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Breakdown of the integer quantum Hall effect studied by Corbino discs
Physica B 249—251 (1998) 93—96
Breakdown of the integer quantum Hall effect studied by Corbino discs Masahide Yokoi!, Tohru Okamoto!, Shinji Kawaji!,*, Takayuki Goto", Tetsuro Fukase" ! Department of Physics, Gakushuin University, Mejiro, Tokyo 171, Japan " Institute for Materials Research, Tohoku University, Katahira, Sendai 980-77, Japan
Abstract Breakdown of non-conductive state of Corbino discs at integer filling factors of Landau levels at high source-drain voltages has been measured for samples fabricated from GaAs/Al Ga As heterostructures. Critical breakdown 0.3 0.7 electric fields at the filling factors of 1, 2, 4 and 6 are in agreement with the critical breakdown Hall electric fields in the quantized Hall plateaus with these quantum numbers measured in butterfly-type Hall bars. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum Hall effect; Breakdown of quantum Hall effect; Corbino disc; GaAs/AlGaAs heterostructures
1. Introduction Breakdown of the integer quantum Hall effect (QHE) at high Hall electric fields was anticipated by early Corbino disc experiments for Si-MOSFETs where the finite gate voltage regions for p " xx 0 at low source-drain fields disappear when the source-drain field increases [1]. In the early 1980s several experimental and theoretical studies of the breakdown of QHE were reported. However, very few papers have been published on experimental results obtained by using Corbino discs [2,3]. Ex-
* Corresponging author. Fax: 81 3 5391 1513; e-mail: [email protected].
perimental researches into the breakdown were mainly carried out for Hall bars because the main concern in standards laboratories was a high current in a Hall bar which generates a high Hall voltage for high precision measurements in resistance standards. In Hall bars, however, it is well known that current electrodes play important roles in the breakdown of QHE. Therefore, we have carried out a systematic experimental study on the breakdown by using butterfly-type Hall bars which reduce effects of source and drain electrodes [4—7]. In respect of the effects of the current electrodes, a Corbino disc is the best electrode structure except non-uniform distribution of radial electric fields F(r) in the conductive ring when diagonal conductivity p is uniform. xx
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 0 7 4 - X
94
M. Yokoi et al. / Physica B 249—251 (1998) 93—96
In a Corbino disc with an inner radius (radius of the source electrode) R and an outer radius */ (radius of the drain electrode) R , the average 065 electric field is given by F(av)"» /(R !R ), (1) SD 065 */ for a given voltage difference » between the SD source and the drain electrode. This average field only has meaning for the case (R !R )@R . 065 */ */ Generally, when the diagonal conductivity p is xx uniform, the highest field appears at the edge of the source electrode as F(R )"» /R ln(R /R ). (2) */ SD */ 065 */ At the limit of (R !R )@R , F(R ) in Eq. (2) 065 */ */ */ agrees with the average field given by Eq. (1). In order to determine the critical field for the breakdown of the non-conductive state, it is necessary to measure the critical breakdown voltage » for #3 a series of Corbino discs with different R or R . 065 */ We have systematically measured source-drain voltage dependences of conductance of two series of Corbino discs to determine the critical electric field for the breakdown of the non-conductive state. In this article we report our experimental results and compare them with the results of critical fields for breakdown of QHE so far obtained by using butterfly-type Hall bars.
2. Experimental results In order to eliminate the effects of non-uniform distribution of radial electric fields, we used a series of Corbino discs whose inner radii are R "140, */ 115 and 75 lm for a fixed outer radius of R " 065 175 lm for a GaAs/Al Ga As heterostructure 0.3 0.7 wafer. In other words, the width ¼"R !R of 065 */ the ring of the two-dimensional electron system is 35, 60 and 100 lm, respectively, in three Corbino discs. Two series of Corbino discs were fabricated from the following two wafers: Wafer HO (k"29 m2/Vs, N "4.6]1015 m~2) and Wafer 4 HL (k"14 m2/Vs, N "5.8]1015 m~2). 4 Source-drain voltage dependences of conductance in the two series of Corbino discs were measured at a temperature of 0.3 or 0.5 K and magnetic fields up to 20 T. As an example, Fig. 1 shows
Fig. 1. Average diagonal conductivity p versus source-drain xx voltage » . SD
change in average diagonal conductivity p at the xx filling factor of i"4 in a magnetic field of B"4.8 T against the source-drain voltage » measured at 0.5 K for a Corbino disc with SD ¼"60 lm in Wafer HO series. From the voltage difference at the breakdown, the average critical breakdown field F (av) and the critical breakdown #3 field at the edge of the source electrode F (R ) #3 */ assuming uniform p are calculated. In Fig. 2, xx these critical breakdown fields, F (av) and #3 F (R ), are plotted against widths of the Corbino #3 */ discs measured for the filling factor i"4 in the Wafer HO series. In Fig. 2, we can see that the extrapolated value of F (av) at ¼P0 coincides #3 with that of F (R ) at ¼P0. We take the value #3 */ F (av)"F (R ) at the limit ¼P0 as the critical #3 #3 */ breakdown field in the Corbino disc fabricated from this wafer. Similarly, the critical field of the breakdown was determined for another series of the Wafer HL. Critical breakdown fields F for the filling fac#3 tors of i"1, 2, 3, 4, and 6 measured in the two series of Corbino discs are plotted in Fig. 3 as a function of the magnetic field B. The results in Fig. 3 show that the critical fields of breakdown of the non-conductive state measured in Corbino
M. Yokoi et al. / Physica B 249—251 (1998) 93—96
95
Fig. 2. Critical breakdown electric fields, F (av) and F (R ), #3 #3 */ versus width ¼ of Corbino rings.
Fig. 4. Critical breakdown Hall electric field F versus mag#3 netic field B in butterfly-type Hall bars. Low mobility: 13.5)k(m2/Vs))27.
much smaller than the critical field measured in the butterfly-type Hall bars.
3. Discussions
Fig. 3. Critical breakdown electric field F versus magnetic #3 field B in Corbino discs.
discs for i"1, 2, 4 and 6 are in agreement with the critical breakdown fields of QHE measured in the butterfly-type Hall bars shown in Fig. 4 [7]. The critical field in Corbino discs for i"3 is, however,
In a magnetic field for an integer filling factor of Landau levels, a Corbino disc does not generate Joule heat at the source-drain voltage smaller than the critical voltage for the breakdown of the nonconductive state. In a Hall bar, however, Joule heat is generated at the source electrode and the drain electrode in any level of electric current. Joule heat at these current electrodes may produce non-equilibrium population of electrons in Landau levels at opposite diagonal corners of the Hall bar. Effects of the non-equilibrium population of electrons may not appear in the central part of the Hall bar when the distance between the central part and the current electrodes are long enough. Agreement between the critical breakdown fields at the filling factors of i"1, 2, 4 and 6 obtained in the Corbino
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M. Yokoi et al. / Physica B 249—251 (1998) 93—96
discs and those obtained in the butterfly-type Hall bars clearly show that effects of the Joule heat at current electrodes are eliminated in the measurements of the breakdown of QHE in our butterflytype Hall bars. In the butterfly-type Hall bars, we have the result F (i"1 and 3)/F (i"2 and 4)&1 as shown in 3 #3 #3 Fig. 4. In the present experiment for Corbino discs, we have a similar result to the butterfly-type Hall bars in F (i"1)/F (i"2 and 4) as shown in #3 #3 Fig. 3. However, we found F (i"3)/F (i"2 and #3 #3 4)& 1 in Fig. 3. This disagreement appearing at 13 the filling factor of i"3 probably arises from the difference between the electron mobility in the present experiment, k"14 and 29 m2/Vs, in Fig. 3 and those in the butterfly-type Hall bars, k"75 and 110 m2/Vs, in Fig. 4. A recent experimental result of the Hall electric field F dependence of activation energy E in H A o (F )"o exp(E /k ¹) in butterfly-type Hall xx H 0 A B bars has shown that F (i"2 and 4) is propor#3 tional to the ratio of the Landau level splitting to the radius of the ground Landau orbit; i.e. F (i"2 #3 and 4)J+u /l [8]. In the breakdown of QHE at # B an odd filling factor, we expect that F (i"odd #3 integer) is proportional to the ratio of the Zeeman splitting to the radius of ground Landau orbit; i.e. F (i"odd integer)J*E /l "g*k B/l . When #3 Z B B B we use g*"0.52, we have bare Zeeman splittings g*k B"2.3 K for B"6.5 T and g*k B"7.0 K B B for B"20 T. Landau level broadenings simply evaluated by C(k)"+/2q from electron mobilities are C(k"14 m2/Vs)"0.71 K, C(k" 29 m2/Vs)"0.34 K and C(k"75 m2/Vs)" 0.13 K. In the case of F (i"3)/F (i"2 and #3 #3 4)& 1 in Fig. 3, the ratio of the Zeeman split13 ting to the level broadening is g*k B/C) B g*k B(B"6.7 T)/C (k"29 m2/Vs)"6.8 whereas B g*k B/C*g*k B(B"6.5 T)/C (k"75 m2/Vs)" B B 17.7 in the case of F (i"3)/F (i"2 and 4)&1 in 3 #3 #3 Fig. 4. In the case of F (i"1)/F (i"2 and 4)&1 3 #3 #3 in Fig. 3, the ratio is g*k B(B"20 T)/C (k" B 14 m2/Vs)"9.9. Difference between *E /C"6.8 Z for i"3 and *E /C"9.9 for i"1 in the present Z experiment is not so large. But such a small difference may result in a large difference between exchange enhancement of Zeeman splitting in the magnetic field at i"3 and that in the magnetic
field at i"1. More elaborate discussions is necessary on effects of disorder in exchange enhancement of Zeeman splitting than a simple argument based on electron mobility. Further experiments are needed to clarify this problem. In conclusion, we have carried out measurements of the critical electric fields for breakdown of the non-conductive state at the integer filling factors of Landau levels in two series of Corbino discs and obtained results which agree with the results so far obtained for the critical breakdown fields of the quantum Hall effect by use of butterfly-type Hall bars at the filling factor of Landau levels i"1, 2, 4, and 6. The agreement shows that the effects of Joule heat at the current electrodes are eliminated in the central part of the butterfly-type Hall bars. Further experiments are needed to make clear the lower breakdown field in Corbino discs which have low electron mobility (29 m2/Vs) than that measured in butterfly-type Hall bars which have high electron mobility (75 m2/Vs).
Acknowledgements The authors wish to thank J. Sakai, and Y. Kurata of ANELVA Corporation for providing us with GaAs/AlGaAs wafers. This work is supported in part by the Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan.
References [1] S. Kawaji, J. Wakabayashi, Surf. Sci. 58 (1976) 254. [2] G. Ebert, K. von Klitzing, K. Ploog, G. Weiman, J. Phys. C 16 (1983) 5441. [3] H.L. Stormer, A.M. Chan, D.C. Tsui, J.C.M. Hwang, in: D.C. Chadi, W.A. Harrison (Eds.), Proc. 17th Int. Conf. Physics of Semicond., San Francisco, 1984, Springer, Berlin, 1985, p. 267. [4] S. Kawaji, K. Hirakawa, M. Nagata, Physica B 184 (1993) 17. [5] S. Kawaji, K. Hirakawa, M. Nagata, T. Okamoto, T. Goto, T. Fukase, J. Phys. Soc. Japan 63 (1994) 2303. [6] T. Okuno, S. Kawaji, T. Ohrui, T. Okamoto, Y. Kurata, J. Sakai, J. Phys. Soc. Japan 64 (1995) 1881. [7] S. Kawaji, J. Semicond. Sci. Technol. 11 (1996) 1546. [8] T. Shimada, T. Okamoto, S. Kawaji, Physica B 249—251 (1998), These Proceedings.
Breakdown of the integer quantum Hall effect studied by Corbino discs Masahide Yokoi!, Tohru Okamoto!, Shinji Kawaji!,*, Takayuki Goto", Tetsuro Fukase" ! Department of Physics, Gakushuin University, Mejiro, Tokyo 171, Japan " Institute for Materials Research, Tohoku University, Katahira, Sendai 980-77, Japan
Abstract Breakdown of non-conductive state of Corbino discs at integer filling factors of Landau levels at high source-drain voltages has been measured for samples fabricated from GaAs/Al Ga As heterostructures. Critical breakdown 0.3 0.7 electric fields at the filling factors of 1, 2, 4 and 6 are in agreement with the critical breakdown Hall electric fields in the quantized Hall plateaus with these quantum numbers measured in butterfly-type Hall bars. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum Hall effect; Breakdown of quantum Hall effect; Corbino disc; GaAs/AlGaAs heterostructures
1. Introduction Breakdown of the integer quantum Hall effect (QHE) at high Hall electric fields was anticipated by early Corbino disc experiments for Si-MOSFETs where the finite gate voltage regions for p " xx 0 at low source-drain fields disappear when the source-drain field increases [1]. In the early 1980s several experimental and theoretical studies of the breakdown of QHE were reported. However, very few papers have been published on experimental results obtained by using Corbino discs [2,3]. Ex-
* Corresponging author. Fax: 81 3 5391 1513; e-mail: [email protected].
perimental researches into the breakdown were mainly carried out for Hall bars because the main concern in standards laboratories was a high current in a Hall bar which generates a high Hall voltage for high precision measurements in resistance standards. In Hall bars, however, it is well known that current electrodes play important roles in the breakdown of QHE. Therefore, we have carried out a systematic experimental study on the breakdown by using butterfly-type Hall bars which reduce effects of source and drain electrodes [4—7]. In respect of the effects of the current electrodes, a Corbino disc is the best electrode structure except non-uniform distribution of radial electric fields F(r) in the conductive ring when diagonal conductivity p is uniform. xx
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 0 7 4 - X
94
M. Yokoi et al. / Physica B 249—251 (1998) 93—96
In a Corbino disc with an inner radius (radius of the source electrode) R and an outer radius */ (radius of the drain electrode) R , the average 065 electric field is given by F(av)"» /(R !R ), (1) SD 065 */ for a given voltage difference » between the SD source and the drain electrode. This average field only has meaning for the case (R !R )@R . 065 */ */ Generally, when the diagonal conductivity p is xx uniform, the highest field appears at the edge of the source electrode as F(R )"» /R ln(R /R ). (2) */ SD */ 065 */ At the limit of (R !R )@R , F(R ) in Eq. (2) 065 */ */ */ agrees with the average field given by Eq. (1). In order to determine the critical field for the breakdown of the non-conductive state, it is necessary to measure the critical breakdown voltage » for #3 a series of Corbino discs with different R or R . 065 */ We have systematically measured source-drain voltage dependences of conductance of two series of Corbino discs to determine the critical electric field for the breakdown of the non-conductive state. In this article we report our experimental results and compare them with the results of critical fields for breakdown of QHE so far obtained by using butterfly-type Hall bars.
2. Experimental results In order to eliminate the effects of non-uniform distribution of radial electric fields, we used a series of Corbino discs whose inner radii are R "140, */ 115 and 75 lm for a fixed outer radius of R " 065 175 lm for a GaAs/Al Ga As heterostructure 0.3 0.7 wafer. In other words, the width ¼"R !R of 065 */ the ring of the two-dimensional electron system is 35, 60 and 100 lm, respectively, in three Corbino discs. Two series of Corbino discs were fabricated from the following two wafers: Wafer HO (k"29 m2/Vs, N "4.6]1015 m~2) and Wafer 4 HL (k"14 m2/Vs, N "5.8]1015 m~2). 4 Source-drain voltage dependences of conductance in the two series of Corbino discs were measured at a temperature of 0.3 or 0.5 K and magnetic fields up to 20 T. As an example, Fig. 1 shows
Fig. 1. Average diagonal conductivity p versus source-drain xx voltage » . SD
change in average diagonal conductivity p at the xx filling factor of i"4 in a magnetic field of B"4.8 T against the source-drain voltage » measured at 0.5 K for a Corbino disc with SD ¼"60 lm in Wafer HO series. From the voltage difference at the breakdown, the average critical breakdown field F (av) and the critical breakdown #3 field at the edge of the source electrode F (R ) #3 */ assuming uniform p are calculated. In Fig. 2, xx these critical breakdown fields, F (av) and #3 F (R ), are plotted against widths of the Corbino #3 */ discs measured for the filling factor i"4 in the Wafer HO series. In Fig. 2, we can see that the extrapolated value of F (av) at ¼P0 coincides #3 with that of F (R ) at ¼P0. We take the value #3 */ F (av)"F (R ) at the limit ¼P0 as the critical #3 #3 */ breakdown field in the Corbino disc fabricated from this wafer. Similarly, the critical field of the breakdown was determined for another series of the Wafer HL. Critical breakdown fields F for the filling fac#3 tors of i"1, 2, 3, 4, and 6 measured in the two series of Corbino discs are plotted in Fig. 3 as a function of the magnetic field B. The results in Fig. 3 show that the critical fields of breakdown of the non-conductive state measured in Corbino
M. Yokoi et al. / Physica B 249—251 (1998) 93—96
95
Fig. 2. Critical breakdown electric fields, F (av) and F (R ), #3 #3 */ versus width ¼ of Corbino rings.
Fig. 4. Critical breakdown Hall electric field F versus mag#3 netic field B in butterfly-type Hall bars. Low mobility: 13.5)k(m2/Vs))27.
much smaller than the critical field measured in the butterfly-type Hall bars.
3. Discussions
Fig. 3. Critical breakdown electric field F versus magnetic #3 field B in Corbino discs.
discs for i"1, 2, 4 and 6 are in agreement with the critical breakdown fields of QHE measured in the butterfly-type Hall bars shown in Fig. 4 [7]. The critical field in Corbino discs for i"3 is, however,
In a magnetic field for an integer filling factor of Landau levels, a Corbino disc does not generate Joule heat at the source-drain voltage smaller than the critical voltage for the breakdown of the nonconductive state. In a Hall bar, however, Joule heat is generated at the source electrode and the drain electrode in any level of electric current. Joule heat at these current electrodes may produce non-equilibrium population of electrons in Landau levels at opposite diagonal corners of the Hall bar. Effects of the non-equilibrium population of electrons may not appear in the central part of the Hall bar when the distance between the central part and the current electrodes are long enough. Agreement between the critical breakdown fields at the filling factors of i"1, 2, 4 and 6 obtained in the Corbino
96
M. Yokoi et al. / Physica B 249—251 (1998) 93—96
discs and those obtained in the butterfly-type Hall bars clearly show that effects of the Joule heat at current electrodes are eliminated in the measurements of the breakdown of QHE in our butterflytype Hall bars. In the butterfly-type Hall bars, we have the result F (i"1 and 3)/F (i"2 and 4)&1 as shown in 3 #3 #3 Fig. 4. In the present experiment for Corbino discs, we have a similar result to the butterfly-type Hall bars in F (i"1)/F (i"2 and 4) as shown in #3 #3 Fig. 3. However, we found F (i"3)/F (i"2 and #3 #3 4)& 1 in Fig. 3. This disagreement appearing at 13 the filling factor of i"3 probably arises from the difference between the electron mobility in the present experiment, k"14 and 29 m2/Vs, in Fig. 3 and those in the butterfly-type Hall bars, k"75 and 110 m2/Vs, in Fig. 4. A recent experimental result of the Hall electric field F dependence of activation energy E in H A o (F )"o exp(E /k ¹) in butterfly-type Hall xx H 0 A B bars has shown that F (i"2 and 4) is propor#3 tional to the ratio of the Landau level splitting to the radius of the ground Landau orbit; i.e. F (i"2 #3 and 4)J+u /l [8]. In the breakdown of QHE at # B an odd filling factor, we expect that F (i"odd #3 integer) is proportional to the ratio of the Zeeman splitting to the radius of ground Landau orbit; i.e. F (i"odd integer)J*E /l "g*k B/l . When #3 Z B B B we use g*"0.52, we have bare Zeeman splittings g*k B"2.3 K for B"6.5 T and g*k B"7.0 K B B for B"20 T. Landau level broadenings simply evaluated by C(k)"+/2q from electron mobilities are C(k"14 m2/Vs)"0.71 K, C(k" 29 m2/Vs)"0.34 K and C(k"75 m2/Vs)" 0.13 K. In the case of F (i"3)/F (i"2 and #3 #3 4)& 1 in Fig. 3, the ratio of the Zeeman split13 ting to the level broadening is g*k B/C) B g*k B(B"6.7 T)/C (k"29 m2/Vs)"6.8 whereas B g*k B/C*g*k B(B"6.5 T)/C (k"75 m2/Vs)" B B 17.7 in the case of F (i"3)/F (i"2 and 4)&1 in 3 #3 #3 Fig. 4. In the case of F (i"1)/F (i"2 and 4)&1 3 #3 #3 in Fig. 3, the ratio is g*k B(B"20 T)/C (k" B 14 m2/Vs)"9.9. Difference between *E /C"6.8 Z for i"3 and *E /C"9.9 for i"1 in the present Z experiment is not so large. But such a small difference may result in a large difference between exchange enhancement of Zeeman splitting in the magnetic field at i"3 and that in the magnetic
field at i"1. More elaborate discussions is necessary on effects of disorder in exchange enhancement of Zeeman splitting than a simple argument based on electron mobility. Further experiments are needed to clarify this problem. In conclusion, we have carried out measurements of the critical electric fields for breakdown of the non-conductive state at the integer filling factors of Landau levels in two series of Corbino discs and obtained results which agree with the results so far obtained for the critical breakdown fields of the quantum Hall effect by use of butterfly-type Hall bars at the filling factor of Landau levels i"1, 2, 4, and 6. The agreement shows that the effects of Joule heat at the current electrodes are eliminated in the central part of the butterfly-type Hall bars. Further experiments are needed to make clear the lower breakdown field in Corbino discs which have low electron mobility (29 m2/Vs) than that measured in butterfly-type Hall bars which have high electron mobility (75 m2/Vs).
Acknowledgements The authors wish to thank J. Sakai, and Y. Kurata of ANELVA Corporation for providing us with GaAs/AlGaAs wafers. This work is supported in part by the Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan.
References [1] S. Kawaji, J. Wakabayashi, Surf. Sci. 58 (1976) 254. [2] G. Ebert, K. von Klitzing, K. Ploog, G. Weiman, J. Phys. C 16 (1983) 5441. [3] H.L. Stormer, A.M. Chan, D.C. Tsui, J.C.M. Hwang, in: D.C. Chadi, W.A. Harrison (Eds.), Proc. 17th Int. Conf. Physics of Semicond., San Francisco, 1984, Springer, Berlin, 1985, p. 267. [4] S. Kawaji, K. Hirakawa, M. Nagata, Physica B 184 (1993) 17. [5] S. Kawaji, K. Hirakawa, M. Nagata, T. Okamoto, T. Goto, T. Fukase, J. Phys. Soc. Japan 63 (1994) 2303. [6] T. Okuno, S. Kawaji, T. Ohrui, T. Okamoto, Y. Kurata, J. Sakai, J. Phys. Soc. Japan 64 (1995) 1881. [7] S. Kawaji, J. Semicond. Sci. Technol. 11 (1996) 1546. [8] T. Shimada, T. Okamoto, S. Kawaji, Physica B 249—251 (1998), These Proceedings.