'j
QUANTUM BALLISTIC AND ADIABATIC ELECTRON TRANSPORT,
STUDIED WITH QUANTUM POINT CONTACTS
Proefschrift ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. drs. P.A. Schenck,
in het openbaar te verdedigen ten overstaan van een commissie
aangewezen door het College van Dekanen, op
ff
dinsdag 3 oktober 1989 te 16.00 uur
door
Bart Jan van Wees
geboren te Nootdorp
natuurkundig ingenieur
Dit proefschrift is goedgekeurd door de promotor prof.dr.ir. J.E. Mooij
Het onderzoek beschreven in dit proefschrift is financieel ondersteund door de
Stichting voor Fundamenteel Onderzoek der Materie (FOM)
TABLE OF CONTENTS
Chapter 1 Introduction 1 Chapter 2 Quantum ballistic and adiabatic electron transport, studied with
quantum point contacts 6
I General introduction 7 II Device layout and experimental set up 9
m Quantum ballistic transport and conductance quantization in single
quantum point contacts 10
A) Introduction 10 B) Conductance quantization in a quantum point contact (experimental) 13
C) Conductance quantization in a quantum point contact (a theoretical model) 14
D) Transmission resonances and deviations from ideal quantization 15
E) Energy averaging of the conductance 18 F) Comparison of the experimental results with model calculations 20
G) Transition from zero-field quantization to quantization in high
magnetic fields 21 H) Concluding remarks 26
IV Quantum adiabatic transport in high magnetic fields 27
A) Introduction 27 B) Quantum transport in high magnetic fields 29
C) High magnetic field transport in quantum point contacts 31
D) Anomalous integer quantum Hall effect 33 E) Anomalous quantization of the longitudinal resistance 43
G) Inter- and intra- Landau level scattering in high magnetic fields 46 H) Description of high magnetic field transport in the presence of
Shubnikov-de Haas backscattering 49 J) Suppression of the Shubnikov-de Haas oscillations due to selective
population or detection of edge channels 51 K) Edge channel mixing by quantum point contacts 57
L) Conclusions and discussion 60
Chapter 3 Quantized conductance of point contacts in a two-dimensional
electron gas 66
Chapter 4 Quantized conductance of magnetoelectricsubbands
in ballistic point contacts 73
Chapter 5 Anomalous integer quantum Hall effect in the ballistic regime
with quantum point contacts 81 Chapter 6 Suppression of the Shubnikov-de Haas resistance oscillations due to
selective population or detection of Landau levels: Absence of
inter-Landau level scattering on macroscopic length scales. 90
Chapter 7 Observation of zero-dimensional states in a one-dimensional
electron interferometer 100
Chapter 8 Transition from Ohmic to adiabatic transport in quantum point
contacts in series 110
Summary 121
CHAPTER 1
INTRODUCTION
The study of electron transport in small conductors has been a hot topic for several years. At low temperatures the resistance of conductors with dimensions in the (sub)micron range shows interesting deviations from the classical behaviour. Quantum effects in the conductance, such as localization, universal conductance fluctuations, and the Aharonov-Bohm effect, have been studied extensively in small metal or semiconductor devices. The majority of these experiments has been performed in the diffusive transport regime. In this regime the elastic mean free path between impurity collisions is less than the dimensions of the conductor.
An important category of quantum phenomena is formed by the so-called quantum size effects. When the electrons in a conductor are confined in one or more directions, the motion in these directions becomes quantized. Subbands are formed, with an energy spacing which increases with increasing confinement. At low temperatures, when the thermal energy becomes less than the subband spacing, the subband structure will be reflected in the (magneto) resistance of the conductor.
The initial aim of this project was to use a two-dimensional electron gas (2DEG), in which the electron motion is already confined in a two-dimensional plane, as the starting point to reduce the electron transport to one dimension. This one-dimensional regime is reached when the electrons are confined in a narrow wire, which has a width comparable to the Fermi wave length of the electrons. The relatively large Fermi wave length (=40 nm) of the 2DEG in a GaAs/AlxGai-xAs
hetero junction means that it is feasible to fabricate these wires with advanced electron beam lithography.
There are basically two methods to define narrow strucures in a 2DEG. The first is the etching technique where an etching mask is fabricated first, and the electron gas which is not covered by the etching mask is subsequently removed. The project was started by fabricating narrow wires with this technique. A special shallow etch technique was used, in which the etching process is stopped before the actual 2DEG layer is reached1. The aim was to minimize etching damage as
The experiments showed that these wires were in the so-called quasi-ballistic regime. This means that the electrons can move between the boundaries of the conductor without being scattered by impurities, but the motion in the direction along the wire remains diffusive. In this quasi-ballistic regime quantum size effects due to the lateral confinement of the electrons can become visible. We have studied this by analyzing the magneto-resistance of the wires. At low magnetic fields deviations from the 1/B periodicity of the Shubnikov-de Haas oscillations were found, which could be attributed to the lateral confinement of the electrons2-3-4. It was found that
the narrowest wire contained about five one-dimensional subbands. However, the search for quantum confinement effects was severely hindered by the presence of irregular resistance fluctuations, which were due to the quantum interference from the randomly distributed impurities in the wires. Another drawback of the etching technique was the fixed width of the wires, which made a detailed study of the width dependence of the quantum size effects difficult.
The split-gate technique offers an alternative possibility for defining narrow wires5-6. A
metallic gate which consists of two sections separated by a narrow gap is fabricated on top of the surface of the heterojunction. A wire can be defined by the application of a negative voltage, whichs depletes the electron gas underneath the gates. An attractive feature of this technique is that the width of the wire can be controlled continuously by means of the gate voltage. We fabricated and studied 10 urn long wires, defined with this split-gate technique. We were able to reproduce some of the results of Thornton et al5. However, unlike Thornton et al., we could not
unequivocally attribute the observed structure in the magneto-resistance to the lateral confinement of the electrons. The reason was that, similar to the etched wires, the structure in the resistance due to quantum size effects is masked by the presence of irregular fluctuations.
To avoid these random fluctuations it was decided to fabricate and study devices through which the electrons could travel ballistically. This was achieved by combining the use of high-mobility material, which has a long mean free path (>10u.m), with the fabrication of narrow and short constrictions. These structures are known as point contacts, and they have been used extensively for the study of electron transport in metals. An essential difference between metallic point contacts and the point contacts defined in the 2DEG is that the transport through metallic point contacts can be described classically. In the 2DEG point contacts however, the width (250 nm or smaller) can become comparable with the Fermi wave length, and quantum effects are expected to
which consisted of two adjacent quantum point contacts. In addition to the quantum transport through the individual point contacts this would allow us to study transverse electron focusing in a 2DEG for the first time.
A wealth of phenomena has been discovered in these devices. Electron focusing was observed7, which illustrates that ballistic transport takes place in between the QPCs. The large
number of observed focusing peaks shows the high degree of specularity of die reflections at the 2DEG boundary. Also large quantum interference structure was observed at low temperatures. This has been explained by the coherent excitation of magnetic edge states by the point contact with which the electrons are injected, and the subsequent interference of these edge states at the collector point contact8. In this thesis we will not deal with electron focusing, and we refer to Van
Houten et al.9 for a detailed study of electron focusing.
In this thesis we present an experimental and theoretical investigation of quantum ballistic and adiabatic tranport, studied with quantum point contacts (QPCs). The chapters consist of separate publications. A survey of our study of quantum ballistic transport in single quantum point contacts is given in the first section of chapter 2. In the second section an experimental and theoretical investigation of high magnetic field transport in a 2DEG, studied with quantum point contacts, is given. Chapter 2 partially covers the results of chapters 3,4,5 and 6. Chapter 3 deals with the discovery of the quantized conductance of point contacts10. In chapter 4 the magnetic
depopulation of subbands in a QPC is studied. It is shown that the zero field quantization and the high field quantization ("quantum Hall effect"), are two extremes of a more general quantization phenomenon11. Chapter 5 deals with the anomalous integer quantum Hall effect The quantization
of the Hall conductance, measured with two adjacent QPCs, was found to be independent of the number of Landau levels in the bulk 2DEG, and determined by the number of Landau levels in the QPCs instead. This result is explained by the adiabatic transport (absence of scattering between Landau levels) in high magnetic fields, combined with the selective population and detection of magnetic edge channels by the QPCs12. Chapter 6 shows that under certain
circumstances the scattering between edge channels can be very weak, even on length scales exceeding 200 u.m. This was concluded from the suppression of the Shubnikov-de Haas resistance oscillations, when they are measured with QPCs which couple selectively to the magnetic edge channels13. In chapter 7 we have employed the one-dimensional nature of the
device consist of two separately controllable point contacts with a cavity in between them. The discrete electronic states which are formed in the cavity show up in the conductance as regular transmission resonances14. Chapter 8 deals with the transition from Ohmic transport in zero
magnetic field to adiabatic transport in high magnetic fields in an identical double point contact device. In agreement with the theoretical expectations, the resistance of the device in the absence of a field is approximately the sum of the individual point contact resistances. In a high magnetic field the device resistance becomes equal to the resistance of the point contact with the highest resistance, which illustrates that adiabatic transport takes place15.
I thank Leo Kouwenhoven, Eric Willems, Kees Harmans, Walter Kool, and Nijs van der Vaart for the pleasant collaboration and their contribution made to the work presented in this thesis, and Dick van der Marel and Evert Haanappel for stimulating discussions.
This research was performed in collaboration with the Philips Research Laboratories Eindhoven. I would like to thank Henk van Houten, Carlo Beenakker, John Williamson, Toine Staring, Laurens Molenkamp, Paul van Loosdrecht, and Rob Eppenga for the fruitful, and certainly exciting collaboration. Finally I thank Leo Lander, Bram van den Enden, Eugene Timmering, Marcel Broekaart, Marines Lagemaat, Steve Phelps of the Philips Mask Centre, and the Delft Centre for Submicron Technology for their contribution in the sample fabrication, and C.T. Foxon and JJ. Harris for the supply of the high-mobility heterojunctions.
REFERENCES
1) H. van Houten, B.J. van Wees, M.G.J. Heijman, and J.P. Andre, Appl. Phys. Lett. 49, 1781 (1986)
2) H. van Houten, B.J. van Wees, J.E. Mooij, G. Roos, and K.F. Berggren, Superlat. and Microstmc. 5,497(1987)
3) H. van Houten, C.W.J. Beenakker, M.E.I. Broekaart, M.G.J. Heijman, B.J. van Wees, J.E. Mooij, and J.P Andre, Acta Electronica 28,27 (1988)
4) H. van Houten, C.W.J. Beenakker, B.J. van Wees, and J.E. Mooij, Surf. Sci. 196, 144 (1988)
6) H.Z. Zheng, H.P. Wei, D.C. Tsui, and G. Weimann, Phys. Rev. B34, 5635 (1986) 7) H. van Houten, B.J. van Wees, J.E. Mooij, C.W.J. Beenakker, J.G. Williamson, and C.T.
Foxon, Europhys. Lett. 5,721 (1988)
8) C.W.J. Beenakker, H. van Houten, and B.J. van Wees, Europhys. Lett. 7, 359 (1988) 9) H. van Houten, C.W.J. Beenakker, J.G. Williamson, M.E.I. Broekaart, P.H.M. van
Loosdrecht, B.J. van Wees, J.E. Mooij, C.T. Foxon, and J.J. Harris, Phys. Rev. B39, 8556 (1989)
10) B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, and C.T. Foxon, Phys. Rev. Lett. 60, 848 (1988)
11) B.J. van Wees, L.P. Kouwenhoven, H. van Houten, C.W.J. Beenakker, J.E. Mooij, C.T. Foxon, and J.J. Harris, Phys. Rev. B38, 3625 (1988)
12) B.J. van Wees, E.M.M. Willems, C.J.P.M. Harmans, C.W.J. Beenakker, H. van Houten, J.G. Williamson, C.T. Foxon, and J.J. Harris, Phys. Rev. Lett. 62, 1181 (1989)
13) B.J. van Wees, E.M.M. Willems, L.P. Kouwenhoven, C.J.P.M. Harmans, J.G. Williamson, C.T. Foxon, and J.J. Harris, Phys. Rev. B39, 8066 (1989)
14) B.J. van Wees, L.P. Kouwenhoven, C.J.P.M. Harmans, J.G. Williamson, C.E. Timmering, M.E.I. Broekaart, C.T. Foxon, and J.J. Harris, Phys. Rev. Lett. 62, 2523 (1989)
15) L.P. Kouwenhoven, B.J. van Wees, W.Kool, C.J.P.M. Harmans, A.A.M. Staring, C.T. Foxon, submitted to Phys. Rev. B.
CHAPTER 2
QUANTUM BALLISTIC AND ADIABATIC ELECTRON TRANSPORT, STUDIED WITH QUANTUM POINT CONTACTS
ABSTRACT
In the first part of the paper we present an experimental and theoretical study of quantum ballistic transport in single quantum point contacts (QPCs), defined in the two-dimensional electron gas (2DEG) of a high-mobility GaAs/Alo.33Gao.67As heterojunction. In zero magnetic field the conductance of quantum point contacts shows the formation of quantized plateaux at multiples of 2e2/h. The experimental results are explained with a simple model. Deviations from
ideal quantization are discussed, including the observation of transmission resonances. The experimental results are compared with model calculations. Energy averaging of the conductance has been studied, both as a function of temperature and voltage across the device. The application of a magnetic field leads to the magnetic depopulation of the one-dimensional subbands in the QPC. It is shown that the zero-field quantization and quantization in high magnetic fields are two extremes of a more general quantization phenomenon.
In the second part we use quantum point contacts to study the high magnetic field transport in a 2DEG. Quantum point contacts are used to selectively populate and detect edge channels. The experiments show that scattering between adjacent edge channels can be very weak, even on length scales longer than 200 nm. This has resulted in the observation of an anomalous integer quantum Hall effect, in which the quantization of the Hall conductance is not determined by the number of Landau levels in the bulk 2DEG, but by the number of Landau levels in the QPCs instead. Related effects are the anomalous quantization of the longitudinal resistance, and the adiabatic transport in series QPCs. A theoretical description for transport in the presence of Shubnikov-de Haas backscattering is given. This model explains the experimentally observed suppression of the SdH oscillations due to the selective population or detection of edge channels. Finally, we demonstrate that the combination of a QPC and a bulk contact can act as a controllable edge channel mixer.
L GENERAL INTRODUCTION
The fundamental properties of electron transport are best studied in the ballistic regime. In this regime the elastic and inelastic mean free paths le and lj are both larger than the dimensions of the conductor through which the electrons travel. The motion of the electrons is completely determined by the (smooth) electrostatic potential which defines the conductor, and is not disturbed by external factors, such as interactions with phonons, impurities etc. A classical description of ballistic transport suffices when the dimensions of the conductor are large compared to the Fermi wave length Xp of the electrons. When the device dimensions become comparable to XF, the quantum ballistic regime is entered. In this regime the wave-like nature of the electrons becomes prominent.
The two-dimensional electron gas (2DEG) of a high-mobility GaAs/Alo.33Gao.67As heterojunction is a very attractive system for the study of quantum ballistic transport. At low temperatures both le and lj can become relatively large (>10 |J.m). Also the Fermi wave length is
relatively large (of the order of 40 nm). With modern microfabrication techniques it is therefore possible to fabricate devices in a 2DEG which operate in the quantum ballistic regime. We have employed a split-gate technique to fabricate quantum point contacts (QPCs). These quantum point contacts are essentially short and narrow constrictions. They are defined by electrostatic depletion underneath a split-gate. An attractive feature of the split-gate technique is that the properties of the quantum point contacts can be controlled continuously by the applied gate voltage. This has enabled us to perform a detailed study of the quantum ballistic transport regime.
This paper consists of two major parts (III and IV). After the description of the device layout and the experimental set up in part II, we study the ballistic transport through single quantum point contacts in part HI. A brief introduction of quantum ballistic transport is given in section HI A. The experiments which reveal the quantization of the ballistic conductance of quantum point contacts in the absence of a magnetic field are presented in section B. The results will be explained with a simple model in section C. Deviations from ideal quantization and the observation of transmission resonances are discussed in section D. A comparison of our results with model calculations will be given in section E. In section F we study the influence of energy averaging due to a finite temperature and finite voltage across the QPCs. The application of a
(a)
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(b)
Fig. 1 (a) Schematic layout of the device. The gates define two adjacent quantum point contacts A and B. (b) Micrograph, showing the gate on top of the heterojunction, which defines two adjacent QPCs. The white bar is 1 [im.
perpendicular magnetic field leads to the magnetic depopulation of the one-dimensional subbands in the QPC. The quantization is preserved, and it is shown that the zero-field quantization and the quantum Hall effect in a QPC are two extremes of a more general quantization phenomenon (section G). Part III of the paper is concluded in section H.
In part IV of the paper we present a detailed theoretical and experimental investigation of high magnetic field transport in a 2DEG, studied with quantum point contacts. This part is introduced in section IV A.
Part of our results has been published in earlier papers: Quantized conductance of point contacts in a two-dimensional electron gas1, Quantized conductance of magnetoelectric subbands
in ballistic point contacts2, Anomalous integer quantum Hall effect in the ballistic regime with
quantum point contacts3, Suppression of Shubnikov-de Haas resistance oscillations due to
selective population or detection of Landau levels: Absence of inter-Landau level scattering on macroscopic length scales4.
IL DEVICE LAYOUT AND EXPERIMENTAL SET UP
In fig. 1 we show the schematic layout and a micrograph of the devices. Identical devices have been used for the study of coherent electron focusing5-6-7, non-linear transport in QPCs8, and
the Aharonov-Bohm effect in singly connected point contacts9. The starting material is a
high-mobility two-dimensional electron gas, which is present in a GaAs/Alo.33Gao.67As heterojunction. The electron density is 3.6 lO^/m2, which results in a Fermi energy Ep = 12
meV, and a Fermi wave length XF ~ 40 nm. The elastic mean free path (at 4.2K) is 9 p.m (The mobility is 85 m2/Vs). Ohmic bulk contacts 1 - 6 are fabricated by alloying Au/Ge/Ni. A Hall bar
(200 (i.m wide and 600 urn long) is defined by optical lithography and wet chemical mesa etching. Gates A and B are fabricated by combination of optical lithography (hatched section), electron lithography (solid section) and lift off techniques.
The QPCs are defined by a split-gate technique, which was pioneered by Thornton et al10. and
Zheng et al11, for the study of low-dimensional electron transport12. An attractive feature of this
technique is that contact with the 2DEG, which is located about 60 nm below the surface, is avoided during the fabrication process. This prevents a possible reduction of the electron mobility due to surface damage. Application of a negative gate voltage Vg = -0.6 V depletes the
electron gas underneath the gate. As a result two quantum points contacts A and B are defined, with a lithographic width of 250 nm and a separation of 1.5 (im. A further reduction of the gate voltage creates a saddle-shaped potential at the QPCs, and reduces their width and electron density. The QPCs are completely pinched-off at ~ -2.2 V. The two separate gates make it possible to control the QPCs individually. As can be seen in fig. 1, QPC B is controlled by the gate voltage VB , whereas QPC A is controlled by both VA and VB . It was found experimentally that the properties of QPC A are approximately determined by the effective gate voltage (VA + VB)/2.
We have investigated several nominally identical samples. In section III we present experimental results of sample 1. Thermal cycling between room and helium temperature resulted in a gradual deterioration of the quality of the quantization in zero magnetic field in this sample. Therefore the conductance of this sample obtained in different measurement runs shows a different quality of quantization as well as different fine structure. However, the overall behaviour of the sample did not change. In section IV results on sample 2 are presented. The results obtained from these samples are typical for the remainder of the investigated samples.
The experiments were performed either in a pumped He4 cryostat or in a HeVHe4 dilution
refrigerator. The measurement leads were filtered to prevent RF interference. A phase-sensitive lock-in technique was used, with the voltages across the device kept below kT/e to prevent energy averaging of the conductance.
ffl. QUANTUM BALLISTIC TRANSPORT AND QUANTIZED CONDUCTANCE IN SINGLE QUANTUM POINT CONTACTS.
A. INTRODUCTION
An important feature of ballistic transport is its non-locality. The electron distribution (both in energy and momentum space) in a given section of the conductor is determined by scattering processes which have occurred in other sections of the conductor. This is the reason that a description of electron transport in which a local electric field is the driving agent is not suitable for the description of ballistic transport
In an alternative description current flows as a result of the difference in electrochemical potentials in the different parts of the conductor. The electrochemical potential ^ indicates up to which energy (kinetic + electrostatic) the electronic states are occupied. A net current flows when the electron states which carry current in one direction are occupied up to a different energy than the electron states which carry current in the opposite direction. In this description of electron transport the resistance is caused by the backscattering of electrons. Landauer13 has proposed
that the resistance can be described with transmission and reflection probabilities, which indicate the fraction of the current which is transmitted or reflected by an obstacle. In the diffusive regime, where the mean free path between collisions with impurities is smaller than the dimensions of the conductor, the backscattering results from these impurity collisions. In the ballistic regime the backscattering is caused by the boundaries of the conductor itself.
The most elementary devices to study ballistic transport are so-called point contacts. A point contact, first proposed by Sharvin14, basically consist of a narrow and short constriction which
connects two wider conductors15. Both its width and length are less than the elastic and inelastic
mean free paths. The description of the electron transport is as follows: The two wide conductors on either side of the constriction act as electron reservoirs which emit and absorb electrons. A voltage difference V which is applied between the two regions creates a difference in electrochemical potential eV = |!L-WR- AS a result electrons will impinge on the point contact from the right with energies up to |1R and from the left with energies up to U.L- The net current I through the point contact is therefore determined by the transmission probability of electrons in the energy interval between ^.R and (XL- When the applied voltage is low enough (eV«EF), the two-terminal conductance Gc of the point contact is determined by the transmission probability at
the Fermi energy T(Ep):
Gc(EF) = - ^ - = T p T(EF), (1)
HL-HR
in which we have introduced the conductance quantum 2e2/h. The ballistic point contact
resistance is exclusively determined by elastic processes. Dissipative processes in the wide reservoirs will equilibrate the electron distribution. In the ballistic regime these processes occur sufficiently far away from the point contact, and do not influence the resistance.
In metals the Fermi wave length is typically a few A, and is usually much smaller than the width of the point contact. This means that the transmission probability T(Ep) can be evaluated
1 0
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o
4
2
-0
-2.0 -1.8 -1.6 -1.4 -1.2
GATE VOLTAGE (V)
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Fig. 2 Quantized conductance of a quantum point contact at 0.6 K. The conductance was obtained from the measured resistance after subtraction of a constant series resistance of 400 Q.
Fig. 3 Occupied electron states in the channel at two different gate voltages in the case of a current flow through the channel. In equilibrium the electron states are occupied up to the bulk
classically, and the point contact conductance is expected to be proportional to its width. In the experiments described in the next section we will measure the conductance of a quantum point contact as a function of its width. The fact that the width (250 nm or less) is comparable to A.F (=40 nm) produces a result which is dramatically different from the classical result.
B. CONDUCTANCE QUANTIZATION IN A QUANTUM POINT CONTACT (EXPERIMENTAL)
The resistance of QPC A is measured in zero magnetic field as a function of applied gate voltage V A = V B at 0.6 K. A three-terminal set up is used, with voltage contacts 1 and 5 and current contacts 4 and 5 (see fig. 1). This three-terminal measurement eliminates a possible series resistance from 2DEG region I, and measures the QPC resistance in series with the resistance of 2DEG region II and a possible resistance of bulk contact 5. Fig. 2 shows the conductance Gc,
which was obtained from the measured resistance after subtraction of a constant series resistance of 400 Q. This resistance was chosen to match the plateaux with their corresponding quantized values, and is in reasonable agreement with the estimated series resistance, based on the sheet resistance (20 Q) and the geometry (=16 squares) of region II.
The conductance of the QPC shows a sequence of quantized plateaux1 at multiples of 2e2/h.
In the gate voltage interval between the formation of the QPC at -0.6 V to pinch-off at -2.2 V 16 plateaux are observed. A close examination of fig. 2 shows that several plateaux are quite flat, whereas others show some fine-structure. Similar results have been reported by Wharam et al., who observed the conductance quantization in short (= 0.6 nm) and narrow channels, also defined with a split gate technique16.
We have studied several nominally identical QPCs. They all show the step-like structure in Gc(Vg). However, the fine-structure in between the plateaux is different for each device. Also
some devices show structure on the plateaux themselves. In our device geometry it is difficult to determine the accuracy of the quantization at the plateaux, because the series resistance may depend slightly on the applied gate voltage17. However, a prerequisite for accurate quantization
is that the plateaux are flat, and do no show fine-structure. The results therefore show that the quantization is not exact. We will discuss the deviations from exact quantization in detail in sections D and F.
C. CONDUCTANCE QUANTIZATION IN A QUANTUM POINT CONTACT (A THEORETICAL MODEL)
The elementary explanation for the observed conductance quantization is in principle very simple. We assume that we can model the QPC as a channel with finite length, in which the electrons are confined laterally by a parabolic potential 1/2 m*C0o2x2, in which m*= 0.067 mo is
the effective mass of the electrons, and co0 indicates the strength of the lateral confinement. This
choice of confinement is not essential for the result, but is a realistic approximation when the QPCs are near pinch-off18 (A calculation of the QPC conductance for the case of an
infinite-square well potential is given in refs. 1 and 2). The lateral confinement leads to the quantization of the lateral motion, and the formation of one-dimensional subbands. We obtain the following dispersion relation for the electron states in the QPC:
En(ky) = (n - \ ) hco0 + ^ N r + eV0 (2)
which is the sum of the quantized lateral motion (n=l,2,.. is the index of the ID subbands), the kinetic energy along the channel (ky is the wave number for the motion along the channel), and
the electrostatic energy eV0 in the QPC. Fig. 3 shows the occupied electron states at two different
gate voltages. The analysis of the magneto-resistance of the QPCs in section G shows that the effect of the gate voltage is twofold: A more negative gate voltage increases the confinement and thus the energy separation hcOo. As a second effect the electrostatic potential V0 in the QPC is
raised. As can be seen in fig. 3, both effects reduce the number of occupied subbands Nc.
For the evaluation of the conductance Gc we assume that all electron states with positive
velocity vy =l/h (dEn(ky)/dky) are occupied to (1L and all electron states with negative vy are
occupied to |IR. This is equivalent to the assumption that no reflection occurs at both ends of the channel. Furthermore we assume that the channel is long enough to prevent a contribution of evanescent waves to the conductance (In sections D and F we will discuss what the consequences are if these conditions are not satisfied). The expression for Gc reads:
i Nc HL 1
The product of the ID density of states (including both spin orientations) Nn(E) = 2/rc
(dEn(ky)/dky)"1 and the group velocity vn(E) =1/K (dEn(ky)/dky) is energy independent, and
equal to 4/h. This is an important feature of ID transport and gives the result:
GC= T T NC .with Nc = Int{ E F'e VQ + y } (4)
n ho)o z
in which Int denotes the truncation to an integer. The conductance is simply given by the conductance quantum 2e2/h, multiplied by the number of occupied subbands in the QPC. Prior to
the experimental discovery of the quantized point contact conductance, the possibility of a quantized contact resistance between two reservoirs has been anticipated by Imry1^. However, it
was not expected at that time that an experimental system would show conductance quantization in such a clear and convincing way.
It can be shown that a classical evaluation of the point contact conductance gives the result: Gc
= 2e2/h (Ep - eV0)/hco0. A comparison with eq.(4) shows that the difference between classical
and quantum results does not exceed 2e2/h. This shows that in the limit of Gc»2e2/h the
difference between quantum and classical results becomes unimportant.
D. TRANSMISSION RESONANCES AND DEVIATIONS FROM IDEAL QUANTIZATION
Although the model of a channel with a finite length is clearly oversimplified, we can nevertheless use it to explain some of the features of the data. In this section we focus on the transition regions in between the quantized plateaux. We will explain the absence of quantization in these regions by the (partial) reflection of electron waves at both ends of the channel. A sudden widening of the channel, or change in electrostatic potential, at both ends of the channel will induce a partial reflection of the electron waves. This can be compared with the reflection of waves at an open-ended waveguide. In a first order approximation the electron waves in a particular subband (or wave-guide mode) are reflected in the same subband. We can define a reflection probability R which describes the fraction of the current carried by a subband which is reflected at the ends of the channel. In a one-dimensional model the reflection probability for an abrupt potential step is given by:
-2 -1.95 -1.9 GATE VOLTAGE (V)
Fig. 4 Temperature averaging of the transmission resonances of the second subband. The values for the energy averaging parameter AE are given. The curves have been offset for clarity.
-2 -1.95 -1.9 GATE VOLTAGE (V)
Fig. 5 Voltage averaging of the transmission resonances of the second subband. The values for the energy averaging parameter AE are given. The curves have been offset for clarity.
with kyi and ky2 are the longitudinal wave numbers inside and outside the channel. The
transition regions between the quantized plateaux can now be understood with eq.(5). The threshold for transmission of the n^ subband is given by Ep = eV0 + (n-l/2)hoo0 • Slightly above
the threshold, kyi = (2m* (EF-eV0-(n-l/2)hcüo)/h2)1''2 is very small, and eq. (5) shows that R is
near unity. The n* subband does not yet contribute significantly to the conductance. When eV0 +
(n-l/2)nc0ois reduced further by increasing the gate voltage, kyi increases, R slowly drops to
zero, and the conductance gradually reaches its quantized value.
A prerequisite for the observation of quantized plateaux is therefore that the reflection probability of a given subband has dropped to zero, before the threshold value for the transmission of the next subband is reached20. Evaluation of (5) with the values for hco0 and eV0
obtained in section G, shows that this condition is by far not satisfied for an abrupt potential step. This means that in the actual devices a gradual widening of the channel as well as a gradual reduction of the electrostatic potential must take place at both ends of the channel. This matching of the narrow channel to the wide regions will make it possible that R approaches zero, and that a quantized plateau is formed.
Due to the possibility of multiple reflections at both ends of the channel, we also expect to observe transmission resonances21. When we assume equal reflection probabilities R at both
ends of the channel we can write the conductance of the QPC as:
Gc = Jc
2e2 f (1-R)2 l
h {N+ 1 - 2R cos(2ky lL) + R2 ' W
in which L is the length of the channel. This equation expresses that Gc can be written as the sum
of the quantized conductance of N low-lying subbands (with low quantum number n) and the (resonant) transmission of the upper subband. Eq. (6) predicts transmission resonances in the transition regions between the quantized plateaux, where R*0. An important feature of Eq.(6) is that even in the case of a finite reflection probability R, the conductance can still be quantized, provided that the condition for resonant transmission is satisfied: 2kyiL = integer * 2TC.
Figs. 4 and 5 (upper traces) show experimental results. The data illustrate the transition from the first to the second plateau. Three maxima and two minima are observed, of which the second and the third maximum approach the quantized value 4e2/h. The fact that the first maximum does
the wire (Note that the geometry of the QPCs is not symmetric (see fig.1)). The number of observed resonances allows us to make an estimate of the length L of the channel. At the threshold for the transmission of the third subband the longitudinal wave number of the second subband is given by kyi = (2m*E/R2)1/2, with the subband spacing E=Kco0 = 2.5 meV (see
section G). From the resonance condition 2kyiL = 3 * 2%, we find L= 140 nm, which is a
reasonable value.
We emphasize that, although several devices showed structure in the transition regions between the plateaux, clear resonances have been observed in only two devices. The fact that the devices usually show transition regions without clear resonant structure is probably a sign of the inadequacy of our simple model. Model calculations (section F) show that tapering at both ends of the channel can result in a smooth transition region without pronounced resonance structure.
E. ENERGY AVERAGING OF THE CONDUCTANCE
In the previous sections it was shown that at low voltages across the device and low temperatures the conductance of a QPC can be described by the transmission probabilities T0(EF)
of the different subbands at the Fermi energy. At a finite temperature, or finite voltage across the device, the current will be carried by a energy interval of finite width. This leads to energy averaging of the point contact conductance22. The conductance at a finite voltage V is given by:
2 e l Nc EF+ e V
GcOO = f J ^ J T„(E) dE (7)
At a finite temperature T the conductance is given by: 9e2 Nc o» HffETI
G c ( T ) =
T ~ £ J (
2iS
LL)
Tn(E)dE (8)
n=l 0
in which f(E,T) = (1+ exp((E-EF)/kT)"1 is the Fermi-Dirac distribution function. Eqs. (7) and (8)
show that in both cases the physics is the same, and only the weighing factors are different. The temperature averaging has a Gaussian weighing factor, which has an effective width AE = 3.5 keT. For voltage averaging AE =eV.
resonances when the temperature is increased, and fig. 5 shows how they disappear when the AC current through the device is increased. The currents and temperatures have been selected such that each set of traces has approximately the same energy averaging parameter AE. (The effective AE due to the AC current with RMS value I is estimated to be AE = 1.4 eI/Gc). The
results show that the effects of elevated temperature and voltage are similar. This means that other processes which may introduce the loss of phase coherence, and destroy the resonances, such as inelastic scattering, are not important. The transport remains ballistic, at least up to temperatures of IK, and voltages of 40n.V across the device. Recent experiments show that ballistic and phase-coherent transport in a 2DEG can even occur up to energies in the meV range8,23.
Figs. 4 and 5 show that an energy interval AE=0.5 meV is sufficient to wash out the transmission resonances. We now investigate how the quantized plateaux themselves are destroyed when the temperature is raised further. Fig. 6 shows that temperature averaging becomes effective above =0.6 K. At 4.2 K the plateaux have almost disappeared. The mechanism for the destruction of the plateaux is that at high temperatures electron states of the next subband become occupied, and not all electron states of the low-lying subbands are occupied anymore (eq.8). A comparison of the effective energy averaging parameter at 4.2K: AE = 1.6 meV with the subband spacing obtained in section G (= 2.5 meV) confirms that the mechanism for the destruction of the quantized plateaux is energy averaging24. The 4.2 K trace
shows that the plateaux near pinch-off are less rounded than the other plateaux. This is in agreement with section G, which shows that the subband spacing increases when the gate voltage is reduced. Finally, we mention that the breakdown of the conductance quantization as a function of applied voltage has been studied by Kouwenhoven et al8. They showed that the
conductance quantization breaks down at a voltage which is approximately equal to the subband spacing.
F. COMPARISON OF THE EXPERIMENTAL RESULTS WITH MODEL CALCULATIONS.
After the discovery of the quantized conductance of point contacts, many calculations of the conductance of narrow constrictions have been performed25"4**. in this section we make a
comparison between these model calculations and our experimental results. We do not give an exhaustive discussion, but focus on the aspects which are relevant for the experimental results.
An interesting question is whether an actual channel of finite length is required to observe quantization of the conductance, or whether a "hole in a screen" point contact is already sufficient. Calculations26'27-29'34-35 show that the conductance of a "hole in a screen" point
contact, calculated as a function of its width W, already shows a modulation with a period 2e2/h.
Van der Marel and Haanappel27 obtained the surprising result that the conductance at the points
of inflection in the GC(W) curve is exactly equal to multiples of 2e2/h. When the point contact is
given a finite length, the structure rapidly develops into well-defined plateaux. Van der Marel and Haanappel27 obtained a criterion that the length L of the channel should exceed 0.3 \ 2WX.F, to
prevent the contribution of evanescent waves to the conductance, which destroy the quantized plateaux. Strong transmission resonances are observed when the channel is made longer such that it can accomodate several wave lengths. The absence of clear transmission resonances in all but two of our devices shows that the model of a rectangular wave guide with abrupt ends is not a fully accurate representation of the actual QPCs.
Several authors have calculated the conductance of a constriction with the typical wedge geometry of the lithographic gate (fig 1.) which defines the QPCs29-42. No well-defined plateaux
were observed in this geometry. This clearly shows that the actual electrostatic potential which defines the QPCs is substantially different from the geometry of the lithographic gate.
If the change in width and electrostatic potential at both ends of the channel is sufficiently smooth, adiabatic transport can occur. In this case the electrons move with conservation of subband index, and no mode-mixing takes place. Adiabatic transport through QPCs was studied by Glazman et al.31, who obtained a condition for the radius of curvature of the boundaries of
the constriction, required for adiabatic tranport. However, it is difficult to compare this criterion with the experimental results, since the actual QPCs also contain a potential barrier (see section
Several authors have included scattering in their model calculations, which, as expected, destroys the quantization27.38,40. However, it should be noted that not only the strength of the
scattering potential will be important, but also the length scale on which it varies. It is well known that the electron mobility in a 2DEG is limited by the scattering induced by ionized donor atoms, which are located about 50 nm away from the 2DEG. We therefore think that the actual potential in the QPC is the sum of the smooth potential from the gate, and a (screened) corrugated potential formed by the superposition of the potentials of the individual donor atoms. The latter is expected to vary on a scale of = 50 nm, which is the average distance between the donor atoms. Although a slight variation in the gate geometry for different devices cannot be excluded, it may well be that the differences in the fine-structure of the devices is due to the statistical distribution of the donor atoms.
We conclude that the model calculations can account for several features observed in the experiment, but do not yet fully explain the results. It would therefore be interesting to calculate the conductance of a QPC, which is formed by a saddle shaped potential with the experimentally determined parameters given in section G.
G. TRANSITION FROM ZERO-FIELD QUANTIZATION TO QUANTIZATION IN HIGH MAGNETIC FIELDS
In this section we study the effect of a perpendicular magnetic field on the conductance quantization. It is shown that the application of a magnetic field preserves the quantization and a gradual transition is observed from the conductance quantization due to the lateral confinement of the electrons, to the quantization in high magnetic fields. We deliberately do not use the term quantum Hall effect, since this is restricted to four-terminal measurements, whereas we study a two-terminal conductance. However, as we will show, the origin of the quantum Hall effect and the zero-field quantization is closely related.
The presence of a perpendicular magnetic field does not change the one-dimensional nature of the transport in the QPC. Because of the translational invariance of the Hamiltonian in the direction along the channel, the transport can still be described by electron waves travelling in a wave guide. The dispersion of these waves now becomes:
CM C\J UJ O Ü Q Z
o
o
- 2 - 1 . 8 GATE VOLTAGE (V) -1.6 12 10 QJ ™ 6 W A O 4z
%* 3 Qz
o
ü °
0 . ... . , . , , / . . r . . . 1 . . . 1 ■ ■ ' 1" ■"■ 1 ' ' ' 1 ' B=0 T J ^ 0.7 "P; ^J . .—/. . J O j ■ ^ 2 . 5 T •■ 1 . . . I . . . 1 . . . . L . . . . - 2 - 1 . 8 - 1 . 6 - 1 . 4 - 1 . 2 - 1 GATE VOLTAGE (V)Fig. 6 Breakdown of the conductance quantization due to temperature averaging. The curves have been offset for clarity.
Fig. 7 Transition from quantization in zero-field to quantization in high magnetic fields. The curves have been offset for
1 h2|f 2 i
En(ky) = (n - j ) li© + ^ - + eV0 ± j guBB
with m = m* — - , (ü = "J 0)02 + coc2 , and coc = c—g (9)
COo2 m
The magnetic field creates hybrid magneto-electric subbands, and changes the dispersion relation of the waves49-50. However, because of the one-dimensional nature of the transport, the
essential relation between the ID density of states Nn(E) and the group velocity vn(E) still holds:
Nn(E) vn(E) = 4/h. Ignoring spin-splitting we obtain the result:
GC(B) = ~ Nc , with Nc = Int { § £ ^ 2 - + * } ( 1 0 )
n hco z
Eq. (10) shows that there is a gradual transition between the quantization in zero field (o)c=0) to the quantization in high field (coc»coo)2,51.
Fig. 7 presents experimental results on the transition from zero-field to high field quantization2, obtained at 0.6 K. The top trace reproduces the B=0 result. When a magnetic field
is applied, the width of the plateaux is widened compared to the B=0 case. This reflects the increase of subband spacing with magnetic field (eq.9). It takes a larger variation of the gate voltage to populate (or depopulate) a new subband. The quantization is preserved, in agreement with eq.(10). At high fields the spin-degeneracy is lifted (gUsB exceeds keT), and plateaux at uneven multiples of e2/h become visible. We can make a coarse estimate for the effective g factor
when we assume that the ratio between the width of the spin-split plateaux and the regular plateaux is given by (g|a.BB)/(hcoc). The experimental ratio is « 0.15 - 0.2, which gives g = 2 - 3.
This shows that the g factor may be enhanced above its bear GaAs value (0.5).
Eq.(10) predicts that at high magnetic fields (coc»co0) Nc is determined exclusively by the
combination of the potential barrier V0 and coc, and is proportional to 1/B. At low fields
however, the number of subbands is limited by the lateral confinement, and determined by co0
-We have determined the number of occupied subbands Nc as a function of magnetic field at
several fixed values of the gate voltage from fig. 7. The result is shown in fig. 8 (square dots). From the fit of eq.(10) we have obtained the values of V0 and co0 at these values of the gate
voltage. They are given in table 1 (A similar analysis for a infinite square-well potental is given in ref.2).
12 10 Q
z
< m m CO LL O CC LU CD . . . . . . . . • j*J -M— J _ . .u
(-•— 1 ■ —» ■ . . 1 m - I V . - 1 . 3 V .. m m ■ ■ - 1 . 6 / .. - 1 . B 5 V .. . . - 2 V ' 1 2 3 1/B (1/T) CM LU O 2-O Z5 Q§°
O ■ 1 ' ' ' ' 1 1 ■ 1 ' ' 1 'r
•f
0 T ƒ / _y 0.05 T / // 0. 1 T j J
J
°'
3 T
/
- 2 . 1 - 2 - 1 . 9 - 1 . 8 GATE VOLTAGE (V)Fig. 8. Number of occupied subbands as a function of inverse magnetic field (square dots). The solid curves correspond with fits with eq. 9. The parameters are given in table 1. The curves have been offset for clarity.
Fig. 9 Improvement of the conductance quantization by the application of a magnetic field, measured at 40 mK.
Table 1. Values for the subband spacing H co0 and potential barrier eV0 at several values of the
gate voltage Vg, obtained from a fit ofeq. 9 to the experimental data.
Vg(V) KcOo(meV) eV0(meV) -1.0 -1.3 -1.5 -1.85 -2.0 1.0 1.1 1.5 1.8 3.0 0 2.0 3.5 5.5 6.5
The results show that a reduction of the gate voltage increases both the confinement (measured by <x>o), and the potential barrier V0 in the QPC. The results show that the maximum subband
spacing, which is achieved in our QPCs is about 3 meV. Similar results have been also been obtained by Wharam et al. for a split-gate wire52.
A characteristic feature of QPCs in a magnetic field is that the quality of the quantization is improved when a magnetic field is applied. This is most clearly observed in QPCs whose zero-field quantization is poor. In this case the quality of the quantization has deteriorated due to several thermal cycles. Fig. 9 shows how a relatively small magnetic field already improves the quantization (Notice the improvement of the second and third plateaux). The mechanism is probably that the backscattering near or in the QPC is reduced in the presence of a magnetic field. Because the quantization is already improved at a very low field (the cyclotron radius at 0.1 T is about 1 |im) it is possible that part of the backscattering occurs near the QPC (possibly by impurities), and not in the QPC itself.
As discussed by Biittiker53, a sufficiently large magnetic field can completely prevent the
back-scatttering induced by impurities or irregularities in the confining potential. This absence of backscattering in high magnetic fields is probably the main reason for the extreme accuracy of the quantum Hall effect, compared to the limited accuracy of the conductance quantization in zero field.
H. CONCLUDING REMARKS
The conductance of quantum point contacts was found to display quantized plateaux at multiples of the conductance quantum 2e2/h. This quantization can be explained by the formation
of one-dimensional subbands in the point contacts, each occupied subband contributing 2e2/h to
the conductance. Both experiments and model calculations show that the accuracy of the quantization is sensitive to the detailed shape of the confining potential and the possible presence of impurities. Nevertheless, we estimate that it may be possible to obtain accuracies exceeding 0.1% in properly designed geometries. However, the fact that the quantization can probably be destroyed by a single impurity, located at an unfavourable position, will exclude the possible use of QPCs as a resistance standards.
The experiments show that the transport through the QPCs remains ballistic up to at least 4.2K. This means that inelastic processes are not yet important at 4.2K. The conductance quantization breaks down due to energy averaging. It is shown that the application of a magnetic field leads to a gradual transition to magnetic quantization. The major difference between the quantization in the absence of a field and the quantum Hall effect is the nature of the scattering. In the absence of a magnetic field the backscattering from impurities or irregularities in the confining potential will destroy the quantization. As discussed in the next part of the paper, backscattering is suppressed by a sufficiently high magnetic field.
IV. QUANTUM ADIABATIC TRANSPORT IN HIGH MAGNETIC FIELDS
A. INTRODUCTION
Electron transport in a two-dimensional electron gas (2DEG) subjected to high magnetic fields has been the subject of many experimental and theoretical investigations. An important incentive for these investigations has been the discovery of the quantum Hall effect (QHE)54'55. In high
magnetic fields the Hall resistance exhibits quantized plateaux at values h/(e2 NjJ, with NL the
number of occupied Landau levels. The formation of these plateaux is accompanied by a vanishing of the longitudinal resistance. In particular the extreme accuracy of the quantization (at present known to be better than one part in 107) has presented a major theoretical challenge.
Recently a simple and appealing model for electron transport in the QHE regime has been proposed53-56-57. The main ingredients of this model are the so-called edge channels. These edge
channels consist of the current carrying electron states of each Landau level, and are located at the boundaries of the 2DEG. In his paper Biittiker particularly emphasizes that the voltage and current contacts which are attached to the 2DEG can play an important role in the establishment of the quantum Hall effect53. We will give a brief description of this model in section B.
With the advent of the quantum point contacts (QPCs) it has become possible to perform a detailed experimental investigation of transport in the two-dimensional electron gas. The power of the quantum point contact technique is that the transmission properties of QPCs can be controlled by the applied gate voltage. Also their transmission can be measured first, and the QPCs can subsequently be used as voltage and current probes with well-known properties. In summary the most important property of QPCs in high magnetic fields is that they, when used as current probes, can selectively inject current into specific edge channels. When used as voltage probes they can selectively measure the occupation of specific edge channels. A description of the transport in single QPCs is given in section C.
The selective properties of the QPCs allow us to perform a detailed study of the role of contacts in the QHE. An important result of our investigation is that scattering between adjacent edge channels (located at the same 2DEG boundary) can be very weak, which means that adiabatic transport takes place in high magnetic fields. Electrons travel through the 2DEG with conservation of their quantized magnetic energy, with only little chance of being scattered into
Fig. 10 Cross-section of a 2DEG, showing the occupied electron states of two Landau levels, in the presence of a current flow, (a) shows the regular situation. The arrow indicates intra-Landau level scattering, (b) shows the occupied electron states when current is injected selectively with a QPC. The arrow illustrates inter-Landau level scattering between adjacent edge channels.
other edge channels. The combination of this quantum adiabatic transport with the selective population and detection of edge channels by QPCs has resulted in the observation of an anomalous integer QHE3 (section D). The quantization of the Hall conductance is not determined
by the number of Landau levels in the bulk 2DEG, but by the number of Landau levels in the QPCs instead. Related phenomena are the anomalous quantization of the longitudinal resistance (section E) and the quantum adiabatic transport in QPCs in series, (section F)
Next we use QPCs to perform a detailed study of the scattering between edge channels. In section G we give a description of the scattering processes in the 2DEG. We make a distinction between intra-Landau level scattering (scattering between edge channels belonging to the same Landau level) and inter-Landau level scattering (scattering between edge channels belonging to different Landau levels). In section H we give a theoretical description of electron transport in the presence of Shubnikov-de Haas (SdH) backscattering. In this model the SdH oscillations arise from backscattering of electrons in the upper (highest occupied) Landau level. In section J we present experimental results which show that the SdH oscillations can be suppressed, either by selective population or by selective detection of edge channels. These results show that under certain circumstances the scattering between adjacent edge channels can be weak even on macroscopic (>200|im) length scales4. Another illustration of the non-local transport is given in
section K, where we demonstrate that the voltage measured with a particular voltage probe can be strongly affected by the presence of an adjacent voltage probe. This shows that QPCs can act as controllable edge channel mixers. Finally, we give a discussion of our results in section L.
B. QUANTUM TRANSPORT IN HIGH MAGNETIC FIELDS
In this section we give a brief description of transport in high magnetic fields in a 2DEG free of imperfections. We assume that the electrons are laterally confined in the 2DEG by the electrostatic potential given in fig. 10. The electrostatic potential V(x) has a flat part in the middle, and rises at the edges of the 2DEG. The width W of these depletion regions at the edges is usually of the order of 100 to 500 nm in actual devices. The following dispersion relation is obtained for the electron states in the 2DEG55:
1 1 1 E
The energy of an electron consists of four terms: the electrostatic energy eV(x) at the center coordinate x = lb2 ky of the electron wave function flb= V h/eB), the quantized cyclotron energy
(n is the Landau level index), the kinetic energy associated with the drifting motion of the electrons in crossed E and B fields, and the Zeeman spin splitting term. Evaluation of the third term in (11) with a typical value E = EpAeW) = 104 - 105 V/m for the electric field at the
boundary of the 2DEG, shows that this term can usually be neglected in high magnetic fields (B>1T).
The relevant electrons for the transport are those at the Fermi energy EF- We now obtain a very simple picture for electron transport when we note that electrons with different Landau level indices n flow along different equipotential lines V(x), which are given by the condition:
eV(x) = EF - (n - j ) hcoc + i-guBB (12)
Because this condition is usually satisfied at the edges of the 2DEG one speaks about transport in edge channels. These edge channels are located at the intersections of the Landau levels and the Fermi energy. (The validity of the edge channel description will be discussed at the end of this section). Fig. 10(a) shows the occupied electron states of two Landau levels when a net current I flows in the 2DEG. This current is a result of the difference in occupation of the right and left hand edge channels, which carry current in opposite directions. It can be shown57 that the net
current I is independent of the details of the dispersion of the Landau levels and is given by:
I = NL £-aiL-uR) (13)
The current carried by each Landau level is simply given by e/h multiplied by the electrochemical potential difference HL-M-R between right and left edge channels. Voltage probes attached to either side of the 2DEG will measure this electrochemical potential difference and the Hall resistance is:
This is the elementary explanation for the quantum Hall effect53-57. It is important to note that
that this explanation does not require localization. A necessary condition is that the right-hand contact exclusively measures the electrochemical potential of the right-hand edge channels and vice versa.
A major deficiency of the above description is that it does not take into account screening. A description of the transport in terms of edge channels is only possible by assuming that the Fermi level EF in the interior of the 2DEG can be positioned in between the flat parts of two consecutive Landau levels. In an actual 2DEG this is not possible, because the electron density is fixed, and the Fermi level will be pinned to the upper Landau level. Selfconsistent calculations, which take into account the finite width of the 2DEG, support this picture.58-59 They show that the large
degeneracy of the Landau levels can result in perfect screening, and the upper Landau level may be pinned to the Fermi energy in a considerable region of the 2DEG. Because all electron states of the low-lying Landau levels remain occupied in the interior of the 2DEG, the edge channel description will remain valid for these Landau levels. We will assume in the remainder of this paper that the edge channel description can also be used for the upper Landau level.
C. HIGH MAGNETIC FIELD TRANSPORT IN QUANTUM POINT CONTACTS
The transport properties of QPCs in zero and non-zero magnetic field have been discussed in section III. In this section we focus on the high field regime, in which the electron transport can be described in terms of edge channels. We first note that the electrostatic potential landscape at the QPCs has a saddle shape. Besides the lateral confinement of the electrons, the potential in the QPCs is also raised relative to the bulk 2DEG. This potential barrier V0 is a function of the
applied gate voltage (see section IH G). In high magnetic fields (when coc » C0Q) the transport is exclusively determined by V0 and Cue, and independent of co0. The number of occupied Landau
levels in the QPC is reduced relative to the bulk and given by: N = Int {(EF -eV0)/(hrac) + 1/2}
Fig. 11 illustrates the current flow in edge channels through the QPC for three different values of the potential barrier V0. In fig. 11(a) no potential barrier is present, and all edge channels are
transmitted. The QPC does not influence the electron transport. This is the case when the QPC is formed at -0.6V. In fig. 11(b) the gate voltage is reduced, and a potential barrier is created. In this particular example a fraction T of the electrons in the second edge channel is transmitted through the QPC and a fraction R=l-T is reflected. Note that the electrons in the edge channel with the highest Landau level index are the first to be reflected, since this edge channel follows the lowest equipotential line. In fig. 11(c) the potential barrier is such that this edge channel is
(a)
2
Hï
Fig. 11 High magnetic field transport in a QPCfor three different values of the potential barrier V0, illustrated for the case of two occupied Landau levels (see text).
completely reflected, whereas the other is still completely transmitted. We now write the two terminal resistance Gc of the QPC as:
el 2e2
Gc= - 2 J - = ^ - ( N + T ) (15)
in this expression N denotes the number of edge channels which are fully transmitted through the QPC, and T the transmission of the partially transmitted edge channel. We assume that at the QPC only one edge channel can be partially transmitted, and all others are either completely reflected or completely transmitted. Also we assume that no scattering between edge channels occurs in or near the QPC. The observation of an anomalous integer quantum Hall effect (section D), shows that these assumptions are justified for B>1.5T in our device geometry.
By considering the edge channels which flow away from the QPC it can be seen that they are occupied up to different electro chemical potentials HA or (1B , depending on wether they have been transmitted or reflected at the QPC (see fig. 11(c)). This means that a QPC, when used as a current probe, can selectively inject current into only those edge channels which are transmitted by the QPC. Similarly, when used as a voltage probe, a QPC will exclusively measure the electro chemical potentials of those edge channels which are transmitted through the QPC. We will employ these properties to study the role of contacts in the QHE (sections D-F), and also to perform a detailed study of the scattering processes in a 2DEG (sections G-K)
D. ANOMALOUS INTEGER QUANTUM HALL EFFECT
In this section we investigate the (quantization of the) Hall conductance, when it is measured with QPCs which couple selectively to specific edge channels. In the regular QHE, when the Hall conductance is measured with ideal bulk contacts (which couple ideally to all available NL edge channels), the quantization of G H is determined by the number of bulk Landau levels N L5 3.
The formation of a quantized plateau in GH is accompanied by a vanishing of the longitudinal resistance RL. It is shown in this section that the selective coupling of the QPCs, combined with the absence of scattering between edge channels, leads to an anomalous quantization of the Hall conductance, in which GH is not determined by the number of bulk Landau levels NL, but by the number of Landau levels in the QPCs instead3. At the same time the longitudinal resistance
(a)
(b)
-2.4QPC A
QPC B
J_ i . . . i . -2 - 1 . 6 - 1 . 2 GATE VOLTAGE (V)Fig. 12 (a) Electron flow in edge channels, resulting in an anomalous quantization of the Hall conductance GH = 2e2/h. (b) Comparison between the two-terminal conductances GA and GB of
the point contacts with the Hall conductance GH- The Hall conductance shows an anomalous plateau at 2e2lh, in agreement with eqs. (17) - (20) The rapid rise in GH below -2.2V is an
Hall and longitudinal resistances, as well as the adiabatic transport in series QPCs (section F) have the same origin: the selective population and detection of edge channels, combined with the absence of scattering between edge channels in the region between the QPCs.
In an identical device Van Houten et al.5-6-7 have studied coherent electron focusing at low
fields. Electron focusing peaks were observed in both Hall and longitudinal resistances as a result of the ballistic transport in between the QPCs. At low fields many edge channels are occupied, and the focusing peaks can be explained with a classical calculation60. In this paper we
are interested in the high field regime, where a fully quantum mechanical description in terms of edge channels is required.
We calculate the Hall conductance GH, which is defined as the ratio of the current I and the voltage difference between contacts 1 and 6, when 5 and 4 are used as current probes (see fig. 1 and fig. 12(a)). The two QPCs serve as adjacent current and voltage probes. We first perform the calculation for a forward directed magnetic field. We assume that all bulk contacts are ideal. This means that these contacts absorb the total current which flows along the 2DEG boundary, and that all NL edge channels which leave a bulk contact are equally occupied and have the same electrochemical potential53. An ideal contact therefore has a two-terminal conductance G = 2e2/h
NL. In the calculation we put Hi=0 for convenience. By employing the general Biittiker-Landauer formula for four-terminal measurements61, an expression for G H can be given in terms
of transmission probabilities between the bulk contacts. However, we prefer to give a step-by-step derivation of the result, which brings out the physics involved more clearly.
The two terminal conductance of the current QPC A can be written as:
el 2e2
GA = —£ £— = T r ( N A + TA), (16)
(H5-M-1) h
in which NA denotes the number of fully transmitted (spin-degenerate) edge channels, and TA the transmission of the partially transmitted edge channel through QPC A. Whenever NA < NL the injected current is distributed unequally over the available N L bulk edge channels. (Fig. 12(a) illustrates the electron flow for the case N L = 2 , and NA.NB =1). The lowest N A channels are fully occupied up to us, and carry a current (2e/h) NA H5- Channel N A + 1 is only partially occupied, and carries a current (2e/h) \i$ T\. Channels NA + 2 up to N L are not populated at all, and carry no current. The injected current flows towards the voltage QPC B. At this point we assume that no scattering between edge channels takes place in the region between the QPCs.
