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Hall electric field-dependent broadening of extended state bands in Landau levels and breakdown of the quantum Hall effect
Physica B 249—251 (1998) 107—110
Hall electric field-dependent broadening of extended state bands in Landau levels and breakdown of the quantum Hall effect Takako Shimada, Tohru Okamoto, Shinji Kawaji* Department of Physics, Gakushuin University, Mejiro, Tokyo 171, Japan
Abstract Current dependence of activation energy has been measured in the diagonal resistance of two standard butterfly-type Hall bars and a butterfly-type Hall bar with a short and narrow channel in the central part made from GaAs/AlGaAs heterostructures. The activation energy E in each sample decreases with increasing current and is expressed by A E "E !ael F where E "+u /2 at the plateau center for i"2 and 4, a"38$8, l is the radius of the ground A A0 B H A0 C B Landau orbit and F the Hall electric field. The result suggests that broadening of extended state bands in Landau levels H due to the Hall electric field give rise to the magnetic field dependence of the critical Hall electric field, F JB3@2, #3 obtained in various butterfly-type Hall bars. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum Hall effect; Breakdown of quantum Hall effect; GaAs/AlGaAs hetrostructures
1. Introduction Since 1992, we have carried out experiments on the breakdown of the quantum Hall effect (QHE) with an interest to observe the phenomenon as a local property of a two-dimensional electron system (2DES) in a strong magnetic field [1—5]. The breakdown of the QHE which we want to observe is the appearance of dissipation in the central part of a Hall bar. In order to reduce the effects of dissipation at the current electrodes (source and drain) on the central part, we designed special Hall bars called butterfly-type Hall bars fabricated from GaAs/Al Ga As heterostructure wafers [1,3]. 0.3 0.7 * Corresponding author. Fax: [email protected].
81-3-5391-1513;
e-mail:
A typical butterfly-type Hall bar has wide current electrodes of ¼"400 lm and a large total length of ¸"2900 lm. Three pairs of potential probes are made in the central measurement part which is 600 lm long and has a width w much smaller than ¼. The width of each Hall bar is linearly narrowed from both current electrodes to the ends of the rectangular central part. Our results of the critical breakdown fields F for i"2 and 4 are no smaller #3 than other author’s results at the same magnetic field B and F is proportional to B3@2 [3,4]. Our #3 recent experiments on the critical breakdown field of non-conductive state in Corbino discs results in an agreement with the critical breakdown field measured by butterfly-type Hall bars [6]. In order to clarify the mechanism of the relation F JB3@2, we have measured current dependence #3
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 7 - 5
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T. Shimada et al. / Physica B 249—251 (1998) 107—110
of activation energy of diagonal resistivity o in xx the QHE state. Based on the results, we report in this article that the relation F JB3@2 arises from #3 the broadening of extended state bands at the center of Landau levels due to Hall electric field.
2. Experimental results Two standard butterfly-type Hall bars used are fabricated from two GaAs/Al Ga As wafers, 0.3 0.7 Wafer 4 (l"70 m2/Vs and N "3.5]1015 m~2) 4 and Wafer H (k"40 m2/Vs and N "1.6] 4 1015 m~2). Width of the central part of each sample is w"35 lm. A butterfly-type Hall bar with a 7 lm long (l@) and 20 lm (w@) wide channel in a 120 lm wide central part is fabricated from the wafer H. We call these samples 4B, HB and HS, respectively. In the sample HS, we obtained k"48 m2/Vs and N "1.9]1015 m~2 which are different from the 4 characteristics of HB probably due to the inhomogenity of the wafer. Current dependences of diagonal resistance were measured at different temperatures between 0.3 and 7.5 K. Results obtained in the sample 4B for the quantized Hall plateau center with i"4 at B" 3.92 T is shown in Fig. 1. Similar results obtained in the sample HS for i"2 at B"3.93 T are shown in Fig. 2. In these figures, the Hall electric field F is H
Fig. 2. Diagonal resistivity versus Hall electric field at different temperatures in the sample HS.
calculated by F "I R (i)/w and I R (i)/w@, reH SD H SD H spectively, where I is the current, w and w@ are SD widths of each sample given in their photolithographic mask patterns. Actual electrical width of the samples are probably about 2 lm smaller than the mask width due to chemical etching and spacecharge formation near the sample edges [4]. Resistivities are also calculated based on dimensions of the samples in their photolithographic mask patterns. Activation energy E (K) is determined by fitting A the experimental results at each fixed current to the following formula: o (¹)"o exp(!E /¹)#o exp(!(¹ /¹)1@3). xx 1 A 2 0 (1) In the samples 4B and HB, temperature dependence of o in high F regions cannot be described xx H by Eq. (1). Activation energies measured at each plateau center are shown as functions of F in Fig. 3. H 3. Discussions
Fig. 1. Diagonal resistivity versus Hall electric field at different temperatures in the sample 4B.
Our results of activation energies E in o at A xx different currents are summarized as follows: (1) In standard butterfly-type Hall bars 4B and HB,
T. Shimada et al. / Physica B 249—251 (1998) 107—110
109
Fig. 3. Activation energy versus Hall electric field. The vertical bar shows the critical beakdown field in each sample.
Fig. 4. Normalized activation energy versus normalized Hall electric field.
E (F ) decreases linearly with increasing F up to A H H the field where E (F )&E (0)/2. (2) In a short A H A channel sample HS, the linear decrease in E (F ) is A H observed up to the field where E (F ) reaches A H zero. In the samples 4B and HB, electron heating disturbed to obtain E at the high F region. HowA H ever, the electron heating does not appear in the sample HS because of short transit time of electrons in the short channel in this sample. The data in Fig. 3 are replotted in Fig. 4 where E is norA malized by +u /2 and F is normalized by C H +u /2el . Here l "(+/eB)1@2 is the radius of the C B B ground Landau orbit. Results in Fig. 4 show that the activation energies in these samples measured for i"2 and 4 plateau centers are approximately expressed by a simple relation as follows:
a possible mechanism for the relation in Eq. (2) where al is connected to the localization length B of electrons for short-range scatterers. Recently Boisen et al. reported a"9 to describe their experimental results obtained for the i"2 plateau in rectangular Hall bars with different widths between 50 and 400 lm made from a wafer having k" 5 m2/Vs and N "2.5]1015 m~2 [9]. These 4 authors [7—9] discussed the breakdown by the electron heating picture. Our results obtained by the short channel sample HS show that the breakdown appears when the activation energy decreases to zero due to the increase in the Hall electric field. Activation energies in standard butterfly-type samples, 4B and HB, cannot be obtained near the breakdown fields. But the fields for E "0 extrapolated from the low field A data, F (E P0), are close to the critical fields F . H A #3 In the sample 4B, F (i"2) is in accord with #3 F (E P0) and F (i"4) is close to 0.9F H A #3 H (E P0). These results suggest that the breakdown A due to Hall electric field is mainly caused by creation of mobile carriers caused by the decrease in the activation energy due to the Hall electric field. When we assume a symmetric density of states in Landau levels at the integer filling factor, the activation energy can be given by E (0)"+u /2!c A C where c is the half-width of the extended state band.
E (F )"E !ael F , (2) A H A0 B H where E "+u /2 and a"38$8. A0 C In the early 1980s Ebert et al. reported a similar relation to Fig. 1 observed in a Corbino disc though they did not give any quantitative description related to activation energy [7]. Komiyama et al. reported a similar dependence of the activation energy on the Hall electric field in a rectangular Hall bar fabricated from a wafer which has k" 25 m2/Vs and N "3.77]1015 m~2 [8]. Their re4 sult is a"25 in the i"4 plateau. They discussed
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T. Shimada et al. / Physica B 249—251 (1998) 107—110
Experimental results of E (F ) can be described A H by c"c #ael F . Then it is probable that 0 B H overlapping of the wave functions between electrons at the Fermi level and the lower mobility edge in the higher Landau level gives rise to the breakdown of the QHE. Then the critical field is given by F "(+u /2!c )/ael . (3) #3 C 0 B The magnetic field dependence of the critical field F (i"2, 4) we observed for various samples is #3 reproduced when we take a"38 and c " 0 0.34 meV [5]. Mechanism of the broadening of extended state bands due to the Hall electric field is not understood. We have to take into account long-range scatterers for localization in Landau levels in GaAs/AlGaAs 2DES. In conclusion, we have carried out measurements of the current dependence of activation energy in the diagonal resistance at the integer filling of Landau levels in two standard butterfly-type Hall bars and a butterfly-type Hall bar with a short and narrow channel and have found that the activation energy decreases with increasing current. The results suggest that the broadening of extended state bands in a Landau level due to Hall electric field gives rise to the breakdown of the QHE.
Acknowledgements The authors would like to thank colleagues in GU for assistance in the experiments, in particular A. Fukano for sample preparations. They also thank the ANELVA Corporation for supplying heterostructure wafers. This work is supported by Grants-in-Aid from the Ministry of Education, Science, Sports and Culture, Japan.
References [1] S. Kawaji, K. Hirakawa, M. Nagata, Physica B 184 (1993) 17. [2] S. Kawaji, K. Hirakawa, M. Nagata, T. Okamoto, T. Goto, T. Fukase, J. Phys. Soc. Japan 63 (1994) 2303. [3] T. Okuno, S. Kawaji, T. Ohrui, T. Okamoto, Y. Kurata, J. Sakai, J. Phys. Soc. Japan 64 (1995) 1881. [4] S. Kawaji, J. Semicond. Sci. Technol. 11 (1996) 1546. [5] S. Kawaji, in: G. Landwehr, W. Ossau (Eds.), Proc. 12th Int. Conf. High Magnetic Fields in the Physics of Semiconductors, vol. 1, World Scientific, Singapore, 1997, p. 147. [6] H. Yokoi, T. Okamoto, S. Kawaji, T. Goto, T. Fukase, Physica B 249—251 (1998), ¹hese Proceedings. [7] G. Ebert, K. von Klitzing, K. Ploog, G. Weiman, J. Phys. C 16 (1983) 5441. [8] S. Komiyama, T. Takamasu, T. Hiyamizu, S. Sasa, Solid State Commun. 54 (1958) 479. [9] A. Boisen, P. Boggild, A. Kristensen, P.E. Lindelof, Phys. Rev. B 50 (1994) 1967.
Hall electric field-dependent broadening of extended state bands in Landau levels and breakdown of the quantum Hall effect Takako Shimada, Tohru Okamoto, Shinji Kawaji* Department of Physics, Gakushuin University, Mejiro, Tokyo 171, Japan
Abstract Current dependence of activation energy has been measured in the diagonal resistance of two standard butterfly-type Hall bars and a butterfly-type Hall bar with a short and narrow channel in the central part made from GaAs/AlGaAs heterostructures. The activation energy E in each sample decreases with increasing current and is expressed by A E "E !ael F where E "+u /2 at the plateau center for i"2 and 4, a"38$8, l is the radius of the ground A A0 B H A0 C B Landau orbit and F the Hall electric field. The result suggests that broadening of extended state bands in Landau levels H due to the Hall electric field give rise to the magnetic field dependence of the critical Hall electric field, F JB3@2, #3 obtained in various butterfly-type Hall bars. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum Hall effect; Breakdown of quantum Hall effect; GaAs/AlGaAs hetrostructures
1. Introduction Since 1992, we have carried out experiments on the breakdown of the quantum Hall effect (QHE) with an interest to observe the phenomenon as a local property of a two-dimensional electron system (2DES) in a strong magnetic field [1—5]. The breakdown of the QHE which we want to observe is the appearance of dissipation in the central part of a Hall bar. In order to reduce the effects of dissipation at the current electrodes (source and drain) on the central part, we designed special Hall bars called butterfly-type Hall bars fabricated from GaAs/Al Ga As heterostructure wafers [1,3]. 0.3 0.7 * Corresponding author. Fax: [email protected].
81-3-5391-1513;
e-mail:
A typical butterfly-type Hall bar has wide current electrodes of ¼"400 lm and a large total length of ¸"2900 lm. Three pairs of potential probes are made in the central measurement part which is 600 lm long and has a width w much smaller than ¼. The width of each Hall bar is linearly narrowed from both current electrodes to the ends of the rectangular central part. Our results of the critical breakdown fields F for i"2 and 4 are no smaller #3 than other author’s results at the same magnetic field B and F is proportional to B3@2 [3,4]. Our #3 recent experiments on the critical breakdown field of non-conductive state in Corbino discs results in an agreement with the critical breakdown field measured by butterfly-type Hall bars [6]. In order to clarify the mechanism of the relation F JB3@2, we have measured current dependence #3
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 7 - 5
108
T. Shimada et al. / Physica B 249—251 (1998) 107—110
of activation energy of diagonal resistivity o in xx the QHE state. Based on the results, we report in this article that the relation F JB3@2 arises from #3 the broadening of extended state bands at the center of Landau levels due to Hall electric field.
2. Experimental results Two standard butterfly-type Hall bars used are fabricated from two GaAs/Al Ga As wafers, 0.3 0.7 Wafer 4 (l"70 m2/Vs and N "3.5]1015 m~2) 4 and Wafer H (k"40 m2/Vs and N "1.6] 4 1015 m~2). Width of the central part of each sample is w"35 lm. A butterfly-type Hall bar with a 7 lm long (l@) and 20 lm (w@) wide channel in a 120 lm wide central part is fabricated from the wafer H. We call these samples 4B, HB and HS, respectively. In the sample HS, we obtained k"48 m2/Vs and N "1.9]1015 m~2 which are different from the 4 characteristics of HB probably due to the inhomogenity of the wafer. Current dependences of diagonal resistance were measured at different temperatures between 0.3 and 7.5 K. Results obtained in the sample 4B for the quantized Hall plateau center with i"4 at B" 3.92 T is shown in Fig. 1. Similar results obtained in the sample HS for i"2 at B"3.93 T are shown in Fig. 2. In these figures, the Hall electric field F is H
Fig. 2. Diagonal resistivity versus Hall electric field at different temperatures in the sample HS.
calculated by F "I R (i)/w and I R (i)/w@, reH SD H SD H spectively, where I is the current, w and w@ are SD widths of each sample given in their photolithographic mask patterns. Actual electrical width of the samples are probably about 2 lm smaller than the mask width due to chemical etching and spacecharge formation near the sample edges [4]. Resistivities are also calculated based on dimensions of the samples in their photolithographic mask patterns. Activation energy E (K) is determined by fitting A the experimental results at each fixed current to the following formula: o (¹)"o exp(!E /¹)#o exp(!(¹ /¹)1@3). xx 1 A 2 0 (1) In the samples 4B and HB, temperature dependence of o in high F regions cannot be described xx H by Eq. (1). Activation energies measured at each plateau center are shown as functions of F in Fig. 3. H 3. Discussions
Fig. 1. Diagonal resistivity versus Hall electric field at different temperatures in the sample 4B.
Our results of activation energies E in o at A xx different currents are summarized as follows: (1) In standard butterfly-type Hall bars 4B and HB,
T. Shimada et al. / Physica B 249—251 (1998) 107—110
109
Fig. 3. Activation energy versus Hall electric field. The vertical bar shows the critical beakdown field in each sample.
Fig. 4. Normalized activation energy versus normalized Hall electric field.
E (F ) decreases linearly with increasing F up to A H H the field where E (F )&E (0)/2. (2) In a short A H A channel sample HS, the linear decrease in E (F ) is A H observed up to the field where E (F ) reaches A H zero. In the samples 4B and HB, electron heating disturbed to obtain E at the high F region. HowA H ever, the electron heating does not appear in the sample HS because of short transit time of electrons in the short channel in this sample. The data in Fig. 3 are replotted in Fig. 4 where E is norA malized by +u /2 and F is normalized by C H +u /2el . Here l "(+/eB)1@2 is the radius of the C B B ground Landau orbit. Results in Fig. 4 show that the activation energies in these samples measured for i"2 and 4 plateau centers are approximately expressed by a simple relation as follows:
a possible mechanism for the relation in Eq. (2) where al is connected to the localization length B of electrons for short-range scatterers. Recently Boisen et al. reported a"9 to describe their experimental results obtained for the i"2 plateau in rectangular Hall bars with different widths between 50 and 400 lm made from a wafer having k" 5 m2/Vs and N "2.5]1015 m~2 [9]. These 4 authors [7—9] discussed the breakdown by the electron heating picture. Our results obtained by the short channel sample HS show that the breakdown appears when the activation energy decreases to zero due to the increase in the Hall electric field. Activation energies in standard butterfly-type samples, 4B and HB, cannot be obtained near the breakdown fields. But the fields for E "0 extrapolated from the low field A data, F (E P0), are close to the critical fields F . H A #3 In the sample 4B, F (i"2) is in accord with #3 F (E P0) and F (i"4) is close to 0.9F H A #3 H (E P0). These results suggest that the breakdown A due to Hall electric field is mainly caused by creation of mobile carriers caused by the decrease in the activation energy due to the Hall electric field. When we assume a symmetric density of states in Landau levels at the integer filling factor, the activation energy can be given by E (0)"+u /2!c A C where c is the half-width of the extended state band.
E (F )"E !ael F , (2) A H A0 B H where E "+u /2 and a"38$8. A0 C In the early 1980s Ebert et al. reported a similar relation to Fig. 1 observed in a Corbino disc though they did not give any quantitative description related to activation energy [7]. Komiyama et al. reported a similar dependence of the activation energy on the Hall electric field in a rectangular Hall bar fabricated from a wafer which has k" 25 m2/Vs and N "3.77]1015 m~2 [8]. Their re4 sult is a"25 in the i"4 plateau. They discussed
110
T. Shimada et al. / Physica B 249—251 (1998) 107—110
Experimental results of E (F ) can be described A H by c"c #ael F . Then it is probable that 0 B H overlapping of the wave functions between electrons at the Fermi level and the lower mobility edge in the higher Landau level gives rise to the breakdown of the QHE. Then the critical field is given by F "(+u /2!c )/ael . (3) #3 C 0 B The magnetic field dependence of the critical field F (i"2, 4) we observed for various samples is #3 reproduced when we take a"38 and c " 0 0.34 meV [5]. Mechanism of the broadening of extended state bands due to the Hall electric field is not understood. We have to take into account long-range scatterers for localization in Landau levels in GaAs/AlGaAs 2DES. In conclusion, we have carried out measurements of the current dependence of activation energy in the diagonal resistance at the integer filling of Landau levels in two standard butterfly-type Hall bars and a butterfly-type Hall bar with a short and narrow channel and have found that the activation energy decreases with increasing current. The results suggest that the broadening of extended state bands in a Landau level due to Hall electric field gives rise to the breakdown of the QHE.
Acknowledgements The authors would like to thank colleagues in GU for assistance in the experiments, in particular A. Fukano for sample preparations. They also thank the ANELVA Corporation for supplying heterostructure wafers. This work is supported by Grants-in-Aid from the Ministry of Education, Science, Sports and Culture, Japan.
References [1] S. Kawaji, K. Hirakawa, M. Nagata, Physica B 184 (1993) 17. [2] S. Kawaji, K. Hirakawa, M. Nagata, T. Okamoto, T. Goto, T. Fukase, J. Phys. Soc. Japan 63 (1994) 2303. [3] T. Okuno, S. Kawaji, T. Ohrui, T. Okamoto, Y. Kurata, J. Sakai, J. Phys. Soc. Japan 64 (1995) 1881. [4] S. Kawaji, J. Semicond. Sci. Technol. 11 (1996) 1546. [5] S. Kawaji, in: G. Landwehr, W. Ossau (Eds.), Proc. 12th Int. Conf. High Magnetic Fields in the Physics of Semiconductors, vol. 1, World Scientific, Singapore, 1997, p. 147. [6] H. Yokoi, T. Okamoto, S. Kawaji, T. Goto, T. Fukase, Physica B 249—251 (1998), ¹hese Proceedings. [7] G. Ebert, K. von Klitzing, K. Ploog, G. Weiman, J. Phys. C 16 (1983) 5441. [8] S. Komiyama, T. Takamasu, T. Hiyamizu, S. Sasa, Solid State Commun. 54 (1958) 479. [9] A. Boisen, P. Boggild, A. Kristensen, P.E. Lindelof, Phys. Rev. B 50 (1994) 1967.