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Studies of the quantum Hall to quantum Hall insulator transition in InSb-based 2DESs
Physica E 6 (2000) 293–296
www.elsevier.nl/locate/physe
Studies of the quantum Hall to quantum Hall insulator transition in InSb-based 2DESs S.Q. Murphy ∗ , J.L. Hicks, W.K. Liu, S.J. Chung, K.J. Goldammer, M.B. Santos Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA
Abstract The temperature dependence of xx is studied in the vicinity of the quantum Hall to quantum Hall insulator transition ( = 1 →√ 0) in InSb=InAlSb based 2DESs. √ xx displays a symmetric temperature dependence about the transition with xx ˙ e−
T0 =T
on the QH side and xx ˙ e+ − 1
T0 =T
on the insulating side. A plot of 1=T0 for successive displays √ power-law
divergence with 1=T0 ˙ | − c | , with = 2:2 ± 0:3. This critical behavior in addition to the xx ˙ e− 1=T behavior expected of the quantum transport regime con rms that the QH=QHI transition is indeed a good quantum phase transition. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Transport; Insulator transition; InSb
The classi cation of the integer quantum Hall (QH) to quantum Hall insulator (QHI) transition ( = 1 → 0) as a quantum phase transition has been called into question as the result of recent transport and microwave experiments [1,2]. The inference from these GaAs and InAs 2DES transport studies where xx ˙ e−1=( T + ) , is that the critical region of the transition stays of nite width as T → 0, hence the transition cannot be a true quantum phase transition. Disturbingly, the demonstrated similarity between the QH=QHI transition and the localization=delocalization transitions between quantum Hall (QH) plateaus [3]
∗
Corresponding author. (We use the to represent the exponent more commonly known as z to avoid confusion with the lling factor). 1
would suggest that these latter, well-understood transitions are not quantum phase transitions either. Motivated by these experiments, we have undertaken a study of the QH=QHI transition in InSb=InAlSb based 2DESs grown via molecular beam epitaxy [4]. InSb is unique amongst the III–V family of semiconductors with the smallest energy gap (236 meV at 4.2 K), the smallest electron effective mass (m∗ = 0:0139m0 ) and the largest Lande g-factor (−51). These extreme parameters make the InSb=InAlSb system an interesting alternative to those systems in which the quantum Hall e ect has previously been studied. The small e ective mass and large g factor translate to extremely large cyclotron (!C ˙ 1=m∗ ) and spin gaps (s ˙ g); not unexpectedly, this results in an integer QHE which persists at high temperature (¿ 25 K) and high excitation currents
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 1 5 7 - 5
294
S.Q. Murphy et al. / Physica E 6 (2000) 293–296
(¿ 200 A). More subtly, these large gaps a ect the nature of the quantum Hall transport itself. Whereas in most 2DESs the quantum Hall transport displays Arrhenius behavior xx ˙ e−(=T ) , where 2 is √ the − T0 =T e cyclotron or spin gap, in our samples xx ˙ p at all but the highest temperatures. This T0 =T behavior is the signature of the Coulomb gap (GG) [5] and has been observed in all of our InSb samples. Because the spin and cyclotron gaps in InSb-based 2DESs are so large (¿ 100 K), Arrhenius behavior is highly suppressed and the transport at integer is dominated by CG behavior over virtually our entire experimental temperature range (0:5–20 K). Since CG behavior is inherently an interaction e ect, the parameter T0 conveniently provides an e ective temperature proportional to the scale of the electron– electron interaction, kB T0 ˙ e2 =, where is the dielectric constant of InSb ( = 17:70 ), and is the electronic localization length. Due to this connection, we have made extensive use of CG behavior to study various localization=delocalization transitions in our InSb samples. As a measure of the method’s usefulness, we have tracked the CG behavior of xx as a function of , sweeping through the localization=delocalization transition between QH states ( = 1 → 32 and = 2 → 52 ) for a number of InSb samples. While the assignment of CG behavior at integer is unambiguous, we can only assume this temperature dependence closer to the transition where T0 should vanish. Reassuringly, it is observed that 1=T0 diverges as is swept towards c , the critical lling factor of the interleaving delocalized states, with characteristic power-law behavior, | − c |− . This con rms that 1=T0 ˙ , as the correlation length should diverge as the delocalized states are approached. Indeed, our resultant measurements of in several samples ( ranges from 2:1 to 2:7 with an uncertainty of ±0:35 [6]) are in excellent agreement with theoretical predictions ( = 73 ) [7–12] and previous experimental measurements [13,14]. The method continues to be useful at the QH=QHI transition. Fig. 1a displays xx versus T for a series of applied magnetic elds spanning the = 1 → 0 transition for a sample with ns = 7:25 × 1010 =cm2 and a = 6300 cm2 =V s. 2 The transition is close 2 While samples with mobilities as high as 300,000 cm2 =V s are available, we have focussed our studies on low mobility samples.
to a magnetic eld of 5.06 T (c = 0:595) where xx should intersect the T = 0 axis with p zero slope. Fig. 1=T . As can be 1b displays the same data versus p seen, 1=T behavior can be extrapolated across the transition, persisting with a sign change on the QH insulator side. Surprisingly such a symmetry has also been observed at the B = 0 metal=insulator transition in low mobilty 2DESs such as SiMOSFETs √ and − T0 =T is SiGe heterostructures [15 –17] where ˙ e √ observed in the metallic phase and ˙ e+ T0 =T is observed in the insulating phase. The extracted 1=T0 s from both sides of the transition are displayed in Fig. 2a with the solid and open squares representing data from the QH and QHI sides of the transition, respectively. The 1=T0 s from both sides of the transition fall on the same divergent power-law with the same exponent ( = 2:37 ± 0:15) as observed in the higher lling factor transitions, in good agreement with the view that the QH=QHI transition is in the same universality class as the localization=delocalization transitions [3]. Traditionally scaling exponents have also been extracted in a number of studies which scale the data by sliding sets at xed (or ns in the case of B = 0) to form a common scaling function. This is done in Fig. 2b for our data for all 16 magnetic elds with 0:5 ¡ T ¡ 2 K where the value of which results in the best t is closer to 3. The scaling function method can be problematic since the scaling function is unknown. It is easy to weigh higher-temperature points which are outside the critical region too heavily and hence extract an erroneous value of the scaling exponent. While the power-law behavior of 1=T0 and the strong scaling behavior are suggestive, they are not sucient evidence that this is a good quantum phase transition; even the data which suggests a nite size critical region as T → 0 collapses onto a common scaling function. 3 Rather the evidence that we have a good quantum phase transition lies in the temperature dependence of xx . A recent preprint by Shimshoni [18] examines the crossover from classical to quantum transport at the QH=QHI transition. This paper explains the 1=( T + ) behavior observed in the nite-sized critical regime experiments [1,2] as 3
Ref. [19] of Shahar et al. [1].
S.Q. Murphy et al. / Physica E 6 (2000) 293–296
295
Fig. 1. (a) Displays xx versus T on a log-linear scale for 16 magnetic elds (3.97– 6.78 T) in the vicinity of the QH=QHI transition (the transition eld is close to 5.06 T). The sample has an ns = 0:725 × 1011 =cm2 and a = 6300 cm2 =V s; (b) displays the same data versus √ 1= T . The Coulomb gap behavior seen on the QH side of the transition at low elds can be extrapolated through the transition to the insulating side with a change of sign.
Fig. 2. (a) displays the 1=T00 s extracted from the ts to Fig. 1b versus | − c | on a log–log scale. The solid symbols represent data from the QH side of the transition and the open symbols that from the insulating side. The graph on the right displays the magnetic eld sweeps for T ¡ 2 K scaled onto a common scaling function. The exponent which results in the best data reduction is = 3.
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S.Q. Murphy et al. / Physica E 6 (2000) 293–296
a high-temperature transport regime which is dominated by hopping between nearest-neighbor QH puddles. According to Shimshoni, the quantum regime √ is indicated by variable range hopping (i.e. 1= T behavior) which occurs when the size of the QH puddles becomes smaller than the dephasing length. √ Hence our observation that xx ˙ e− 1=T implies that we are in the the requisite low-temperature limit to see quantum behavior. In the model the crossover between the classical and quantum regimes moves to lower temperatures as the critical point is approached. While we do not see a clear crossover to classical behavior, we do observe that the temperature range √ over which we can credibly assign 1= T behavior shrinks towards lower temperatures as we approach the transition. Suggestions of such a crossover can be seen in a recent preprint reporting QH transitions studies in p-SiGe 2DESs by Coleridge et al. [19]. In conclusion, the observation of strong scaling behavior in the low-temperature limit (i.e. the quantum regime) suggests that the QH=QHI transition is indeed a good quantum phase transition. And the symmetry in the temperature dependence is a tantalizing suggestion that the transition has a commonality with the transition at B = 0 for low-mobility samples. Acknowledgements We acknowledge the support of the Sloan Foundation (SQM), and the NSF through grants DMR-9624699 and DMR-9631709.
References [1] D. Shahar, M. Hilke, C.C. Li, D.C. Tsui, S.L. Sondhi, J.E. Cunningham, M. Razeghi, Solid State Commun. 107 (1998) 19. [2] N.Q. Balaban, U. Meirav, I. Bar-Joseph, Phys. Rev. Lett. 81 (1998) 4967. [3] D. Shahar, D.C. Tsui, M. Shayegan, E. Shimshoni, S.L. Sondhi, Phys. Rev. Lett. 79 (1997) 479. [4] K.J. Goldammer, W.K. Liu, G.A. Khodaparast, S.C. Lindstrom, M.B. Johnson, R.E. Doezema, M.B. Santos, J. Vac. Sci. Technol. B 16 (1998) 1367. [5] A.L. Efros, B.I. Shklovskii, J. Phys. C 8 (1975) L49. [6] S.Q. Murphy, J.L. Hicks, S. Raymond, E. Waters, W.K. Liu, S. Chung, K.J. Goldammer, M.B. Santos, in preparation. [7] G.V. Mil’nikov, I.M. Sokolov, JETP Lett. 48 (1988) 536. [8] J.T. Chalker, P.D. Coddington, J. Phys. C 21 (1988) 2655. [9] Bodo Huckestein, Bernhard Kramer, Phys. Rev. Lett. 64 (1990) 1437. [10] Y. Huo, R.N. Bhatt, Phys. Rev. Lett. 68 (1992) 1375. [11] Dung-Hai Lee, Ziqiang Wang, Steven Kivelson, Phys. Rev. Lett. 70 (1993) 4130. [12] Hui Lin Zhao, Shechao Feng, Phys. Rev. Lett. 70 (1993) 4134. [13] H.P. Wei, D.C. Tsui, M.A. Paalanen, A.M.M. Pruisken, Phys. Rev. Lett. 61 (1988) 1294. [14] S. Koch, R.J. Haug, K.v. Klitzing, K. Ploog, Phys. Rev. Lett. 67 (1991) 883. [15] S.V. Kravchenko, Whitney Mason, G.E. Bowker, J.E. Furneaux, V.M. Pudalov, M. D’Iorio, Phys. Rev. B 51 (1995) 7038. [16] Dragana Popovic, A.B. Fowler, S. Washburn, Phys. Rev. Lett. 59 (1997) 1543. [17] P.T. Coleridge, R.L. Williams, Y. Feng, P. Zawadzki, Phys. Rev. B 56 (1997) R12 764. [18] Efrat Shimshoni, cond-mat=9904145. [19] P.T. Coleridge, P. Zawaszki, cond-mat=9903246.
www.elsevier.nl/locate/physe
Studies of the quantum Hall to quantum Hall insulator transition in InSb-based 2DESs S.Q. Murphy ∗ , J.L. Hicks, W.K. Liu, S.J. Chung, K.J. Goldammer, M.B. Santos Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA
Abstract The temperature dependence of xx is studied in the vicinity of the quantum Hall to quantum Hall insulator transition ( = 1 →√ 0) in InSb=InAlSb based 2DESs. √ xx displays a symmetric temperature dependence about the transition with xx ˙ e−
T0 =T
on the QH side and xx ˙ e+ − 1
T0 =T
on the insulating side. A plot of 1=T0 for successive displays √ power-law
divergence with 1=T0 ˙ | − c | , with = 2:2 ± 0:3. This critical behavior in addition to the xx ˙ e− 1=T behavior expected of the quantum transport regime con rms that the QH=QHI transition is indeed a good quantum phase transition. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Transport; Insulator transition; InSb
The classi cation of the integer quantum Hall (QH) to quantum Hall insulator (QHI) transition ( = 1 → 0) as a quantum phase transition has been called into question as the result of recent transport and microwave experiments [1,2]. The inference from these GaAs and InAs 2DES transport studies where xx ˙ e−1=( T + ) , is that the critical region of the transition stays of nite width as T → 0, hence the transition cannot be a true quantum phase transition. Disturbingly, the demonstrated similarity between the QH=QHI transition and the localization=delocalization transitions between quantum Hall (QH) plateaus [3]
∗
Corresponding author. (We use the to represent the exponent more commonly known as z to avoid confusion with the lling factor). 1
would suggest that these latter, well-understood transitions are not quantum phase transitions either. Motivated by these experiments, we have undertaken a study of the QH=QHI transition in InSb=InAlSb based 2DESs grown via molecular beam epitaxy [4]. InSb is unique amongst the III–V family of semiconductors with the smallest energy gap (236 meV at 4.2 K), the smallest electron effective mass (m∗ = 0:0139m0 ) and the largest Lande g-factor (−51). These extreme parameters make the InSb=InAlSb system an interesting alternative to those systems in which the quantum Hall e ect has previously been studied. The small e ective mass and large g factor translate to extremely large cyclotron (!C ˙ 1=m∗ ) and spin gaps (s ˙ g); not unexpectedly, this results in an integer QHE which persists at high temperature (¿ 25 K) and high excitation currents
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 1 5 7 - 5
294
S.Q. Murphy et al. / Physica E 6 (2000) 293–296
(¿ 200 A). More subtly, these large gaps a ect the nature of the quantum Hall transport itself. Whereas in most 2DESs the quantum Hall transport displays Arrhenius behavior xx ˙ e−(=T ) , where 2 is √ the − T0 =T e cyclotron or spin gap, in our samples xx ˙ p at all but the highest temperatures. This T0 =T behavior is the signature of the Coulomb gap (GG) [5] and has been observed in all of our InSb samples. Because the spin and cyclotron gaps in InSb-based 2DESs are so large (¿ 100 K), Arrhenius behavior is highly suppressed and the transport at integer is dominated by CG behavior over virtually our entire experimental temperature range (0:5–20 K). Since CG behavior is inherently an interaction e ect, the parameter T0 conveniently provides an e ective temperature proportional to the scale of the electron– electron interaction, kB T0 ˙ e2 =, where is the dielectric constant of InSb ( = 17:70 ), and is the electronic localization length. Due to this connection, we have made extensive use of CG behavior to study various localization=delocalization transitions in our InSb samples. As a measure of the method’s usefulness, we have tracked the CG behavior of xx as a function of , sweeping through the localization=delocalization transition between QH states ( = 1 → 32 and = 2 → 52 ) for a number of InSb samples. While the assignment of CG behavior at integer is unambiguous, we can only assume this temperature dependence closer to the transition where T0 should vanish. Reassuringly, it is observed that 1=T0 diverges as is swept towards c , the critical lling factor of the interleaving delocalized states, with characteristic power-law behavior, | − c |− . This con rms that 1=T0 ˙ , as the correlation length should diverge as the delocalized states are approached. Indeed, our resultant measurements of in several samples ( ranges from 2:1 to 2:7 with an uncertainty of ±0:35 [6]) are in excellent agreement with theoretical predictions ( = 73 ) [7–12] and previous experimental measurements [13,14]. The method continues to be useful at the QH=QHI transition. Fig. 1a displays xx versus T for a series of applied magnetic elds spanning the = 1 → 0 transition for a sample with ns = 7:25 × 1010 =cm2 and a = 6300 cm2 =V s. 2 The transition is close 2 While samples with mobilities as high as 300,000 cm2 =V s are available, we have focussed our studies on low mobility samples.
to a magnetic eld of 5.06 T (c = 0:595) where xx should intersect the T = 0 axis with p zero slope. Fig. 1=T . As can be 1b displays the same data versus p seen, 1=T behavior can be extrapolated across the transition, persisting with a sign change on the QH insulator side. Surprisingly such a symmetry has also been observed at the B = 0 metal=insulator transition in low mobilty 2DESs such as SiMOSFETs √ and − T0 =T is SiGe heterostructures [15 –17] where ˙ e √ observed in the metallic phase and ˙ e+ T0 =T is observed in the insulating phase. The extracted 1=T0 s from both sides of the transition are displayed in Fig. 2a with the solid and open squares representing data from the QH and QHI sides of the transition, respectively. The 1=T0 s from both sides of the transition fall on the same divergent power-law with the same exponent ( = 2:37 ± 0:15) as observed in the higher lling factor transitions, in good agreement with the view that the QH=QHI transition is in the same universality class as the localization=delocalization transitions [3]. Traditionally scaling exponents have also been extracted in a number of studies which scale the data by sliding sets at xed (or ns in the case of B = 0) to form a common scaling function. This is done in Fig. 2b for our data for all 16 magnetic elds with 0:5 ¡ T ¡ 2 K where the value of which results in the best t is closer to 3. The scaling function method can be problematic since the scaling function is unknown. It is easy to weigh higher-temperature points which are outside the critical region too heavily and hence extract an erroneous value of the scaling exponent. While the power-law behavior of 1=T0 and the strong scaling behavior are suggestive, they are not sucient evidence that this is a good quantum phase transition; even the data which suggests a nite size critical region as T → 0 collapses onto a common scaling function. 3 Rather the evidence that we have a good quantum phase transition lies in the temperature dependence of xx . A recent preprint by Shimshoni [18] examines the crossover from classical to quantum transport at the QH=QHI transition. This paper explains the 1=( T + ) behavior observed in the nite-sized critical regime experiments [1,2] as 3
Ref. [19] of Shahar et al. [1].
S.Q. Murphy et al. / Physica E 6 (2000) 293–296
295
Fig. 1. (a) Displays xx versus T on a log-linear scale for 16 magnetic elds (3.97– 6.78 T) in the vicinity of the QH=QHI transition (the transition eld is close to 5.06 T). The sample has an ns = 0:725 × 1011 =cm2 and a = 6300 cm2 =V s; (b) displays the same data versus √ 1= T . The Coulomb gap behavior seen on the QH side of the transition at low elds can be extrapolated through the transition to the insulating side with a change of sign.
Fig. 2. (a) displays the 1=T00 s extracted from the ts to Fig. 1b versus | − c | on a log–log scale. The solid symbols represent data from the QH side of the transition and the open symbols that from the insulating side. The graph on the right displays the magnetic eld sweeps for T ¡ 2 K scaled onto a common scaling function. The exponent which results in the best data reduction is = 3.
296
S.Q. Murphy et al. / Physica E 6 (2000) 293–296
a high-temperature transport regime which is dominated by hopping between nearest-neighbor QH puddles. According to Shimshoni, the quantum regime √ is indicated by variable range hopping (i.e. 1= T behavior) which occurs when the size of the QH puddles becomes smaller than the dephasing length. √ Hence our observation that xx ˙ e− 1=T implies that we are in the the requisite low-temperature limit to see quantum behavior. In the model the crossover between the classical and quantum regimes moves to lower temperatures as the critical point is approached. While we do not see a clear crossover to classical behavior, we do observe that the temperature range √ over which we can credibly assign 1= T behavior shrinks towards lower temperatures as we approach the transition. Suggestions of such a crossover can be seen in a recent preprint reporting QH transitions studies in p-SiGe 2DESs by Coleridge et al. [19]. In conclusion, the observation of strong scaling behavior in the low-temperature limit (i.e. the quantum regime) suggests that the QH=QHI transition is indeed a good quantum phase transition. And the symmetry in the temperature dependence is a tantalizing suggestion that the transition has a commonality with the transition at B = 0 for low-mobility samples. Acknowledgements We acknowledge the support of the Sloan Foundation (SQM), and the NSF through grants DMR-9624699 and DMR-9631709.
References [1] D. Shahar, M. Hilke, C.C. Li, D.C. Tsui, S.L. Sondhi, J.E. Cunningham, M. Razeghi, Solid State Commun. 107 (1998) 19. [2] N.Q. Balaban, U. Meirav, I. Bar-Joseph, Phys. Rev. Lett. 81 (1998) 4967. [3] D. Shahar, D.C. Tsui, M. Shayegan, E. Shimshoni, S.L. Sondhi, Phys. Rev. Lett. 79 (1997) 479. [4] K.J. Goldammer, W.K. Liu, G.A. Khodaparast, S.C. Lindstrom, M.B. Johnson, R.E. Doezema, M.B. Santos, J. Vac. Sci. Technol. B 16 (1998) 1367. [5] A.L. Efros, B.I. Shklovskii, J. Phys. C 8 (1975) L49. [6] S.Q. Murphy, J.L. Hicks, S. Raymond, E. Waters, W.K. Liu, S. Chung, K.J. Goldammer, M.B. Santos, in preparation. [7] G.V. Mil’nikov, I.M. Sokolov, JETP Lett. 48 (1988) 536. [8] J.T. Chalker, P.D. Coddington, J. Phys. C 21 (1988) 2655. [9] Bodo Huckestein, Bernhard Kramer, Phys. Rev. Lett. 64 (1990) 1437. [10] Y. Huo, R.N. Bhatt, Phys. Rev. Lett. 68 (1992) 1375. [11] Dung-Hai Lee, Ziqiang Wang, Steven Kivelson, Phys. Rev. Lett. 70 (1993) 4130. [12] Hui Lin Zhao, Shechao Feng, Phys. Rev. Lett. 70 (1993) 4134. [13] H.P. Wei, D.C. Tsui, M.A. Paalanen, A.M.M. Pruisken, Phys. Rev. Lett. 61 (1988) 1294. [14] S. Koch, R.J. Haug, K.v. Klitzing, K. Ploog, Phys. Rev. Lett. 67 (1991) 883. [15] S.V. Kravchenko, Whitney Mason, G.E. Bowker, J.E. Furneaux, V.M. Pudalov, M. D’Iorio, Phys. Rev. B 51 (1995) 7038. [16] Dragana Popovic, A.B. Fowler, S. Washburn, Phys. Rev. Lett. 59 (1997) 1543. [17] P.T. Coleridge, R.L. Williams, Y. Feng, P. Zawadzki, Phys. Rev. B 56 (1997) R12 764. [18] Efrat Shimshoni, cond-mat=9904145. [19] P.T. Coleridge, P. Zawaszki, cond-mat=9903246.