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Non-local magnetoresistance at the crossover between the classical and quantum transport regimes
...:~,,.:::::i,‘;iii::j,:::::j+
,,.
:.. : :9.;,Y::/:.:,....:..~: ..:. ..,:,,..:..:..:... ::: ,, : :
Surface Science North-Holland
263 (1992) 298-302
“’
,.....,
:: +,)).$
.
;
;. ,..
‘surface science .: ,.:::::,... :j::::,::::( j.,‘.:j......>,jZ,, ::. ~,/:):~~,..:: .:.’ :j/....:,j:j ,,,.::: ,.,,. .::>: :y:: ;: ..;,.. 1’.:
Non-local magnetoresistance at the crossover between the classical and quantum transport regimes A.K.
Geim
Depurtmmt
C.D.W. Depurtment Received
(Heym)
‘, P.C.
Main,
P.H.
of Physics. Unirwsify of‘ Nottinglwn, Wilkinson and S.P. Beaumont of Electronics & Ekctricul Engineering, 3 June
1901: accepted
for publication
Beton,
P. Stieda
Nottinghmn
Uniwrsity
26 August
‘, L. Eaves
NC;7 2KD. l/K
of Glusgyn.v, (;luv~ow Gl2 8QQ. tiK 1YYl
A new type of non-local oscillatory magnetoresiatancc effect is observed in heavily-doped II ‘-GaAs wires over a limited range 01 temperature (10 5 T 5 SO K) and at magnetic fields sufficiently large to give rise to Landau quantisation. The effect is observed when ouantum ballistic transport along the edges coexists with diffusive and dissipative conduction in the bulk. In a similar range of magnetic field, universal conductance fluctuations observed in the wires cannot be explained hy current theory
The observation of non-local effects in the resistance and magnetoresistance of small, multiterminal metallic and semiconducting structures has provided new insights into the physics of electrical conduction [l-7]. To date, observations of this effect fall into three distinct categories: (1) non-local universal conductance fluctuations (UCF) in “dirty” semiconductors [2] and metals [3] in which the electrical conduction is diffusive; (2) ballistic transfer resistance in two-dimensional electron gases (2DEGs) [1,4]; (3) non-local Shubnikov-de Haas (SdH) oscillations in the quantum Hall effect regime of 2DEG’s [S-7]. In all three categories the non-local effect is best seen at low temperatures. We report the existence of a new type of non-local oscillatory magnetoresistance under conditions in which diffusive transport would be expected to occur but which is not reliant on the
’ Permanent address: Institute of Microelectronics Technology, USSR Academy of Sciences, Chernogolovka 142432. USSR. ’ Permanent address: Institute of Physics, Cukrovarnicka 10. CS-162 00 Prague. Czechoslovakia. OO~Y-~O2X/Y2/$OS.oO CJ 1902 - Elsevier
Science
Publishers
phase coherence of the electrons. The effect has two striking features. First, it is completely quenched at low temperatures (T < 4 K) and, second, it corresponds to a 100% modulation of the non-local magnetoresistance even though the local SdH oscillations are relatively weak ( < I c/r). We attribute this effect to the coexistcncc of classical bulk conduction and quantum conduction in edge states [8]. The edge states are able to carry the current to classically inaccessible regions but the voltage appears only when there is coupling between the edge states and bulk with subsequent dissipation. The structure used in our experiment is shown schematically in fig. 1 (inset). It was fabricated using electron-beam lithography and dry etching from an n+-GaAs epilaycr (n = 1.1 X 10” mm’) grown by molecular beam epitaxy on a semi-insulating substrate. We estimate an effective conducting channel thickness - 30 nm with four electrically-quantised, two-dimensional subbands occupied. Channel widths, M’.of 150, 250. 350 and 450 nm, all allowing for sidewall depletion, have been investigated. Adjacent pairs of probes arc separated by L = I pm although there are addi-
H.V. and Yamada
Science
Foundation.
All Irights I-cscrved
A.K. Geim et al. / Non-local magnetoresistance in heacily doped n +-GaAs
299
2 1
0 g -’ @z
1
0 -1 -2
10
Fig. 2. Non-local UCF The range of magnetic
0
i
4
B6(T)8
1’0
1'2
Fig. 1. The non-local magnetoresistance traces for .L.= 1 pm and w = 150 nm at different temperatures (a, b and c; d is a threefold magnification of c). Curves b and c are offset for clarity.
tional pairs (not shown in fig. 1) beyond e and f with separation 2 and 50 pm. Each voltage probe has the same thickness and width as the main wire and is at least 7 pm long before opening out to a large contact pad. We refer to this pad as the contact and the length of wire between it and the main wire as the probe. The electron mobility, p, is 0.16 m2 V-’ s-l giving a bulk elastic mean free path, An, of 50 nm and W,T = 1 at 6 T. The bulk resistivity of the material, p = 800 1R 0 -l. We have measured the non-local resistance with V,, the voltage difference between Rij,kl contacts k and 1 due to a current lij between contacts i and j. Standard low-frequency (12 Hz) lock-in techniques were employed with currents down to 50 nA to avoid electric heating [9]. For voltage probes a distance L from the current probes, a and b, in our devices the classical voltage drop is Vkl = pZ,,[(l + CYW,T)/~] exp(-rl/wl [lo] where (Y is a geometric factor and 1CY1 < 1. For w = 150 and 250 nm, this voltage is vanishingly small [(V,,/Z,,) < 0.01 a]. Magnetoresistance Rab,cd (i.e., L = 1 pm> traces are shown in fig. 1 for three temperatures and w = 150 nm. For T < 10 K non-local UCF
12
B 07
for w = 250 nm, L = 1 Km at 4.2 K. field is O-2 T (upper) and lo-12 T (lower).
are clearly visible; their rms amplitude, AR, is well described by AR N p2(e2/h) exp( - a L/l,) with a = 1.1 f 0.2 and a temperature dependent phase-coherence length, I, [2]. At 4.2 K, I, = 0.3 pm and decreases as T- ‘I2 at higher temperatures. In low magnetic fields (B < 3 T), with W,T < 1, the Lee-Stone correlation field [ll], A B, is AB = B(h/e>/{l,[min(w, 1,>]], with B = 0.4. For w,r > 1, AB increases as can be seen in the raw data of fig. lb and more clearly in fig. 2. From 0 to 2 T the typical magnetic field period of the UCF is much shorter than for 10 to 12 T. In fig. 3 we plot AB against magnetic field for w = 250 nm (nominal width 0.3 pm> and another wire with w = 450 nm (nominal width 0.5 pm). The solid line is A B = A B(O)[l + (w,r>*] which appears to be a reasonable empirical fit to the data. This variation of AB with magnetic field is not predicted by theory [ll].
(1 0.3q
n+ wire O.!$m n’ wire
l
0.00
LI 0
4
2
4
6
8
10
12
B CT) Fig. 3. Lee-Stone correlation field, AE, at 4.2 K, plotted against B for wires of nominal width 0.3 pm (W = 250 nm) and 0.5 wrn (w = 450 nm). The solid line is AE = AB(O)[l +
(w,T)21.
300
A.K.
Geim et ul. / Non-local
mugnetoresistatm
For T > 10 K the UCF are exponentially damped and we observe a new type of oscillation as shown in fig. lc and, with threefold magnification, in fig. Id. Note that SdH oscillations measured in a standard four probe geometry (e.g., Rac,bd) have amplitudes only y 1% of the local resistance in 10 T at 30 K. This also distinguishes our effect from the non-local SdH oscillations investigated in high mobility 2DEGs [S-7]. Oscillations, such as those shown in fig. lc are seen in all our samples and for many different probe configurations. The amplitude of the peaks decreases with increasing L and no effect is observable when the probes are separated by 50 pm. Comparison of figs. lb and lc shows clearly that the non-local oscillations do not increase in size upon cooling from 30 to 9 K although the effect is masked by the presence of the UCF at the lower temperature. For comparison the local SdH oscillations incwase by a factor of 4 in the same temperature interval. The behaviour with temperature is shown more clearly in fig. 4 where Knh,,,r (i.e., L = 2 pm> at high magnetic field is shown for five different temperatures. In this configuration the UCF are sufficiently small, even at liquid-helium temperatures, to allow a quanti-
I----
-0.1
i 8
/
I
I
/
9
B Fig. 4. The non-local
I
---’
‘?T)”
high field magnetoresistance
sured at voltage probes with dashed lines represent
” R,,,,,,, mca-
L = 2 pm and w = 150 nm. The
the respective zero resistances.
iti heur,ily doped II ‘-(TOA\
T (K) Fig. 5. The temperature
dependence
of R;,,,,,,
~OI- two peaks
shown in fig. 4. Solid lines are the best fit to the temperature dependence
using eq. (1); dots are the experimental
inset shows the temperature
dependence
local SdlI oscillations (solid lint
data. The
of the amplitude
of
theory).
tativc study of the new effect. Fig. 5 shows the amplitudes of the peaks at 8.7 and 11 T plotted against temperature. The amplitude is defined as the actual value of the resistance at the relevant field. Figs. 4 and 5 show clearly that the effect disappears at both low and high temperatures. The inset of fig. 5 shows the temperature dependence of the relative amplitude of the 1~~1 SdH oscillations for a section of length 1 pm. The solid line in the inset is the expected temperature dependence [lo] [(kT/hw,) cosech(2$k,,T/ izo,)]. Individual non-local magnctoresistancc traces may be different from each other. Peak amplitudes are not generally the same for equal probe spacing and may differ by up to a factor of 2. Also peak positions are not always in the same field; sometimes maxima in one trace occur at the same field as minima in another. However, K,,,x,(B) = K, ,,,, ( -B) always, as required by the Onsagcr relations. In contrast the local SdH oscillations have no change in phase and only very small changes in amplitude for different sections of the wire. Although the local and non-local oscillations always have equal ( 1/B) periodicity. phase shifts as large as 180’ can occur. It is well-known that in high mobility 2DEG systems the edge states are decoupled from the bulk. We may consider the electrons in the edge
A.K. Geim et al. / Non-local magnetoresistance in heavily doped n ‘-GaAs
states to have a mean free path, A,, which in those systems may be of the order of 1 mm [5]. In our low mobility structure we may still expect AE = 1 pm if we scale A, with the number of scattering centres. This is reasonable if we assume that both A, and A, scale with the density of scattering centres. The large difference between A, and A, induces a difference in chemical potential between the edge and the bulk state which is proportional to the applied current [12]. We attribute the observed effects to this absence of local equilibrium between edge and bulk states. Referring to the inset of fig. 1, to measure a voltage difference between contacts c and d, when the current is applied between contacts a and b, involves dissipation occurring somewhere between c and d. In the high mobility case the dissipation of the edge current occurs in the contact regions. This cannot happen in our low mobility structures because the edge electrons which are injected sideways from the main current path cannot reach the voltage contacts. Instead they are either transferred to the opposite edge without energy losses or thermalised within the bulk somewhere in the wire and probe regions. The voltage in our experiment, therefore, measures the dissipation which occurs in and around the probe regions. At low temperatures, our effect disappears when the electrons entering the bulk region can traverse the width of the wire without suffering an inelastic collision and dissipating energy. We expect the voltage drop Vcd to have the form Vcda&P(T)
301
term, P(T), will be small. On the other hand, there will be strong coupling between edge states and bulk when the Fermi energy lies within a bulk Landau level but A, will be correspondingly small. The maximum values of Vcd, given by eq. (11, have the same periodicity but not necessarily the same phase as the Landau levels crossing the Fermi energy, in agreement with experiment. We introduce the dissipation term in a form resulting from the energy balance equation: P(T) where AE is the differa [l + exp(AE/k,T)]-’ ence between the Fermi energy and the energy of the maximum of the density of states in the bulk. It can be determined directly from the phase shift between the local SdH maxima and the corresponding non-local magnetoresistance peaks. The solid lines in fig. 5 are the best fit to the temperature dependence of eq. (1) using AE = 4 meV for the upper curve and AE = 3.2 meV for the lower curve, calculated as discussed above. The only adjustable parameter is the overall amplitude. The fitted values for the amplitudes of both peaks shown in fig. 5 scale with the amplitude of the corresponding local SdH maxima. Given the approximate derivation of eq. (1) the agreement between theory and experiment is very good. To summarise, we have observed a new nonlocal magnetoresistance effect in the intermediate regime between quantum edge conduction and classical bulk transport. The observed resistance is due to dissipation in the bulk rather than in the electrical contacts. However, quantum conduction in the edge states plays an important role in carrying the current to classically inaccessible regions.
exp( -L/An)
x ( kBT/fiw,)cosech(2rr2kBT/tiw,).
(1)
At high T the voltage is limited by the same factor as for local SdH oscillations. The exponential factor is the probability that injected edge currents reach the intersection of the wire with the voltage probes. P(T) is the fraction of these electrons which thermalise. Thermalisation involves a transition from edge to bulk followed by a relaxation to local equilibrium. We expect that A, will be a maximum when the Fermi energy is in between bulk Landau levels but the dissipation
This work is supported by SERC. A.K.G., P.S. and P.H.B. wish to thank the Royal Society for financial support.
References [ll See papers by: M. Roukes, M. Biittiker and C.W.J. Beenakker, in: Electronic Properties of Multilayers and Low-Dimensional Semiconductor Structures, Eds. J.M. Chamberlain, L. Eaves and J.-C. Portal, NATO ASI Series B 231 (Plenum, London, 1990).
302
A. K. Geim et al. / Non-local magnetoresistartce
[2] A.D. Benoit, C.P. Umbach, R.B. Laibowitz and R.A. Webb, Phys. Rev. Lett. 5X (1987) 2343; H. Haucke, S. Washburn, A.D. Benoit, C.P. Umbach and R.A. Webb, Phys. Rev. B 41 (1990) 12454. [3] W.J. Skocpol. P.M. Mankiewich, R.E. Howard. L.D. Jackel, D.M. Tennant and A.D. Stone, Phys. Rev. Lett. 58 (1987) 2347. 141 H. van Houten, C.W.J. Beenakker. J.G. Williamson, M.E. Brockaart, P.H.M. van Loosdrecht, B.J. van Wees, J.E. Mooij, C.T. Foxon and J.J. Harris, Phys. Rev. B 39 (1989) 85.56: J. Spector, H.L. Stormer, K.W. Baldwin. L.N. Pfeiffer and K.W. West, Surf. Sci. 22X (1990) 283; U. Sivan, M. Heiblum, C.P. Umbach and H. Shtrikman. Phys. Rev. B 41 (1990) 7937. [5] P.L. McEuen, A. Szafer, C.A. Richter, B.W. Alphenaar, J.K. Jain, A.D. Stone, R.G. Wheeler and R.N. Sacks. Phys. Rev. Lett. h4 t 1990) 2062. [6] S. Takaoka, H. Kubota, K. Murase, Y. Takagaki, K. Gamo and S. Namba. Solid State Commun. 75 (1990) 293.
in iwa~~ily doped )I +-&As
[7] J.K. Wang. V.J. Goldman, M. Santos and M. Shayegan, Bull. Am. Phys. Sot. 3h (1991) 913. [8] B.1. Halperin, Phys. Rev. B 75 (1982) 2185; P. Stieda. J. Kucera and A.H. MacDonald. Phys. Rev. Lett. SY (1987) 1973; M. Biittiker. Phys. Rev. B 38 (198X) 9375. [Y] R.P. Taylor. P.C. Main, L. Eaves, S.P. Beaumont. S. Thorns and C.D.W. Wilkinson. Superlattices Microstruct. s (1989) s7s. [IO] K. Seeger, Semiconductor Physics (Springer. Berlin, I973 ). [II] P.A. Lee and A.D. Stone. Phys. Rev. Lett. 55 (1985) 1622: B.L. Altshuler. Pis’ma Zh. Eksp. Teor. Fiz. 41 (lY85) 530 [JETP Lett. 41 (iY85) 64x1. [121 R.J. Haug, J. Kucera. P. Stfeda and K. von Klitzing, Phys. Rev. B 3Y (1989) 10892. The difference I/L~,, follows from eq. (3.5) in which the transition probability p is related to I/A,..
,,.
:.. : :9.;,Y::/:.:,....:..~: ..:. ..,:,,..:..:..:... ::: ,, : :
Surface Science North-Holland
263 (1992) 298-302
“’
,.....,
:: +,)).$
.
;
;. ,..
‘surface science .: ,.:::::,... :j::::,::::( j.,‘.:j......>,jZ,, ::. ~,/:):~~,..:: .:.’ :j/....:,j:j ,,,.::: ,.,,. .::>: :y:: ;: ..;,.. 1’.:
Non-local magnetoresistance at the crossover between the classical and quantum transport regimes A.K.
Geim
Depurtmmt
C.D.W. Depurtment Received
(Heym)
‘, P.C.
Main,
P.H.
of Physics. Unirwsify of‘ Nottinglwn, Wilkinson and S.P. Beaumont of Electronics & Ekctricul Engineering, 3 June
1901: accepted
for publication
Beton,
P. Stieda
Nottinghmn
Uniwrsity
26 August
‘, L. Eaves
NC;7 2KD. l/K
of Glusgyn.v, (;luv~ow Gl2 8QQ. tiK 1YYl
A new type of non-local oscillatory magnetoresiatancc effect is observed in heavily-doped II ‘-GaAs wires over a limited range 01 temperature (10 5 T 5 SO K) and at magnetic fields sufficiently large to give rise to Landau quantisation. The effect is observed when ouantum ballistic transport along the edges coexists with diffusive and dissipative conduction in the bulk. In a similar range of magnetic field, universal conductance fluctuations observed in the wires cannot be explained hy current theory
The observation of non-local effects in the resistance and magnetoresistance of small, multiterminal metallic and semiconducting structures has provided new insights into the physics of electrical conduction [l-7]. To date, observations of this effect fall into three distinct categories: (1) non-local universal conductance fluctuations (UCF) in “dirty” semiconductors [2] and metals [3] in which the electrical conduction is diffusive; (2) ballistic transfer resistance in two-dimensional electron gases (2DEGs) [1,4]; (3) non-local Shubnikov-de Haas (SdH) oscillations in the quantum Hall effect regime of 2DEG’s [S-7]. In all three categories the non-local effect is best seen at low temperatures. We report the existence of a new type of non-local oscillatory magnetoresistance under conditions in which diffusive transport would be expected to occur but which is not reliant on the
’ Permanent address: Institute of Microelectronics Technology, USSR Academy of Sciences, Chernogolovka 142432. USSR. ’ Permanent address: Institute of Physics, Cukrovarnicka 10. CS-162 00 Prague. Czechoslovakia. OO~Y-~O2X/Y2/$OS.oO CJ 1902 - Elsevier
Science
Publishers
phase coherence of the electrons. The effect has two striking features. First, it is completely quenched at low temperatures (T < 4 K) and, second, it corresponds to a 100% modulation of the non-local magnetoresistance even though the local SdH oscillations are relatively weak ( < I c/r). We attribute this effect to the coexistcncc of classical bulk conduction and quantum conduction in edge states [8]. The edge states are able to carry the current to classically inaccessible regions but the voltage appears only when there is coupling between the edge states and bulk with subsequent dissipation. The structure used in our experiment is shown schematically in fig. 1 (inset). It was fabricated using electron-beam lithography and dry etching from an n+-GaAs epilaycr (n = 1.1 X 10” mm’) grown by molecular beam epitaxy on a semi-insulating substrate. We estimate an effective conducting channel thickness - 30 nm with four electrically-quantised, two-dimensional subbands occupied. Channel widths, M’.of 150, 250. 350 and 450 nm, all allowing for sidewall depletion, have been investigated. Adjacent pairs of probes arc separated by L = I pm although there are addi-
H.V. and Yamada
Science
Foundation.
All Irights I-cscrved
A.K. Geim et al. / Non-local magnetoresistance in heacily doped n +-GaAs
299
2 1
0 g -’ @z
1
0 -1 -2
10
Fig. 2. Non-local UCF The range of magnetic
0
i
4
B6(T)8
1’0
1'2
Fig. 1. The non-local magnetoresistance traces for .L.= 1 pm and w = 150 nm at different temperatures (a, b and c; d is a threefold magnification of c). Curves b and c are offset for clarity.
tional pairs (not shown in fig. 1) beyond e and f with separation 2 and 50 pm. Each voltage probe has the same thickness and width as the main wire and is at least 7 pm long before opening out to a large contact pad. We refer to this pad as the contact and the length of wire between it and the main wire as the probe. The electron mobility, p, is 0.16 m2 V-’ s-l giving a bulk elastic mean free path, An, of 50 nm and W,T = 1 at 6 T. The bulk resistivity of the material, p = 800 1R 0 -l. We have measured the non-local resistance with V,, the voltage difference between Rij,kl contacts k and 1 due to a current lij between contacts i and j. Standard low-frequency (12 Hz) lock-in techniques were employed with currents down to 50 nA to avoid electric heating [9]. For voltage probes a distance L from the current probes, a and b, in our devices the classical voltage drop is Vkl = pZ,,[(l + CYW,T)/~] exp(-rl/wl [lo] where (Y is a geometric factor and 1CY1 < 1. For w = 150 and 250 nm, this voltage is vanishingly small [(V,,/Z,,) < 0.01 a]. Magnetoresistance Rab,cd (i.e., L = 1 pm> traces are shown in fig. 1 for three temperatures and w = 150 nm. For T < 10 K non-local UCF
12
B 07
for w = 250 nm, L = 1 Km at 4.2 K. field is O-2 T (upper) and lo-12 T (lower).
are clearly visible; their rms amplitude, AR, is well described by AR N p2(e2/h) exp( - a L/l,) with a = 1.1 f 0.2 and a temperature dependent phase-coherence length, I, [2]. At 4.2 K, I, = 0.3 pm and decreases as T- ‘I2 at higher temperatures. In low magnetic fields (B < 3 T), with W,T < 1, the Lee-Stone correlation field [ll], A B, is AB = B(h/e>/{l,[min(w, 1,>]], with B = 0.4. For w,r > 1, AB increases as can be seen in the raw data of fig. lb and more clearly in fig. 2. From 0 to 2 T the typical magnetic field period of the UCF is much shorter than for 10 to 12 T. In fig. 3 we plot AB against magnetic field for w = 250 nm (nominal width 0.3 pm> and another wire with w = 450 nm (nominal width 0.5 pm). The solid line is A B = A B(O)[l + (w,r>*] which appears to be a reasonable empirical fit to the data. This variation of AB with magnetic field is not predicted by theory [ll].
(1 0.3q
n+ wire O.!$m n’ wire
l
0.00
LI 0
4
2
4
6
8
10
12
B CT) Fig. 3. Lee-Stone correlation field, AE, at 4.2 K, plotted against B for wires of nominal width 0.3 pm (W = 250 nm) and 0.5 wrn (w = 450 nm). The solid line is AE = AB(O)[l +
(w,T)21.
300
A.K.
Geim et ul. / Non-local
mugnetoresistatm
For T > 10 K the UCF are exponentially damped and we observe a new type of oscillation as shown in fig. lc and, with threefold magnification, in fig. Id. Note that SdH oscillations measured in a standard four probe geometry (e.g., Rac,bd) have amplitudes only y 1% of the local resistance in 10 T at 30 K. This also distinguishes our effect from the non-local SdH oscillations investigated in high mobility 2DEGs [S-7]. Oscillations, such as those shown in fig. lc are seen in all our samples and for many different probe configurations. The amplitude of the peaks decreases with increasing L and no effect is observable when the probes are separated by 50 pm. Comparison of figs. lb and lc shows clearly that the non-local oscillations do not increase in size upon cooling from 30 to 9 K although the effect is masked by the presence of the UCF at the lower temperature. For comparison the local SdH oscillations incwase by a factor of 4 in the same temperature interval. The behaviour with temperature is shown more clearly in fig. 4 where Knh,,,r (i.e., L = 2 pm> at high magnetic field is shown for five different temperatures. In this configuration the UCF are sufficiently small, even at liquid-helium temperatures, to allow a quanti-
I----
-0.1
i 8
/
I
I
/
9
B Fig. 4. The non-local
I
---’
‘?T)”
high field magnetoresistance
sured at voltage probes with dashed lines represent
” R,,,,,,, mca-
L = 2 pm and w = 150 nm. The
the respective zero resistances.
iti heur,ily doped II ‘-(TOA\
T (K) Fig. 5. The temperature
dependence
of R;,,,,,,
~OI- two peaks
shown in fig. 4. Solid lines are the best fit to the temperature dependence
using eq. (1); dots are the experimental
inset shows the temperature
dependence
local SdlI oscillations (solid lint
data. The
of the amplitude
of
theory).
tativc study of the new effect. Fig. 5 shows the amplitudes of the peaks at 8.7 and 11 T plotted against temperature. The amplitude is defined as the actual value of the resistance at the relevant field. Figs. 4 and 5 show clearly that the effect disappears at both low and high temperatures. The inset of fig. 5 shows the temperature dependence of the relative amplitude of the 1~~1 SdH oscillations for a section of length 1 pm. The solid line in the inset is the expected temperature dependence [lo] [(kT/hw,) cosech(2$k,,T/ izo,)]. Individual non-local magnctoresistancc traces may be different from each other. Peak amplitudes are not generally the same for equal probe spacing and may differ by up to a factor of 2. Also peak positions are not always in the same field; sometimes maxima in one trace occur at the same field as minima in another. However, K,,,x,(B) = K, ,,,, ( -B) always, as required by the Onsagcr relations. In contrast the local SdH oscillations have no change in phase and only very small changes in amplitude for different sections of the wire. Although the local and non-local oscillations always have equal ( 1/B) periodicity. phase shifts as large as 180’ can occur. It is well-known that in high mobility 2DEG systems the edge states are decoupled from the bulk. We may consider the electrons in the edge
A.K. Geim et al. / Non-local magnetoresistance in heavily doped n ‘-GaAs
states to have a mean free path, A,, which in those systems may be of the order of 1 mm [5]. In our low mobility structure we may still expect AE = 1 pm if we scale A, with the number of scattering centres. This is reasonable if we assume that both A, and A, scale with the density of scattering centres. The large difference between A, and A, induces a difference in chemical potential between the edge and the bulk state which is proportional to the applied current [12]. We attribute the observed effects to this absence of local equilibrium between edge and bulk states. Referring to the inset of fig. 1, to measure a voltage difference between contacts c and d, when the current is applied between contacts a and b, involves dissipation occurring somewhere between c and d. In the high mobility case the dissipation of the edge current occurs in the contact regions. This cannot happen in our low mobility structures because the edge electrons which are injected sideways from the main current path cannot reach the voltage contacts. Instead they are either transferred to the opposite edge without energy losses or thermalised within the bulk somewhere in the wire and probe regions. The voltage in our experiment, therefore, measures the dissipation which occurs in and around the probe regions. At low temperatures, our effect disappears when the electrons entering the bulk region can traverse the width of the wire without suffering an inelastic collision and dissipating energy. We expect the voltage drop Vcd to have the form Vcda&P(T)
301
term, P(T), will be small. On the other hand, there will be strong coupling between edge states and bulk when the Fermi energy lies within a bulk Landau level but A, will be correspondingly small. The maximum values of Vcd, given by eq. (11, have the same periodicity but not necessarily the same phase as the Landau levels crossing the Fermi energy, in agreement with experiment. We introduce the dissipation term in a form resulting from the energy balance equation: P(T) where AE is the differa [l + exp(AE/k,T)]-’ ence between the Fermi energy and the energy of the maximum of the density of states in the bulk. It can be determined directly from the phase shift between the local SdH maxima and the corresponding non-local magnetoresistance peaks. The solid lines in fig. 5 are the best fit to the temperature dependence of eq. (1) using AE = 4 meV for the upper curve and AE = 3.2 meV for the lower curve, calculated as discussed above. The only adjustable parameter is the overall amplitude. The fitted values for the amplitudes of both peaks shown in fig. 5 scale with the amplitude of the corresponding local SdH maxima. Given the approximate derivation of eq. (1) the agreement between theory and experiment is very good. To summarise, we have observed a new nonlocal magnetoresistance effect in the intermediate regime between quantum edge conduction and classical bulk transport. The observed resistance is due to dissipation in the bulk rather than in the electrical contacts. However, quantum conduction in the edge states plays an important role in carrying the current to classically inaccessible regions.
exp( -L/An)
x ( kBT/fiw,)cosech(2rr2kBT/tiw,).
(1)
At high T the voltage is limited by the same factor as for local SdH oscillations. The exponential factor is the probability that injected edge currents reach the intersection of the wire with the voltage probes. P(T) is the fraction of these electrons which thermalise. Thermalisation involves a transition from edge to bulk followed by a relaxation to local equilibrium. We expect that A, will be a maximum when the Fermi energy is in between bulk Landau levels but the dissipation
This work is supported by SERC. A.K.G., P.S. and P.H.B. wish to thank the Royal Society for financial support.
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in iwa~~ily doped )I +-&As
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