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InAlAs single quantum well
Solid State Communications 142 (2007) 393–397 www.elsevier.com/locate/ssc
Pseudospin in Si δ-doped InAlAs/InGaAs/InAlAs single quantum well W.Z. Zhou a,b , Z.M. Huang a , Z.J. Qiu a,e , T. Lin a , L.Y. Shang a , D.L. Li c , H.L. Gao c , L.J. Cui c , Y.P. Zeng c , S.L. Guo a , Y.S. Gui a , N. Dai a , J.H. Chu a,d,∗ a National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China b Physical Science & Technology College, Guangxi University, Nanning, Guangxi 530004, China c Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China d ECNU-SITP Joint Laboratory for Imaging Information, East China Normal University, Shanghai 200062, China e School of Microelectronics, Fudan University, Shanghai 200433, China
Received 4 February 2007; received in revised form 7 March 2007; accepted 8 March 2007 by V. Pellegrini Available online 13 March 2007
Abstract Magneto-transport measurements have been carried out on double/single-barrier-doped In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As quantum well samples from 1.5 to 60 K in an applied magnetic field up to 13 T. Beating Shubnikov–de Haas oscillation is observed for the symmetrically double-barrier-doped sample and demonstrated due to a symmetric state and an antisymmetric state confined in two coupled self-consistent potential wells in the single quantum well. The energy separation between the symmetric and the antisymmetric states for the double-barrier-doped sample is extracted from experimental data, which is consistent with calculation. For the single-barrier-doped sample, only beating related to magneto-intersubband scattering shows up. The pesudospin property of the symmetrically double-barrier-doped single quantum well shows that it is a good candidate for fabricating quantum transistors. c 2007 Elsevier Ltd. All rights reserved.
PACS: 72.20.My; 73.40.Kp; 73.21.-b; 71.15.Pd Keywords: A. Quantum wells; D. Electronic transport; D. Tunnelling
1. Introduction In 1990, the spin field-effect transistor (spin FET) was proposed by Datta and Das [1]. The central theme of spintronics is the active manipulation of spin degrees of freedom in solidstate systems [2]. A prerequisite for the realization of many of these devices is the development of solid-state spin injectors. There have been great efforts to develop the injectors [3–5], but realization of the device has proven to be much more difficult than originally anticipated. A promising analogue system is the bilayer two-dimensional electron gas (2DEG), whose layer index is often called pseudospin [6]. Hu and Heitmann proposed such a bilayer system in which the interference effect of the symmetric ∗ Corresponding author at: National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China. E-mail address: [email protected] (J.H. Chu).
c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.03.014
and antisymmetric states in coupled double quantum wells were utilized [7]. The theory of a single wide quantum well acting as double quantum wells due to charge at the interface was proposed by Abolfath et al. [8]. In this letter, we propose a pseudospin bilayer 2DEG based on a single quantum well. In our system, both sides of the barriers are doped symmetrically with Si, which leads to the formation of two coupled and symmetric self-consistent (SC) potential wells at the hetero-interfaces between the well and the two barrier layers. Shubnikov–de Haas (SdH) measurements [9] enable the extraction of detailed information on the 2DEG including the electron effective mass and the subband occupancies, which can be tuned easily through controlling the doping levels or the gate voltage. With heavy δ-doping, however, electrons may occupy more than one subband in a quantum well, leading to much more complicated SdH oscillation patterns, such as beating effects and phase changes. SdH experiments are extremely useful for the extraction of parameters of 2D systems.
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Fig. 2. The calculated conduction band profile and the electron probability distribution in samples A and B. Fig. 1. The configuration of Si δ-doped In0.52 Al0.48 As/In0.53 Ga0.47 As/ In0.52 Al0.48 As single quantum well: (a) sample A; (b) sample B.
In this letter, we propose a pseudospin system consisting of a symmetric state and an antisymmetric state in a single quantum well δ-doped in both barriers. We demonstrate, by SdH measurement, the formation of a symmetric state and an antisymmetric state due to two self-consistent potential wells in In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As interfaces. The experimental data were unravelled using fast Fourier transform (FFT) analysis, giving parameters including electron densities and effective masses in individual subbands. The energy separation of symmetric and antisymmetric states is also estimated from the FFT analysis for the sample in which both sides of the barrier layers were heavily δ-doped, which is consistent with our calculation. 2. Sample structures and theoretical model In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As quantum wells were grown on a semi-insulating InP substrate using a GEN II molecular beam epitaxy system, and the configurations of sample A and sample B are depicted schematically in Fig. 1. SdH measurements were performed using a direct current (dc) technique in a magnetic field range of 0 to 13 T and a temperature range of 1.5 to 60 K. The samples were cut into 5× 5 mm2 squares and four indium ohmic contacts were made on the samples in van der Pauw geometry, and the magnetic field is applied perpendicularly to the plane of the heterointerface. A general picture of a band potential profile is helpful for an understanding of the proposed pseudospin bilayer 2DEG and the detailed experimental results. We calculate the potential profiles of the two samples by solving the Kohn–Sham
Schr¨odinger equation self-consistently in conjunction with the Poisson equation [10]. Fig. 2 presents the calculated conduction band profiles and wave functions of samples A and B at 1.5 K. In the calculation, the effective potential Veff (z) contains the Hartree potential v H (z), the exchange and correlation potential vxc (z) of Hedin and Lundqvist [11], the background potential profile E b (z), and the strain-induced band-edge shifts [12]. The subband charge densities n i , wave functions φi , energy levels E i , and Fermi energy E F of the interacting inhomogeneous electron gas are well described within the framework of the effective-mass approximation. The details of the calculation procedure have been given elsewhere [13]. In sample A, δ-doping in both barriers gives rise to four self-consistent (SC) potential wells for electrons: two in the In0.52 Al0.48 As barriers and two in the In0.53 Ga0.47 As/In0.52 Al0.48 As interfacial regions, as shown in Fig. 2. The two SC wells in the barriers are due to the doped Si+1 ions after each Si gives up one electron and the electron migrates to and drops into the In0.53 Ga0.47 As quantum well. The two SC wells in the In0.53 Ga0.47 As/In0.52 Al0.48 As interfaces are caused by electron accumulation, since the electrons are attracted by positively charged Si+1 in both barriers. The degenerate ground states in the two SC triangular potential wells in the In0.52 Al0.48 As/In0.53 Ga0.47 As interfaces are strongly coupled to form a symmetric state and an antisymmetric state (see Fig. 2), the two SC potential wells acting as an analogue of the psuedospin bilayer system proposed in Ref. [7]. In sample B, however, asymmetric δ-doping leads to one SC well at the In0.52 Al0.48 As/In0.53 Ga0.47 As interface and one SC well in the doped barrier, as illustrated in Fig. 2. Theoretically, coupling between energy states in an SC well in the interface and a neighboring SC well in the barrier is possible. The coupling is, however, expected to be very weak, since the energy difference
W.Z. Zhou et al. / Solid State Communications 142 (2007) 393–397
Fig. 3. The longitudinal resistance R x x and Hall resistance R x y as functions of an applied magnetic field at 1.5 K for sample A. The inset shows the node.
between the two states is around 100 meV. The calculation of the probability distribution of the electrons shows that almost all the electrons are localized in the In0.53 Ga0.47 As wells (see Fig. 2). The doping densities in both samples were chosen in such a way that there are two energy levels below the Fermi level. Our calculation shows that in sample A both the symmetric state and the antisymmetric state are filled with electrons and their energy separation 10 = E 1as − E 1s equals 3.82 meV (where E 1s is the symmetric state level and E 1as is the antisymmetric state level). The symmetric and antisymmetric states confined in two coupled SC potential wells at the In0.52 Al0.48 As/In0.53 Ga0.47 As interfaces are essential for the operation of the pseudospin devices. 3. Experimental results and discussion We now give experimental evidence for the existence of the symmetric and the antisymmetric states in sample A through a study of beating SdH oscillation. Fig. 3 presents the experimental longitudinal resistance Rx x and the Hall resistance Rx y as functions of the applied magnetic field at 1.5 K for sample A, which shows a clear beating SdH oscillation pattern (see the inset of Fig. 3 for the detail at low magnetic fields). The beating of SdH oscillations could originate from (i) the interaction of SdH oscillations of the first and second subbands; (ii) the Zeeman effect; (iii) a slight occupation of the second subband itself [14]; (iv) the magneto-intersubband scattering (MIS) effect; (v) Rashba spin–orbit splitting due to structural inversion asymmetry; and (vi) symmetric and asymmetric states. The relation between the energy level and the SdH oscillation frequency of a subband takes the form of eh¯ f i , (1) m∗ where E i and f i are the energy position and the SdH oscillation frequency of the ith subband respectively, and m ∗ is the effective mass of electrons in the single quantum well. In sample A, the first and second subbands are too far apart in energy, which results in a large difference between f 1 and f 2 . Thus, the observed beating cannot originate from (i). The possibility of (ii) can be ruled out, since SdH beating due to the Zeeman effect usually occurs in a magnetic semiconductor with a large magnetic-field-dependent g-factor. The fact that E F − Ei =
395
Fig. 4. FFT spectra of the SdH oscillations for sample A at several temperatures. The curves are vertically shifted for clarity. The inset presents the mobility spectra for the sample A at 1.5 K.
in sample A the electron concentration in the second subband is not much less than that in the first subband rules out the possibility of (iii). The Hall electron concentration n H is 5.62 × 1012 cm−2 , which is extracted from the experimental Rx y data. From the experimental conductance at zero magnetic field, the Hall mobility is 2.06 × 104 cm2 /V s. The electron effective mass m ∗ is determined to be 0.049m 0 , using the temperature dependence of the SdH amplitudes, and the value is in good agreement with those given in Refs. [15–18]. Fig. 4 illustrates FFT spectra of the longitudinal resistance Rx x for sample A at several temperatures. Three peaks whose frequencies are 18.98, 47.04 and 48.69 T on the FFT curve are observed at 1.5 K. Due to the k B T broadening of the Landau level, the two peaks at 47.04 and at 48.69 T quickly become overlapped, as the temperature increases gradually (see Fig. 4). Obviously, the beating shown in Fig. 3 is due to the interaction of the two oscillations at the frequencies 47.04 and 48.69 T. MIS enhancement is due to the line-up of the magnetic subband ladders of the first and second subbands, and the SdH oscillation frequency derived from MIS equals the frequency difference between the first and second subbands. The fact that 47.06 6= 48.69 − 18.98 indicates that the beating in Fig. 3 is not associated with MIS effect. In addition, the peaks at 18.98, 47.06, and 48.69 T are all strongly temperaturedependent (see Fig. 4), while the MIS effect peak is insensitive to temperature [19–21]. Thus, (iv) is not the mechanism for the observed beating SdH oscillations. The mobility spectrum (MS) analysis [22] shown in the inset of Fig. 4 indicates that there are only two subbands occupied by electrons in sample A. (Here σ is the conductivity and µ the electron mobility.) Therefore, the frequencies at 47.04 and at 48.69 T are associated with the same subband (the first subband) and the frequency at 18.98 T is due to the second subband. Later we will show that the observed beating SdH oscillations do not originate from Rashba spin–orbit splitting. The electron concentration n and the SdH frequency f are related through [23] n = 2e f / h,
(2)
where h is Planck’s constant and f is the frequency corresponding to the peaks in Fig. 4. At a frequency of 18.98 T,
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Fig. 5. FFT spectra of the SdH oscillations for sample B at several temperatures. The inset shows d2 R x x /dB 2 as a function of magnetic field.
the electron concentration in the second subband is calculated to be 9.2 × 1011 cm−2 . In the first subband, calculated electron concentrations are 2.27 × 1012 cm−2 for a frequency of 47.04 T and 2.35 × 1012 cm−2 for 48.69 T. Thus, the total electron concentration in sample A is 5.64 × 1012 cm−2 , which is close to the value of n H = 5.62 × 1012 cm−2 determined by Hall measurements. On the other hand, if the two adjacent energy levels were associated with the spin-up and the spin-down states of the first subband as in the case of Rashba splitting, the electron concentrations should be calculated using n = e f / h, instead of Eq. (2), due to the lifting of the spin degeneracy. Calculations then give 1.14 × 1012 cm−2 and 1.18 × 1012 cm−2 for the spin-down and spin-up states, respectively. The total electron concentration including the spin-up and spin-down subbands is thus far below the measured value of n H = 5.62 × 1012 cm−2 . This indicates that the frequencies at 47.04 and 48.69 T are not associated with spin-up and spin-down states, which rules out the possibility that the observed beating SdH oscillations are caused by Rashba spin–orbit splitting. Now it becomes quite clear that the possible reason for the observed SdH beating has its origin in the quantum interference of the symmetric and antisymmetric states, which is further evidenced by the following discussion. Using Eq. (1), the energy separation 10 between a symmetric state and an antisymmetric state can be obtained from the FFT spectrum of the SdH oscillations in Fig. 4. Using the SdH oscillation frequencies 47.04 and 48.69 T for the antisymmetric and symmetric states, respectively, we obtain 10 = 3.9 meV. The experimental result is consistent with the calculation. In contrast, we studied SdH oscillations for sample B doped asymmetrically in one barrier. Fig. 5 shows the FFT spectra of measured SdH oscillations for sample B at several temperatures. Three distinct frequencies are observed as three peaks on the FFT curves. The peaks at 6.15 T and 34.906 T correspond to the second and first subbands, respectively. The peak at 41.665 T is the sum frequency of the two subbands and the peak at 69.71 T is the doubling frequency of the first subband at 34.906 T. The intensities of the peaks at 28.85 T and 34.906 T become comparable within 30–40 K, leading to a beating SdH oscillation in this temperature range. In order to see the node clearly, we present d2 Rx x /dB 2 as a function of magnetic field B in the inset of Fig. 5. The peaks at 28.85 T
show weak temperature damping behavior, which is a typical feature of MIS oscillation [19–21]. Thus, the observed pattern in sample B has its origin in the SdH oscillation of the first subband on which an MIS oscillation is superimposed. On sample B, a beating pattern associated with a symmetric state and an antisymmetric state is not observed. Obviously, a symmetric state and an antisymmetric state cannot form in sample B, in which only one barrier is doped. The absence of the symmetric/antisymmetric-related beating SdH oscillation in sample B further evidences that the beating in sample A has its origin in quantum interference between the symmetric and antisymmetric states. 4. Conclusion In summary, SdH measurements have been performed on two heavily δ-doped In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As single quantum well samples in an applied perpendicular magnetic field up to 13 T and in a temperature range from 1.5 to 60 K. A beating pattern of longitudinal resistance has been observed in the double-barrier-doped sample and the beating is caused by the symmetric and antisymmetric states confined in the SC potential wells at the In0.52 Al0.48 As/In0.53 Ga0.47 As interfaces. Such a beating pattern is not observed in the singlebarrier-doped sample. Such a double-barrier-doped single quantum well is an analogue of a bilayer psuedospin system. Acknowledgments The authors would like to acknowledge financial support from the State Key Development Program for Basic Research of China (grant no. 001CB309506) and the National Natural Science Foundation of China (60225004, 60221502, 10334030, and 10374094). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. ˇ c, J. Fabian, S. Das Sarma, Rev. Modern Phys. 76 (2004) 323. I. Zuti´ R. Fiederling, et al., Nature (London) 402 (1999) 787. Y. Ohno, et al., Nature (London) 402 (1999) 790. A.T. Hanbicki, et al., Appl. Phys. Lett. 82 (2003) 4092. A. Lorke, U. Merkt, F. Malcher, G. Weimann, W. Schlapp, Phys. Rev. B 42 (1990) 1321. C.M. Hu, D. Heitmann, Appl. Phys. Lett. 77 (2000) 1475. M. Abolfath, L. Belkhir, N. Nafari, Phys. Rev. B 55 (1997) 10643. L. Shubnikov, W.J. de Haas, Leiden Commun. 207(a) (1930) 3. W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. L. Hedin, B.I. Lundqvist, J. Phys. C 4 (1971) 2064. J.M. Gilp´erez, J.L. S´anchez-Rojas, E. Mu˜noz, E. Calleja, J.P.R. David, M. Reddy, G. Hill, J. S´anchez-Dehesa, J. Appl. Phys. 76 (1994) 5931. Z.M Huang, J.H. Chu, Y.N. Wu, Y. Hou, J.Q. Xue, D.Y. Tang, Phys. Rev. B 72 (2005) 052106. T. Koga, J. Nitta, T. Akazaki, H. Takayanagi, Proceedings of NGS 10, in: IPAP Conf. Series, vol. 2, IPAP, Tokyo, 2001, p. 227. W.P. Hong, G.I. Ng, P.K. Bhattacharya, D. Paviidi, S. Willing, B. Das, J. Appl. Phys. 64 (1988) 1945. T. Akazaki, J. Nitta, H. Takayanagi, T. Enoki, K. Arai, J. Electron. Mater. 25 (1996) 745. T.W. Kim, M. Jung, D.U. Lee, J.H. Kim, K.H. Yoo, J.Y. Lee, S.Y. Ryu, J. Appl. Phys. 82 (1997) 4388.
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[21] T.H. Sander, S.N. Holmes, J.J. Harris, D.K. Maude, J.C. Portal, Phys. Rev. B 58 (1998) 13856. [22] W.A. Beck, J.R. Anderson, J. Appl. Phys. 62 (1987) 541. [23] B. Das, D.C. Miller, S. Datta, R. Reifenberger, W.P. Hong, P.K. Bhattacharya, J. Singh, M. Jaffe, Phys. Rev. B 39 (1989) R1411.
Pseudospin in Si δ-doped InAlAs/InGaAs/InAlAs single quantum well W.Z. Zhou a,b , Z.M. Huang a , Z.J. Qiu a,e , T. Lin a , L.Y. Shang a , D.L. Li c , H.L. Gao c , L.J. Cui c , Y.P. Zeng c , S.L. Guo a , Y.S. Gui a , N. Dai a , J.H. Chu a,d,∗ a National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China b Physical Science & Technology College, Guangxi University, Nanning, Guangxi 530004, China c Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China d ECNU-SITP Joint Laboratory for Imaging Information, East China Normal University, Shanghai 200062, China e School of Microelectronics, Fudan University, Shanghai 200433, China
Received 4 February 2007; received in revised form 7 March 2007; accepted 8 March 2007 by V. Pellegrini Available online 13 March 2007
Abstract Magneto-transport measurements have been carried out on double/single-barrier-doped In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As quantum well samples from 1.5 to 60 K in an applied magnetic field up to 13 T. Beating Shubnikov–de Haas oscillation is observed for the symmetrically double-barrier-doped sample and demonstrated due to a symmetric state and an antisymmetric state confined in two coupled self-consistent potential wells in the single quantum well. The energy separation between the symmetric and the antisymmetric states for the double-barrier-doped sample is extracted from experimental data, which is consistent with calculation. For the single-barrier-doped sample, only beating related to magneto-intersubband scattering shows up. The pesudospin property of the symmetrically double-barrier-doped single quantum well shows that it is a good candidate for fabricating quantum transistors. c 2007 Elsevier Ltd. All rights reserved.
PACS: 72.20.My; 73.40.Kp; 73.21.-b; 71.15.Pd Keywords: A. Quantum wells; D. Electronic transport; D. Tunnelling
1. Introduction In 1990, the spin field-effect transistor (spin FET) was proposed by Datta and Das [1]. The central theme of spintronics is the active manipulation of spin degrees of freedom in solidstate systems [2]. A prerequisite for the realization of many of these devices is the development of solid-state spin injectors. There have been great efforts to develop the injectors [3–5], but realization of the device has proven to be much more difficult than originally anticipated. A promising analogue system is the bilayer two-dimensional electron gas (2DEG), whose layer index is often called pseudospin [6]. Hu and Heitmann proposed such a bilayer system in which the interference effect of the symmetric ∗ Corresponding author at: National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China. E-mail address: [email protected] (J.H. Chu).
c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.03.014
and antisymmetric states in coupled double quantum wells were utilized [7]. The theory of a single wide quantum well acting as double quantum wells due to charge at the interface was proposed by Abolfath et al. [8]. In this letter, we propose a pseudospin bilayer 2DEG based on a single quantum well. In our system, both sides of the barriers are doped symmetrically with Si, which leads to the formation of two coupled and symmetric self-consistent (SC) potential wells at the hetero-interfaces between the well and the two barrier layers. Shubnikov–de Haas (SdH) measurements [9] enable the extraction of detailed information on the 2DEG including the electron effective mass and the subband occupancies, which can be tuned easily through controlling the doping levels or the gate voltage. With heavy δ-doping, however, electrons may occupy more than one subband in a quantum well, leading to much more complicated SdH oscillation patterns, such as beating effects and phase changes. SdH experiments are extremely useful for the extraction of parameters of 2D systems.
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Fig. 2. The calculated conduction band profile and the electron probability distribution in samples A and B. Fig. 1. The configuration of Si δ-doped In0.52 Al0.48 As/In0.53 Ga0.47 As/ In0.52 Al0.48 As single quantum well: (a) sample A; (b) sample B.
In this letter, we propose a pseudospin system consisting of a symmetric state and an antisymmetric state in a single quantum well δ-doped in both barriers. We demonstrate, by SdH measurement, the formation of a symmetric state and an antisymmetric state due to two self-consistent potential wells in In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As interfaces. The experimental data were unravelled using fast Fourier transform (FFT) analysis, giving parameters including electron densities and effective masses in individual subbands. The energy separation of symmetric and antisymmetric states is also estimated from the FFT analysis for the sample in which both sides of the barrier layers were heavily δ-doped, which is consistent with our calculation. 2. Sample structures and theoretical model In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As quantum wells were grown on a semi-insulating InP substrate using a GEN II molecular beam epitaxy system, and the configurations of sample A and sample B are depicted schematically in Fig. 1. SdH measurements were performed using a direct current (dc) technique in a magnetic field range of 0 to 13 T and a temperature range of 1.5 to 60 K. The samples were cut into 5× 5 mm2 squares and four indium ohmic contacts were made on the samples in van der Pauw geometry, and the magnetic field is applied perpendicularly to the plane of the heterointerface. A general picture of a band potential profile is helpful for an understanding of the proposed pseudospin bilayer 2DEG and the detailed experimental results. We calculate the potential profiles of the two samples by solving the Kohn–Sham
Schr¨odinger equation self-consistently in conjunction with the Poisson equation [10]. Fig. 2 presents the calculated conduction band profiles and wave functions of samples A and B at 1.5 K. In the calculation, the effective potential Veff (z) contains the Hartree potential v H (z), the exchange and correlation potential vxc (z) of Hedin and Lundqvist [11], the background potential profile E b (z), and the strain-induced band-edge shifts [12]. The subband charge densities n i , wave functions φi , energy levels E i , and Fermi energy E F of the interacting inhomogeneous electron gas are well described within the framework of the effective-mass approximation. The details of the calculation procedure have been given elsewhere [13]. In sample A, δ-doping in both barriers gives rise to four self-consistent (SC) potential wells for electrons: two in the In0.52 Al0.48 As barriers and two in the In0.53 Ga0.47 As/In0.52 Al0.48 As interfacial regions, as shown in Fig. 2. The two SC wells in the barriers are due to the doped Si+1 ions after each Si gives up one electron and the electron migrates to and drops into the In0.53 Ga0.47 As quantum well. The two SC wells in the In0.53 Ga0.47 As/In0.52 Al0.48 As interfaces are caused by electron accumulation, since the electrons are attracted by positively charged Si+1 in both barriers. The degenerate ground states in the two SC triangular potential wells in the In0.52 Al0.48 As/In0.53 Ga0.47 As interfaces are strongly coupled to form a symmetric state and an antisymmetric state (see Fig. 2), the two SC potential wells acting as an analogue of the psuedospin bilayer system proposed in Ref. [7]. In sample B, however, asymmetric δ-doping leads to one SC well at the In0.52 Al0.48 As/In0.53 Ga0.47 As interface and one SC well in the doped barrier, as illustrated in Fig. 2. Theoretically, coupling between energy states in an SC well in the interface and a neighboring SC well in the barrier is possible. The coupling is, however, expected to be very weak, since the energy difference
W.Z. Zhou et al. / Solid State Communications 142 (2007) 393–397
Fig. 3. The longitudinal resistance R x x and Hall resistance R x y as functions of an applied magnetic field at 1.5 K for sample A. The inset shows the node.
between the two states is around 100 meV. The calculation of the probability distribution of the electrons shows that almost all the electrons are localized in the In0.53 Ga0.47 As wells (see Fig. 2). The doping densities in both samples were chosen in such a way that there are two energy levels below the Fermi level. Our calculation shows that in sample A both the symmetric state and the antisymmetric state are filled with electrons and their energy separation 10 = E 1as − E 1s equals 3.82 meV (where E 1s is the symmetric state level and E 1as is the antisymmetric state level). The symmetric and antisymmetric states confined in two coupled SC potential wells at the In0.52 Al0.48 As/In0.53 Ga0.47 As interfaces are essential for the operation of the pseudospin devices. 3. Experimental results and discussion We now give experimental evidence for the existence of the symmetric and the antisymmetric states in sample A through a study of beating SdH oscillation. Fig. 3 presents the experimental longitudinal resistance Rx x and the Hall resistance Rx y as functions of the applied magnetic field at 1.5 K for sample A, which shows a clear beating SdH oscillation pattern (see the inset of Fig. 3 for the detail at low magnetic fields). The beating of SdH oscillations could originate from (i) the interaction of SdH oscillations of the first and second subbands; (ii) the Zeeman effect; (iii) a slight occupation of the second subband itself [14]; (iv) the magneto-intersubband scattering (MIS) effect; (v) Rashba spin–orbit splitting due to structural inversion asymmetry; and (vi) symmetric and asymmetric states. The relation between the energy level and the SdH oscillation frequency of a subband takes the form of eh¯ f i , (1) m∗ where E i and f i are the energy position and the SdH oscillation frequency of the ith subband respectively, and m ∗ is the effective mass of electrons in the single quantum well. In sample A, the first and second subbands are too far apart in energy, which results in a large difference between f 1 and f 2 . Thus, the observed beating cannot originate from (i). The possibility of (ii) can be ruled out, since SdH beating due to the Zeeman effect usually occurs in a magnetic semiconductor with a large magnetic-field-dependent g-factor. The fact that E F − Ei =
395
Fig. 4. FFT spectra of the SdH oscillations for sample A at several temperatures. The curves are vertically shifted for clarity. The inset presents the mobility spectra for the sample A at 1.5 K.
in sample A the electron concentration in the second subband is not much less than that in the first subband rules out the possibility of (iii). The Hall electron concentration n H is 5.62 × 1012 cm−2 , which is extracted from the experimental Rx y data. From the experimental conductance at zero magnetic field, the Hall mobility is 2.06 × 104 cm2 /V s. The electron effective mass m ∗ is determined to be 0.049m 0 , using the temperature dependence of the SdH amplitudes, and the value is in good agreement with those given in Refs. [15–18]. Fig. 4 illustrates FFT spectra of the longitudinal resistance Rx x for sample A at several temperatures. Three peaks whose frequencies are 18.98, 47.04 and 48.69 T on the FFT curve are observed at 1.5 K. Due to the k B T broadening of the Landau level, the two peaks at 47.04 and at 48.69 T quickly become overlapped, as the temperature increases gradually (see Fig. 4). Obviously, the beating shown in Fig. 3 is due to the interaction of the two oscillations at the frequencies 47.04 and 48.69 T. MIS enhancement is due to the line-up of the magnetic subband ladders of the first and second subbands, and the SdH oscillation frequency derived from MIS equals the frequency difference between the first and second subbands. The fact that 47.06 6= 48.69 − 18.98 indicates that the beating in Fig. 3 is not associated with MIS effect. In addition, the peaks at 18.98, 47.06, and 48.69 T are all strongly temperaturedependent (see Fig. 4), while the MIS effect peak is insensitive to temperature [19–21]. Thus, (iv) is not the mechanism for the observed beating SdH oscillations. The mobility spectrum (MS) analysis [22] shown in the inset of Fig. 4 indicates that there are only two subbands occupied by electrons in sample A. (Here σ is the conductivity and µ the electron mobility.) Therefore, the frequencies at 47.04 and at 48.69 T are associated with the same subband (the first subband) and the frequency at 18.98 T is due to the second subband. Later we will show that the observed beating SdH oscillations do not originate from Rashba spin–orbit splitting. The electron concentration n and the SdH frequency f are related through [23] n = 2e f / h,
(2)
where h is Planck’s constant and f is the frequency corresponding to the peaks in Fig. 4. At a frequency of 18.98 T,
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Fig. 5. FFT spectra of the SdH oscillations for sample B at several temperatures. The inset shows d2 R x x /dB 2 as a function of magnetic field.
the electron concentration in the second subband is calculated to be 9.2 × 1011 cm−2 . In the first subband, calculated electron concentrations are 2.27 × 1012 cm−2 for a frequency of 47.04 T and 2.35 × 1012 cm−2 for 48.69 T. Thus, the total electron concentration in sample A is 5.64 × 1012 cm−2 , which is close to the value of n H = 5.62 × 1012 cm−2 determined by Hall measurements. On the other hand, if the two adjacent energy levels were associated with the spin-up and the spin-down states of the first subband as in the case of Rashba splitting, the electron concentrations should be calculated using n = e f / h, instead of Eq. (2), due to the lifting of the spin degeneracy. Calculations then give 1.14 × 1012 cm−2 and 1.18 × 1012 cm−2 for the spin-down and spin-up states, respectively. The total electron concentration including the spin-up and spin-down subbands is thus far below the measured value of n H = 5.62 × 1012 cm−2 . This indicates that the frequencies at 47.04 and 48.69 T are not associated with spin-up and spin-down states, which rules out the possibility that the observed beating SdH oscillations are caused by Rashba spin–orbit splitting. Now it becomes quite clear that the possible reason for the observed SdH beating has its origin in the quantum interference of the symmetric and antisymmetric states, which is further evidenced by the following discussion. Using Eq. (1), the energy separation 10 between a symmetric state and an antisymmetric state can be obtained from the FFT spectrum of the SdH oscillations in Fig. 4. Using the SdH oscillation frequencies 47.04 and 48.69 T for the antisymmetric and symmetric states, respectively, we obtain 10 = 3.9 meV. The experimental result is consistent with the calculation. In contrast, we studied SdH oscillations for sample B doped asymmetrically in one barrier. Fig. 5 shows the FFT spectra of measured SdH oscillations for sample B at several temperatures. Three distinct frequencies are observed as three peaks on the FFT curves. The peaks at 6.15 T and 34.906 T correspond to the second and first subbands, respectively. The peak at 41.665 T is the sum frequency of the two subbands and the peak at 69.71 T is the doubling frequency of the first subband at 34.906 T. The intensities of the peaks at 28.85 T and 34.906 T become comparable within 30–40 K, leading to a beating SdH oscillation in this temperature range. In order to see the node clearly, we present d2 Rx x /dB 2 as a function of magnetic field B in the inset of Fig. 5. The peaks at 28.85 T
show weak temperature damping behavior, which is a typical feature of MIS oscillation [19–21]. Thus, the observed pattern in sample B has its origin in the SdH oscillation of the first subband on which an MIS oscillation is superimposed. On sample B, a beating pattern associated with a symmetric state and an antisymmetric state is not observed. Obviously, a symmetric state and an antisymmetric state cannot form in sample B, in which only one barrier is doped. The absence of the symmetric/antisymmetric-related beating SdH oscillation in sample B further evidences that the beating in sample A has its origin in quantum interference between the symmetric and antisymmetric states. 4. Conclusion In summary, SdH measurements have been performed on two heavily δ-doped In0.52 Al0.48 As/In0.53 Ga0.47 As/In0.52 Al0.48 As single quantum well samples in an applied perpendicular magnetic field up to 13 T and in a temperature range from 1.5 to 60 K. A beating pattern of longitudinal resistance has been observed in the double-barrier-doped sample and the beating is caused by the symmetric and antisymmetric states confined in the SC potential wells at the In0.52 Al0.48 As/In0.53 Ga0.47 As interfaces. Such a beating pattern is not observed in the singlebarrier-doped sample. Such a double-barrier-doped single quantum well is an analogue of a bilayer psuedospin system. Acknowledgments The authors would like to acknowledge financial support from the State Key Development Program for Basic Research of China (grant no. 001CB309506) and the National Natural Science Foundation of China (60225004, 60221502, 10334030, and 10374094). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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