoided-level crossings may provide insight into the role of spin in the interaction between Landau levels from di!erent subbands.
4. Electron spin resonance An asymmetry in the con?nement potential for twodimensional electron systems, has been predicted by Rashba and co-workers [18], to lead to a lifting of the spin degeneracy even in the absence of an applied magnetic ?eld B. Most of the experimental evidence for Rashba spin splitting comes from the beating of SdH oscillations [19] and has been used to obtain a measure of the Rashba parameter characterizing the strength of the B = 0 spin splitting. In GaAs quantum wells, zero-?eld spin splitting has also been inferred from Raman-scattering spectra [20]. We report spin resonance spectroscopy in 20 nm-wide InSb quantum wells with Al0:09 In0:91 Sb barrier layers, which con?rm our earlier observations in 30 nm-wide wells [21]. Fig. 5 shows the observed ESR for an asymmetric sample at low B. The resonance is very narrow, on the order of 40 mT. ESR in the FIR region was observed in bulk InSb by McCombe and Wagner [4] for material with a carrier concentration of 6 × 1015 cm−3 and a mobility of 2×105 cm2 =V s. The measured line width was about 20 mT, similar to the observation by Chen et al. [11]. A much narrower line width was reported in bulk InSb (at lower magnetic ?elds) in the microwave region for a similar bulk carrier concentration [22]. In this work, by using samples with di!erent electron densities, wedges with di!erent tilt, and six laser
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Fig. 6. Measured (symbols) and calculated (solid lines) spin splittings as functions of magnetic ?eld. Filled (open) circles represent asymmetric (symmetric) structures. The calculation includes only the Zeeman e!ect. The Landau level indices for the calculated splittings are labeled.
Fig. 5. Change in resistance of sample S356 due to illumination by FIR radiation. Non-resonant features are seen near integer Landau level ?lling factors . Resonant absorption features due to CR and ESR are marked.
lines producing a suRcient signal-to-noise ratio, we are able to follow the ESR over a range of magnetic ?eld and Landau-level index. We determine the Landau-level index for a given spin resonance from the ?lling factor of the SdH oscillation at the (total) B value at which the resonance occurs. The observed spin splittings for the samples is plotted in Fig. 6. Also plotted in Fig. 6 is the spin splitting Ts(B) expected in the absence of Rashba splitting from the modi?ed (Pidgeon and Brown) bulk band structure which takes into account the non-parabolicity of the InSb band structure. This e!ective energy gap depends on well width, as does the momentum matrix element that we use as the lone ?tting parameter (within its accepted range of values [23]). All the other band parameters are ?xed at their accepted values. We conclude that our symmetric samples behave as expected for wells with no zero-?eld spin splitting based on one-electron energy levels. We also ?nd that the CR positions in both symmetric and asymmetric wells are correctly predicted
by the Pidgeon–Brown model. Similar results have been reported for GaAs quantum wells [24,25]. The asymmetric samples show quite di!erent behavior, especially at low B, as can be seen in Fig. 6. The spin splitting is observed to deviate from those values expected for a symmetric well to values even beyond those calculated using the band-edge g-factor, −48. For the asymmetric samples the spin splitting is determined by the applied ?eld (the Zeeman term) plus an e!ective ?eld from the Rashba splitting. We estimate a Rashba contribution of ∼3 meV by taking the di!erence between the measured Ts(B) in a given sample and the Ts(B) value predicted for a symmetric sample at the same n by the Pidgeon–Brown calculation. For the electron densities used in this study (1.1–2:2 × 1011 cm−2 ), this corresponds to an value ∼1:5 × 10−9 eV cm for the Rashba parameter. This value is comparable to the values determined in gated InAs quantum wells [26], even though the electric ?eld in our ungated asymmetric samples is much smaller. The ability to probe Rashba splitting with spin resonance suggests interesting future work. We are extending our spin resonance studies to gated samples. These should give us the ability to study the spin resonance in the absence of any applied magnetic ?eld. We also plan to study samples in which the asymmetry is due to di!ering Al concentrations in the barriers on either
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side of the quantum well rather than to asymmetric doping as in the present samples. This, together with the ability to vary the electron density, should allow us to distinguish the e!ect of an asymmetric electric ?eld from the role of the barrier asymmetry [5,7] in causing the spin–orbit splitting. In the results we have presented here, both mechanisms may contribute. Acknowledgements We are grateful to Sheena Murphy for helpful comments and to Kory Goldammer for growth of the early samples in this study. This work was supported by the National Science Foundation under Grant Nos. DMR-0209371 and DMR-0080054, and through support of the National High Magnetic Field Laboratory. References [1] W. Zawadzki, Landau level spectroscopy, Modern Problems in Condensed Matter Sciences, Vol. 27.1, North-Holland, Amsterdam, 1991, pp. 485 – 479. [2] B.D. McCombe, R.J. Wagner, Intraband magneto-optical studies of semiconductors in the far-infrared, Advances in Electronics and Electron Physics, Vol. 37, Academic Press, New York, 1975, pp. 1–78. [3] B.D. McCombe, R.J. Wagner, G.A. Prinz, Phys. Rev. Lett. 25 (1970) 87. [4] B.D. McCombe, R.J. Wagner, Phys. Rev. B 4 (1971) 1285; Y.-F. Chen, M. Dobrowolska, J.K. Furdyna, Phys. Rev. B 31 (1985) 7989. [5] S. Lamari, J. Appl. Phys. 91 (2002) 1698. [6] S.J. Chung, K.J. Goldammer, S.C. Lindstrom, M.B. Johnson, M.B. Santos, J. Vac. Sci. Technol. B 17 (1999) 1151. [7] P. Pfe!er, W. Zawadzki, J. Supercond. 16 (2003) 351. [8] N. Dai, F. Brown, R.E. Doezema, S.J. Chung, K.J. Goldammer, M.B. Santos, Appl. Phys. Lett. 73 (1998) 3132.
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[9] N. Dai, G.A. Khodaparast, F. Brown, R.E. Doezema, S.J. Chung, M.B. Santos, Appl. Phys. Lett. 76 (2000) 3905. [10] A. DUarr, J.P. Kotthaus, T. Ando, in: F.G. Fumi (Ed.), Physics of Semiconductors, North-Holland, Amsterdam, 1976, pp. 774–777; A. DUarr, Doctoral Dissertation, Technische UniversitUat MUunchen, 1978, unpublished. [11] Y.-F Chen, M. Dobrowolska, J.K. Furdyna, S. Rodriguez, Phys. Rev. B 32 (1985) 890. [12] Y.-F Chen, M. Dobrowolska, J.K. Furdyna, Phys. Rev. B 31 (1985) 7989. [13] G. Sudha, J.K. Furdyna, S. Rodriguez, Phys. Rev. B 32 (1985) 903. [14] J. Scriba, A. Wixforth, J.P. Kotthaus, C.R. Bolognesi, C. Nguyen, G. Tuttle, J.H. English, H. Kroemer, Semicond. Sci. Technol. 8 (1993) S133. [15] M.J. Yang, R.J. Wagner, B.V. Shanabrook, J.R. Waterman, W.J. Moore, Phys. Rev. B 47 (1992) 6807. [16] C.R. Pidgeon, R.N. Brown, Phys. Rev. 146 (1966) 575. [17] M.A. Brummell, M.A. Hopkins, R.J. Nicholas, J.C. Portal, K.Y. Cheng, A.Y. Cho, J. Phys. C 19 (1986) L107. [18] Y.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039; Y.A. Bychkov, E.E. Rashba, Pis’ma Zh. Eksp. Teor. Fiz. 39 (1984) 66 [JETP Lett. 39 (1984) 78]. [19] Th. SchUapers, G. Engels, J. Lange, Th. Klocke, M. Hollfelder, H. LUuth, J. Appl. Phys. 83 (1998) 4324, and references therein. [20] B. Jusserand, D. Richards, H. Peric, B. Etienne, Phys. Rev. Lett. 69 (1992) 848; B. Jusserand, D. Richards, G. Allan, C. Priester, B. Etienne, Phys. Rev. B 51 (1995) 4707. [21] G.A. Khodaparast, R.E. Doezema, S.J. Chung, K.J. Goldammer, M.B. Santos, IPAP Conf. Ser. 2 (2001) 245. [22] R.A. Isaacson, Phys. Rev. 169 (1968) 312. [23] Landolt-BUornstein, in: O. Madelung (Ed.), Numerical Data and Functional Relationships in Science and Technology, Group III, Vol. 22, Springer, Berlin, 1987. [24] M. Dobers, K.v. Klitzing, G. Weimann, Phys. Rev. B 38 (1988) 5453. [25] R. Meisels, I. Kulac, F. Kuchar, M. Kriechbaum, Phys. Rev. B 61 (2000) 5637. [26] D. Grundler, Phys. Rev. Lett. 84 (2000) 6074.
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Spin e!ects in InSb quantum wells G.A. Khodaparasta;∗ , R.C. Meyera , X.H. Zhanga , T. Kasturiarachchia , R.E. Doezemaa , S.J. Chunga , N. Goela , M.B. Santosa , Y.J. Wangb a Department
of Physics and Astronomy, Center for Semiconductor Physics in Nanostructures, The University of Oklahoma, Norman, OK 73019, USA b National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA
Abstract Among the III–V semiconductors, InSb has the smallest electron e!ective mass and the largest g-factor. We make use of these properties to explore some aspects of electron spin in InSb quantum wells with far-infrared magneto-spectroscopy. We observe the clear signature of spin-resolved cyclotron resonance caused by the non-parabolicity of the conduction band. We observe avoided-level crossings at magnetic ?elds where Landau levels of the same spin are predicted to intersect. We also study electron spin resonance in the far infrared over a wide range of magnetic ?eld. In samples with symmetrically designed quantum wells we ?nd cyclotron masses and observed g-factors in good agreement with a Pidgeon–Brown analysis adapted to the two-dimensional band structure. However, the spin splitting approaches ∼3 meV as the magnetic ?eld approaches zero in samples intentionally asymmetrically doped. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.70.Ej; 78.67.De; 78.20.Ls Keywords: Heterostructures; Magneto-optics; Spin e!ects; Cyclotron resonance; Spin resonance
1. Introduction Recently, there has been growing interest in the study of spin phenomena in semiconductor heterostructures. A new ?eld, “spintronics” has emerged for the purpose of developing devices combining both the charge and spin degrees of freedom. In particular, spin splitting in heterostructures caused by bulk inversion asymmetry and structural inversion asymmetry (SIA—often called Rashba splitting), has attracted much attention. Understanding the spin–orbit ∗ Corresponding author. Department of Electrical and Computer Engineering, Rice Quantum Institute, Rice University, Houston, TX 77005, USA. E-mail address: [email protected] (G.A. Khodaparast).
interaction caused by SIA in heterostructures, which leads to zero ?eld spin splitting, is important for developing spin-based devices. It has been theoretically shown that lack of inversion symmetry and a small energy gap are the two mechanisms responsible for electric-dipole spin resonance. In narrow gap semiconductors with large spin–orbit coupling, the strong interaction between the conduction and valence bands leads to mixing of the electronic energy states. As a result, the wave function for a nominal spin direction and Landau quantum number n depends on both spin quantum numbers as well as n and n ± 1 [1,2]. In this case, the matrix elements for electric-dipole transitions allow the spin to change from up to down or both spin and orbital states can change by ±1. Electric dipole
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.08.042
G.A. Khodaparast et al. / Physica E 20 (2004) 386 – 391
excited electron-spin resonance was ?rst observed in the far-infrared (FIR) spectral range in Hg1−x Cd x Te [3] and later in bulk InSb [4]. For many years, InSb has been considered as an ideal narrow-gap semiconductor. It was ?rst recognized by Kane that the small band gap in this material resulted in strong non-parabolicity in the dispersion relation. There have been numerous experimental and theoretical works to explore the characteristics of this narrow gap system. Because of its large effective g-factor and large predicted Rashba splitting [5], InSb is potentially well-suited for use in spintronic devices. Furthermore, the spin and cyclotron resonances in this material occur at FIR frequencies at an accessible range of magnetic ?elds, making InSb a good material for investigating certain spin phenomena. In this paper, we focus on magneto-optical experiments that are made possible by the large spin-splitting in InSb quantum wells: spin-resolved cyclotron resonance (SRCR), spin-resolved avoided-level crossings, and electron spin resonance (ESR). 2. Experiment The samples studied in this work are InSb single-quantum wells with widths of 20 –30 nm, grown on GaAs (001) substrates [6]. The Alx In1−x Sb (x = 0:09 and 0.15) barrier layers are -doped with Si. The Si -layers are located either on one side of the quantum well (asymmetric samples) or equidistant on both sides of the quantum well (symmetric samples). The -doped layers within the barrier layers are typically located 70 nm from the well center. We expect the shape and symmetry of the wells to be determined only by the well/barrier mismatch (symmetric in all samples) and the mismatch in doping layers. Recently, the inQuence of barrier materials on spin splitting due to SIA has been investigated which demonstrates the importance of the spin-dependent boundary conditions at the interface in narrow-gap materials [7]. The electron concentrations in the wells are in the range (1–4) × 1011 cm−2 , where only the ground-state subband is occupied. The mobility of the samples studied range from 70 000 cm2 =V s to 130 000 cm2 =V s. The energy-gap discontinuity [8] as well as the band o!sets [9] in this strained-layer system has been determined earlier.
387
The magneto-transmission experiments in this work were performed using two measurement systems. The SRCR measurements were carried out at the National High Magnetic Field Laboratory, using a Fourier transform infrared spectrometer (100 –700 cm−1 ) and a superconducting magnet with magnetic ?eld up to B = 17:5 T. FIR radiation was emitted and detected by a glowbar and a Si bolometer, respectively. The sample normal was tilted at small angles (0 –4◦ ) with respect to B. The propagation direction of the incident FIR radiation is the same as the direction of B. The ESR measurements were carried out using a gas laser with wavelengths of 47–458 m and B up to 7 T. The sample was tilted between 30◦ and 50◦ . The laser beam was chopped mechanically at 150 Hz and a Ge:Ga photoconductor was used to detect the transmission signal. The sample temperature was 4:2 K in both measurement systems. In addition to observing ESR through transmission measurements, we observed ESR through a change in the photoconductivity due to resonant absorption of the laser light. In this case, the sample itself functioned as a detector. An AC current of 1 A (∼130 Hz) is applied to the sample while the incident laser light is chopped at a frequency that is suRciently small to ensure an adequate response time (∼13 Hz). Features in the photoconductivity can be attributed to non-resonant phenomena or to resonant mechanisms such as CR and ESR. As in the case of InSb inversion layers [10], the ESR signal is strongest when the sample is tilted. We have been unable to observe the ESR signal with no tilt. Detailed experimental [11,12] and theoretical studies [13] on bulk InSb demonstrated the e!ect of magnetic ?eld orientations on the g-factor anisotropy and magneto-optical selections rules in n-type zinc-blende semiconductors. In our experiment, we vary the tilt angle to extend the range of B where ESR is observable for a given electron density. In this case, the spins (and ESR) respond to the total B but the two-dimensional Landau quantization (and thus the Shubnikov–de Haas (SdH) oscillations and CR) depends only on the component of B along the sample normal. 3. Spin resolved cyclotron resonance SRCR ?rst was observed in InAs quantum wells with AlSb barrier layers [14,15]. The experimental
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Fig. 1. Calculation of Landau level energies for the lowest two subbands, as functions of applied magnetic ?eld. The solid lines (dashed lines) represent spin up (down) states. The three dash–dot lines indicate values of B where the Fermi energy changes discontinuously at low temperature. The three circles indicate values of k and B where Landau levels of the same spin intersect.
observations were explained by using a modi?ed Pidgeon and Brown bulk band structure model [16] that takes into account the non-parabolicity. In the modi?ed model, the bulk energy gap is replaced by an effective energy gap in the two-dimensional band structure equal to the energy spacing between the ground state valence- and conduction-subbands. Fig. 1 shows the calculated Landau level energies for a 30 nm-wide InSb quantum well with Al0:09 In0:91 Sb barriers at zero tilt Em;n = Em + (n + 12 )˝!c ± gB B=2; where Em is the subband energy (m = 0; 1), n = 0; 1; 2; : : : is the Landau level index, and B is the Bohr magneton. The calculated Fermi energy is shown for an electron density of ns =2:2×1011 cm−2 . In general, three spin-conserving transitions with Tn = 1 are possible at a given B, except for Landau level ?lling factors of 1 ¡ ¡ 2 where only two transitions are possible and ¡ 1 where only one transition is possible. The non-parabolicity of the conduction band makes the energies di!erent for each transition at a given B. Fig. 2 shows normalized transmission spectra for sample S499 (ns ∼ 2:2 × 1011 cm−2 ) at several values of B and a tilt angle of 0◦ . The arrows indicate SRCR features. The many features at 180 cm−1 6 k 6 190 cm−1 are artifacts of the strong InSb-like phonon absorption in the Alx In1−x Sb and
Fig. 2. Normalized transmission spectra for sample S499 (ns ∼2:2 × 1011 cm−2 ) at several values of B. The arrows indicate SRCR features.
InSb layers. Two CR peaks are resolved for most values of B. The energies of the CR peaks are plotted as ?lled circles in Fig. 3. The solid lines indicate the calculated positions for the E0; 0; up → E0; 1; up (line at highest k), E0; 0; down → E0; 1; down ; E0; 1; up → E0; 2; up and E0; 1; down → E0; 2; down (line at lowest k) transitions. The crosshatched area indicates the region of InSb-like phonon absorption. It is evident that the experimental data are in agreement with the calculation. Within experimental error, transitions disappear and reappear as predicted at integer values of . The Landau level fan diagram shown in Fig. 1 shows that Landau levels with the same spin intersect at several values of B where the transition energy is ∼(E1; n − E0; n ) or ∼(E1; n − E0; n )=2. These states should interact in a manner similar to Landau levels in GaAs-based two-dimensional systems that are not spin resolved [17]. Since the interaction should be stronger with increased parallel B, an avoided-level crossing is expected when the sample is tilted. For 30 nm-wide InSb quantum well with Al0:09 In0:91 Sb barriers the
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Fig. 4. Normalized transmission spectra for sample S716 (ns ∼ 1:5 × 1011 cm−2 ) between 2.4 and 3 T in steps of 0:1 T. An avoided level crossing for the E0; 1; up → E1; 0; up transition is indicated by the arrows. Points on the lower (upper) branch are indicated by the downward (upward) pointing arrows. The large unlabeled peaks are the E0; 0; down → E0; 1; down transition.
Fig. 3. The ?lled circles indicate the positions of the SRCR features for sample S499. The solid lines indicate the calculated positions for the four highest-energy spin-conserving transitions with n = 1. The crosshatched area indicates the region of InSb-like phonon absorption.
transition energies for the two highest-energy crossings (E0; 0; up → E1; 0; up and E0; 0; down → E1; 0; down ) are calculated to be in GaAs restrahlung band (∼250– 350 cm−1 ), and therefore would not be observable. However, as shown in Fig. 4, an avoided-level crossing is observed (between 2.4 and 3 T) for the third highest-energy (E0; 1; up → E1; 0; up ) crossing in sample S716 (ns ∼ 1:5 × 1011 cm−2 ). The splitting is 6 cm−1 with a 4◦ tilt. We observe the two highest-energy avoided-level crossings in other samples with narrower wells and higher barriers (that increase the subband spacing above GaAs restrahlung band), and higher densities (to populate the second Landau level at the required B). The large Zeeman e!ect and strong non-parabolicity in InSb enables the study of spin-resolved subband/cyclotron combined resonances. Investigation of the avoided-level crossings may provide insight into the role of spin in the interaction between Landau levels from di!erent subbands.
4. Electron spin resonance An asymmetry in the con?nement potential for twodimensional electron systems, has been predicted by Rashba and co-workers [18], to lead to a lifting of the spin degeneracy even in the absence of an applied magnetic ?eld B. Most of the experimental evidence for Rashba spin splitting comes from the beating of SdH oscillations [19] and has been used to obtain a measure of the Rashba parameter characterizing the strength of the B = 0 spin splitting. In GaAs quantum wells, zero-?eld spin splitting has also been inferred from Raman-scattering spectra [20]. We report spin resonance spectroscopy in 20 nm-wide InSb quantum wells with Al0:09 In0:91 Sb barrier layers, which con?rm our earlier observations in 30 nm-wide wells [21]. Fig. 5 shows the observed ESR for an asymmetric sample at low B. The resonance is very narrow, on the order of 40 mT. ESR in the FIR region was observed in bulk InSb by McCombe and Wagner [4] for material with a carrier concentration of 6 × 1015 cm−3 and a mobility of 2×105 cm2 =V s. The measured line width was about 20 mT, similar to the observation by Chen et al. [11]. A much narrower line width was reported in bulk InSb (at lower magnetic ?elds) in the microwave region for a similar bulk carrier concentration [22]. In this work, by using samples with di!erent electron densities, wedges with di!erent tilt, and six laser
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Fig. 6. Measured (symbols) and calculated (solid lines) spin splittings as functions of magnetic ?eld. Filled (open) circles represent asymmetric (symmetric) structures. The calculation includes only the Zeeman e!ect. The Landau level indices for the calculated splittings are labeled.
Fig. 5. Change in resistance of sample S356 due to illumination by FIR radiation. Non-resonant features are seen near integer Landau level ?lling factors . Resonant absorption features due to CR and ESR are marked.
lines producing a suRcient signal-to-noise ratio, we are able to follow the ESR over a range of magnetic ?eld and Landau-level index. We determine the Landau-level index for a given spin resonance from the ?lling factor of the SdH oscillation at the (total) B value at which the resonance occurs. The observed spin splittings for the samples is plotted in Fig. 6. Also plotted in Fig. 6 is the spin splitting Ts(B) expected in the absence of Rashba splitting from the modi?ed (Pidgeon and Brown) bulk band structure which takes into account the non-parabolicity of the InSb band structure. This e!ective energy gap depends on well width, as does the momentum matrix element that we use as the lone ?tting parameter (within its accepted range of values [23]). All the other band parameters are ?xed at their accepted values. We conclude that our symmetric samples behave as expected for wells with no zero-?eld spin splitting based on one-electron energy levels. We also ?nd that the CR positions in both symmetric and asymmetric wells are correctly predicted
by the Pidgeon–Brown model. Similar results have been reported for GaAs quantum wells [24,25]. The asymmetric samples show quite di!erent behavior, especially at low B, as can be seen in Fig. 6. The spin splitting is observed to deviate from those values expected for a symmetric well to values even beyond those calculated using the band-edge g-factor, −48. For the asymmetric samples the spin splitting is determined by the applied ?eld (the Zeeman term) plus an e!ective ?eld from the Rashba splitting. We estimate a Rashba contribution of ∼3 meV by taking the di!erence between the measured Ts(B) in a given sample and the Ts(B) value predicted for a symmetric sample at the same n by the Pidgeon–Brown calculation. For the electron densities used in this study (1.1–2:2 × 1011 cm−2 ), this corresponds to an value ∼1:5 × 10−9 eV cm for the Rashba parameter. This value is comparable to the values determined in gated InAs quantum wells [26], even though the electric ?eld in our ungated asymmetric samples is much smaller. The ability to probe Rashba splitting with spin resonance suggests interesting future work. We are extending our spin resonance studies to gated samples. These should give us the ability to study the spin resonance in the absence of any applied magnetic ?eld. We also plan to study samples in which the asymmetry is due to di!ering Al concentrations in the barriers on either
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side of the quantum well rather than to asymmetric doping as in the present samples. This, together with the ability to vary the electron density, should allow us to distinguish the e!ect of an asymmetric electric ?eld from the role of the barrier asymmetry [5,7] in causing the spin–orbit splitting. In the results we have presented here, both mechanisms may contribute. Acknowledgements We are grateful to Sheena Murphy for helpful comments and to Kory Goldammer for growth of the early samples in this study. This work was supported by the National Science Foundation under Grant Nos. DMR-0209371 and DMR-0080054, and through support of the National High Magnetic Field Laboratory. References [1] W. Zawadzki, Landau level spectroscopy, Modern Problems in Condensed Matter Sciences, Vol. 27.1, North-Holland, Amsterdam, 1991, pp. 485 – 479. [2] B.D. McCombe, R.J. Wagner, Intraband magneto-optical studies of semiconductors in the far-infrared, Advances in Electronics and Electron Physics, Vol. 37, Academic Press, New York, 1975, pp. 1–78. [3] B.D. McCombe, R.J. Wagner, G.A. Prinz, Phys. Rev. Lett. 25 (1970) 87. [4] B.D. McCombe, R.J. Wagner, Phys. Rev. B 4 (1971) 1285; Y.-F. Chen, M. Dobrowolska, J.K. Furdyna, Phys. Rev. B 31 (1985) 7989. [5] S. Lamari, J. Appl. Phys. 91 (2002) 1698. [6] S.J. Chung, K.J. Goldammer, S.C. Lindstrom, M.B. Johnson, M.B. Santos, J. Vac. Sci. Technol. B 17 (1999) 1151. [7] P. Pfe!er, W. Zawadzki, J. Supercond. 16 (2003) 351. [8] N. Dai, F. Brown, R.E. Doezema, S.J. Chung, K.J. Goldammer, M.B. Santos, Appl. Phys. Lett. 73 (1998) 3132.
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