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Hall potential distribution in the quantum Hall regime in the vicinity of a potential probe contact
Physica E 12 (2002) 165 – 168
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Hall potential distribution in the quantum Hall regime in the vicinity of a potential probe contact E. Ahlswede ∗ , J. Weis, K. v. Klitzing, K. Eberl Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Abstract Scanning force microscope measurements of the Hall potential distribution of a two-dimensional electron system (2DES) in the quantum Hall regime reveal the presence of a partial depletion along the border between the 2DES and the good ohmic metal contact: incompressible strips are formed in high magnetic /elds along this border, decoupling the bulk region of the 2DES from the contact. This /nding clari/es the role of potential probe contacts in Hall bar devices and might a2ect the interpretation of conductance measurements on Corbino devices. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.40.H; 07.79 Keywords: Quantum Hall e2ect; Scanning probe microscopy
1. Introduction Within the edge channel model for the integer quantum Hall e2ect, popular for textbooks, current is carried in one-dimensional channels along the edge region of the two-dimensional electron system (2DES) (See for review Refs. [1,2]). These edge channels are identi/ed with one-dimensional extended states at the Fermi level which are present due to the upward bending of the Landau levels at the edge region intersecting the Fermi level. The number of edge channels is directly given by the integer value of the Landau level /lling factor of the 2DES. Applying the Landauer– BButtiker formalism, the quantum Hall e2ect is obtained if perfect transmission of edge channels into the contacts can be assumed [2]. Outside of the quantum ∗ Corresponding author. Tel.: +49-711-689-1545; fax: +49-711689-1010. E-mail address: [email protected] (E. Ahlswede).
Hall plateaus, the potential probe contacts should equilibrate the edge channels and the bulk of the 2DES. Gate electrodes have been used in experiments to change the /lling factor in certain regions of the 2DES between contacts. The experimental result could be described within the Landauer–BButtiker formalism by the selective reGection and transmission of the edge channels by the gates [1]. Later, theoretical works described the formation of compressible and incompressible strips in depletion regions of the 2DES in high magnetic /elds [3]. Recently, several scanning probe techniques [4 –7] have been developed which allow spatially resolved images of the potential distribution in the 2DES under quantum Hall conditions. Our former studies [6,7] revealed that, around integer values of , the Hall potential distribution is very sensitive to the magnetic /eld and the scan position along the Hall bar mesa. But most of the Hall potential drop occurs
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 2 6 7 - 3
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E. Ahlswede et al. / Physica E 12 (2002) 165 – 168
in the bulk regions indicating that the applied current Gows in the mostly incompressible bulk. For /lling factors slightly above integer values, pronounced Hall voltage drops are observed at the mesa edges of the 2DES at the position of the innermost incompressible strip. This result demonstrates that the compressible edge and the compressible bulk are decoupled by this incompressible strip and most of the biased current Gows along the innermost incompressible strips at both sides of the mesa, driven by the Hall potential drops over these innermost incompressible strips. In this study it is shown that the innermost incompressible strip, present in the depletion region along the Hall bar mesa edge, is maintained along the border between the 2DES and ohmic contacts acting as potential probes. We relate this to a partial depletion due to the workfunction di2erence between the alloyed metal and the 2DES. The same result is obtained for Corbino devices. 2. Experimental details The experimental results, presented here, were obtained from Hall bar devices with potential probe contacts and Corbino devices which are based on a modulation-doped Al0:33 Ga0:67 As–GaAs heterostructure grown by molecular beam epitaxy with the 2DES at the heterojunction 40 nm below the surface. Mesa structures were formed by wet etching into the heterostructure and Au=Ge=Ni pads were alloyed in order to contact the 2DES. The contact resistance at zero magnetic /eld scales inversely with the length of the border between the 2DES and the alloyed metal, and is about 10 M for 100 m. The electron concentration of the 2DES is ns =4:2×1015 m−2 for the Hall bar device, ns = 4:4 × 1015 m−2 for the Corbino device, and the electron mobility in both devices is 50 m2 =V s. Data of the Hall potential distribution were taken by a scanning force microscope at a temperature of T = 1:4 K. A low frequency (3:4 Hz) AC voltage modulation of 20 mVpp was applied to the current contacts while the potential probe contacts were Goating in the case of the Hall bar device. Details of the geometry are given in the inset of Fig. 1. Further details of the experimental setup and measurement technique are described elsewhere [6,8].
(a)
(b)
Fig. 1. Hall potential distribution in the vicinity of a potential probe contact. The sample geometry, especially around the contact, is indicated in the insets. (a) At integer /lling factor (here = 2:0), the Hall potential drops rather nonlinearly across the sample width. The Hall potential in the contact region is constant. (b) For /lling factors above integer values (here =2:1), the Hall potential drops mainly at the 2DES edges—not only at the mesa edges, but also in front of the potential probe contact.
3. Experimental ndings Fig. 1 shows the Hall potential distribution in the vicinity of a potential probe contact for bulk /lling factors = 2:0 and 2.1. For = 2:1, the Hall potential drops at the left mesa edge from its maximum to an intermediate value that persists across the 2DES bulk until it drops again at the right mesa edge. This behavior is similarly observed across the mesa width towards the voltage probe contact: the Hall potential drops at the left mesa edge and in front of the potential probe contact. Obviously, the position of the Hall potential drop follows a line along the boundary of the 2DES. The Hall potential pro/le develops remarkably with magnetic /eld similar to what was described in Ref. [7], and is also observed in Corbino devices where the 2DES edge is everywhere de/ned by an interface to a metal contact. A single line scan
E. Ahlswede et al. / Physica E 12 (2002) 165 – 168
167
between the AuGeNi alloy and the 2DES can Guctuate on a local scale. 1 4. Discussion
Fig. 2. (a) Line scan across a Corbino device at bulk /lling factor = 2:2. Visible is a step-like potential drop close to the inner and the outer contact. (b) Evolution of the Hall potential close to the inner contact with magnetic /eld (black indicates low, white intermediate potential). The position of the Hall potential drop follows the theoretical prediction for the shift of the position r of the innermost incompressible strip at these /lling factors : r ˙ (1 − (int()=)2 )−1 (white line).
across such a Corbino device is shown in Fig. 2(a) for the bulk /lling factor = 2:2. Visible are the step-like Hall potential drops close to the contacts and a rather Gat Hall potential pro/le within the bulk region, similar to the /nding when scanning over the width of a Hall bar mesa (see Fig. 1(b)). In Fig. 2(b) the magnetic /eld dependence of the Hall potential pro/le in front of one contact is shown: the position of the drop moves away from the contact with increasing magnetic /eld. This movement is reminiscent of the development of an incompressible strip with magnetic /eld observed at Hall bar mesa edges [7]. This indicates that at least a partial depletion of the electron concentration of the 2DES is present along the border between the 2DES and the metal contact. We attribute this depletion to the workfunction di2erence between the 2DES and the alloyed metal. Due to the formation of various interface phases [9], the Schottky depletion
What is the consequence of this depletion? In Hall bar devices, at bulk /lling factors slightly above an integer value, a well developed incompressible strip is present along the mesa edge, but also along the metal contact. It encloses the whole bulk region of the 2DES and therefore decouples the compressible bulk from the compressible edge region. Only electron scattering via this incompressible strip couples both compressible regions of the 2DES. Contacts along the Hall bar probe the electrochemical potential of the local edge region of the 2DES. Perfect equilibration between compressible edge and compressible bulk by the contacts does not happen. With a current applied at the ends of the Hall bar, a Hall potential drop is observed over the innermost incompressible strip—also in front of the potential probing contact (see Fig. 1(b)). Obviously, at least some part of the biased current Gows without dissipation along this incompressible strip—passing in front of the metal contact without getting into it and causing dissipation within the contact. Since for integer bulk /lling factors the Hall potential drops across the bulk of the 2DES, a bulk current is driven within the 2DES. The electrochemical potential in the vicinity of the potential probe contact is constant (see Fig. 1(a)). Actually, a weak coupling between the edge of the 2DES and the metal (bad contact) would be enough to probe the local electrochemical potential of the 2DES under stationary condition. For the case of a Corbino device, the conductance is usually measured by a two-terminal measurement. But due to the presence of incompressible strips in the 2DES in front of the metal contacts for bulk /lling factors slightly above integer values, the conductance in the device might be limited by the scattering across these incompressible strips. Values for the local conductivity component xx , calculated from such conductance measurements on small Corbino devices under the assumption of a homogeneous 2DES, have to be doubted. 1
Note, that a good contact quality was con/rmed by transport measurements.
168
E. Ahlswede et al. / Physica E 12 (2002) 165 – 168
Acknowledgements The authors thank M. Riek, T. Reindl, and M. Hauser for technical support. This work has been supported by the Bundesministerium fBur Bildung und Forschung (BMBF) under the grant 01 BM 624=7 and by the Deutsche Forschungsgemeinschaft (Grant No. WE 1902=1-1). References [1] R.J. Haug, Semicond. Sci. Technol. 8 (1993) 131. [2] M. BButtiker, Phys. Rev. B 38 (1988) 9375, and references therein.
[3] D.B. Chklovskii, B.I. Shklovskii, L.I. Glazman, Phys. Rev. B 46 (1992) 4026; K. Lier, R.R. Gerhardts, Phys. Rev. B 50 (1994) 7757. [4] A. Yacoby, H.F. Hess, T.A. Fulton, L.N. Pfei2er, K.W. West, Solid State Commun. 111 (1999) 1. [5] K.L. McCormick, M.T. Woodside, M. Huang, M. Wu, P.L. McEuen, C. Duruoz, J.S. Harris, Phys. Rev. B 59 (1999) 464. [6] P. Weitz, A. Ahlswede, J. Weis, K. v. Klitzing, K. Eberl, Physica E 6 (2000) 247. [7] E. Ahlswede, P. Weitz, J. Weis, K. v. Klitzing, K. Eberl, Physica B 298 (2001) 562. [8] P. Weitz, E. Ahlswede, J. Weis, K. v. Klitzing, K. Eberl, Appl. Surf. Sci. 157 (2000) 349. [9] J.M. Woodall, J.L. Freeouf, J. Vac. Sci. Technol. 19 (1981) 794.
www.elsevier.com/locate/physe
Hall potential distribution in the quantum Hall regime in the vicinity of a potential probe contact E. Ahlswede ∗ , J. Weis, K. v. Klitzing, K. Eberl Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Abstract Scanning force microscope measurements of the Hall potential distribution of a two-dimensional electron system (2DES) in the quantum Hall regime reveal the presence of a partial depletion along the border between the 2DES and the good ohmic metal contact: incompressible strips are formed in high magnetic /elds along this border, decoupling the bulk region of the 2DES from the contact. This /nding clari/es the role of potential probe contacts in Hall bar devices and might a2ect the interpretation of conductance measurements on Corbino devices. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.40.H; 07.79 Keywords: Quantum Hall e2ect; Scanning probe microscopy
1. Introduction Within the edge channel model for the integer quantum Hall e2ect, popular for textbooks, current is carried in one-dimensional channels along the edge region of the two-dimensional electron system (2DES) (See for review Refs. [1,2]). These edge channels are identi/ed with one-dimensional extended states at the Fermi level which are present due to the upward bending of the Landau levels at the edge region intersecting the Fermi level. The number of edge channels is directly given by the integer value of the Landau level /lling factor of the 2DES. Applying the Landauer– BButtiker formalism, the quantum Hall e2ect is obtained if perfect transmission of edge channels into the contacts can be assumed [2]. Outside of the quantum ∗ Corresponding author. Tel.: +49-711-689-1545; fax: +49-711689-1010. E-mail address: [email protected] (E. Ahlswede).
Hall plateaus, the potential probe contacts should equilibrate the edge channels and the bulk of the 2DES. Gate electrodes have been used in experiments to change the /lling factor in certain regions of the 2DES between contacts. The experimental result could be described within the Landauer–BButtiker formalism by the selective reGection and transmission of the edge channels by the gates [1]. Later, theoretical works described the formation of compressible and incompressible strips in depletion regions of the 2DES in high magnetic /elds [3]. Recently, several scanning probe techniques [4 –7] have been developed which allow spatially resolved images of the potential distribution in the 2DES under quantum Hall conditions. Our former studies [6,7] revealed that, around integer values of , the Hall potential distribution is very sensitive to the magnetic /eld and the scan position along the Hall bar mesa. But most of the Hall potential drop occurs
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 2 6 7 - 3
166
E. Ahlswede et al. / Physica E 12 (2002) 165 – 168
in the bulk regions indicating that the applied current Gows in the mostly incompressible bulk. For /lling factors slightly above integer values, pronounced Hall voltage drops are observed at the mesa edges of the 2DES at the position of the innermost incompressible strip. This result demonstrates that the compressible edge and the compressible bulk are decoupled by this incompressible strip and most of the biased current Gows along the innermost incompressible strips at both sides of the mesa, driven by the Hall potential drops over these innermost incompressible strips. In this study it is shown that the innermost incompressible strip, present in the depletion region along the Hall bar mesa edge, is maintained along the border between the 2DES and ohmic contacts acting as potential probes. We relate this to a partial depletion due to the workfunction di2erence between the alloyed metal and the 2DES. The same result is obtained for Corbino devices. 2. Experimental details The experimental results, presented here, were obtained from Hall bar devices with potential probe contacts and Corbino devices which are based on a modulation-doped Al0:33 Ga0:67 As–GaAs heterostructure grown by molecular beam epitaxy with the 2DES at the heterojunction 40 nm below the surface. Mesa structures were formed by wet etching into the heterostructure and Au=Ge=Ni pads were alloyed in order to contact the 2DES. The contact resistance at zero magnetic /eld scales inversely with the length of the border between the 2DES and the alloyed metal, and is about 10 M for 100 m. The electron concentration of the 2DES is ns =4:2×1015 m−2 for the Hall bar device, ns = 4:4 × 1015 m−2 for the Corbino device, and the electron mobility in both devices is 50 m2 =V s. Data of the Hall potential distribution were taken by a scanning force microscope at a temperature of T = 1:4 K. A low frequency (3:4 Hz) AC voltage modulation of 20 mVpp was applied to the current contacts while the potential probe contacts were Goating in the case of the Hall bar device. Details of the geometry are given in the inset of Fig. 1. Further details of the experimental setup and measurement technique are described elsewhere [6,8].
(a)
(b)
Fig. 1. Hall potential distribution in the vicinity of a potential probe contact. The sample geometry, especially around the contact, is indicated in the insets. (a) At integer /lling factor (here = 2:0), the Hall potential drops rather nonlinearly across the sample width. The Hall potential in the contact region is constant. (b) For /lling factors above integer values (here =2:1), the Hall potential drops mainly at the 2DES edges—not only at the mesa edges, but also in front of the potential probe contact.
3. Experimental ndings Fig. 1 shows the Hall potential distribution in the vicinity of a potential probe contact for bulk /lling factors = 2:0 and 2.1. For = 2:1, the Hall potential drops at the left mesa edge from its maximum to an intermediate value that persists across the 2DES bulk until it drops again at the right mesa edge. This behavior is similarly observed across the mesa width towards the voltage probe contact: the Hall potential drops at the left mesa edge and in front of the potential probe contact. Obviously, the position of the Hall potential drop follows a line along the boundary of the 2DES. The Hall potential pro/le develops remarkably with magnetic /eld similar to what was described in Ref. [7], and is also observed in Corbino devices where the 2DES edge is everywhere de/ned by an interface to a metal contact. A single line scan
E. Ahlswede et al. / Physica E 12 (2002) 165 – 168
167
between the AuGeNi alloy and the 2DES can Guctuate on a local scale. 1 4. Discussion
Fig. 2. (a) Line scan across a Corbino device at bulk /lling factor = 2:2. Visible is a step-like potential drop close to the inner and the outer contact. (b) Evolution of the Hall potential close to the inner contact with magnetic /eld (black indicates low, white intermediate potential). The position of the Hall potential drop follows the theoretical prediction for the shift of the position r of the innermost incompressible strip at these /lling factors : r ˙ (1 − (int()=)2 )−1 (white line).
across such a Corbino device is shown in Fig. 2(a) for the bulk /lling factor = 2:2. Visible are the step-like Hall potential drops close to the contacts and a rather Gat Hall potential pro/le within the bulk region, similar to the /nding when scanning over the width of a Hall bar mesa (see Fig. 1(b)). In Fig. 2(b) the magnetic /eld dependence of the Hall potential pro/le in front of one contact is shown: the position of the drop moves away from the contact with increasing magnetic /eld. This movement is reminiscent of the development of an incompressible strip with magnetic /eld observed at Hall bar mesa edges [7]. This indicates that at least a partial depletion of the electron concentration of the 2DES is present along the border between the 2DES and the metal contact. We attribute this depletion to the workfunction di2erence between the 2DES and the alloyed metal. Due to the formation of various interface phases [9], the Schottky depletion
What is the consequence of this depletion? In Hall bar devices, at bulk /lling factors slightly above an integer value, a well developed incompressible strip is present along the mesa edge, but also along the metal contact. It encloses the whole bulk region of the 2DES and therefore decouples the compressible bulk from the compressible edge region. Only electron scattering via this incompressible strip couples both compressible regions of the 2DES. Contacts along the Hall bar probe the electrochemical potential of the local edge region of the 2DES. Perfect equilibration between compressible edge and compressible bulk by the contacts does not happen. With a current applied at the ends of the Hall bar, a Hall potential drop is observed over the innermost incompressible strip—also in front of the potential probing contact (see Fig. 1(b)). Obviously, at least some part of the biased current Gows without dissipation along this incompressible strip—passing in front of the metal contact without getting into it and causing dissipation within the contact. Since for integer bulk /lling factors the Hall potential drops across the bulk of the 2DES, a bulk current is driven within the 2DES. The electrochemical potential in the vicinity of the potential probe contact is constant (see Fig. 1(a)). Actually, a weak coupling between the edge of the 2DES and the metal (bad contact) would be enough to probe the local electrochemical potential of the 2DES under stationary condition. For the case of a Corbino device, the conductance is usually measured by a two-terminal measurement. But due to the presence of incompressible strips in the 2DES in front of the metal contacts for bulk /lling factors slightly above integer values, the conductance in the device might be limited by the scattering across these incompressible strips. Values for the local conductivity component xx , calculated from such conductance measurements on small Corbino devices under the assumption of a homogeneous 2DES, have to be doubted. 1
Note, that a good contact quality was con/rmed by transport measurements.
168
E. Ahlswede et al. / Physica E 12 (2002) 165 – 168
Acknowledgements The authors thank M. Riek, T. Reindl, and M. Hauser for technical support. This work has been supported by the Bundesministerium fBur Bildung und Forschung (BMBF) under the grant 01 BM 624=7 and by the Deutsche Forschungsgemeinschaft (Grant No. WE 1902=1-1). References [1] R.J. Haug, Semicond. Sci. Technol. 8 (1993) 131. [2] M. BButtiker, Phys. Rev. B 38 (1988) 9375, and references therein.
[3] D.B. Chklovskii, B.I. Shklovskii, L.I. Glazman, Phys. Rev. B 46 (1992) 4026; K. Lier, R.R. Gerhardts, Phys. Rev. B 50 (1994) 7757. [4] A. Yacoby, H.F. Hess, T.A. Fulton, L.N. Pfei2er, K.W. West, Solid State Commun. 111 (1999) 1. [5] K.L. McCormick, M.T. Woodside, M. Huang, M. Wu, P.L. McEuen, C. Duruoz, J.S. Harris, Phys. Rev. B 59 (1999) 464. [6] P. Weitz, A. Ahlswede, J. Weis, K. v. Klitzing, K. Eberl, Physica E 6 (2000) 247. [7] E. Ahlswede, P. Weitz, J. Weis, K. v. Klitzing, K. Eberl, Physica B 298 (2001) 562. [8] P. Weitz, E. Ahlswede, J. Weis, K. v. Klitzing, K. Eberl, Appl. Surf. Sci. 157 (2000) 349. [9] J.M. Woodall, J.L. Freeouf, J. Vac. Sci. Technol. 19 (1981) 794.