- Email: [email protected]
Resonant transmission through a double-point contact device in high magnetic field
Solid State Communications, Printed in Great Britain.
RESONANT
Vol. 78, No. 3, pp. 215-217,
TRANSMISSION
0038-1098/91 $3.00 + .OO Pergamon Press plc
1991.
THROUGH A DOUBLE-POINT IN HIGH MAGNETIC FIELD
CONTACT
DEVICE
Lydia Jaeger Institut
fur Theoretische
Physik,
Ziilpicher
StraBe 77, 5000 Koln 41, Germany
(Received 4 January 1991 by B. Miihlschlegel) Recently, oscillations have been observed in the conductance of doublepoint contact devices. They are explained microscopically in the experimentally relevant limit of a slowly varying potential and high magnetic field using a WKB-type approximation. The condition for resonant transmission is given; in the limit of low transmittancy of the single barriers, resonant transmission occurs whenever a half-integer number of flux quanta is enclosed in the path of the electron between the two quantum point contacts. RECENTLY, oscillations have been observed in the conductance of coupled quantum point contacts in the interface of semiconductor heterostructures which are similar in structure to Aharonov-Bohm oscillations in mesoscopic quasi-one-dimensional metallic rings. The oscillations occur in magnetic field ranges where one transport channel is neither fully transmitting nor fully reflecting, i.e. in the transition region between two plateaus in the conductance [l, 21. As the transport in the considered experiments is ballistic and adiabatic [3], the scattering is conveniently described in the transition region by a two-dimensional Unitarian scattering matrix s
=
r it
t’ (1)
r’ 1 ’
(2) In terms of the reflection and transmission the transfer matrix takes the form T
=
If T,
rt-’
and
/,-iy’
n \ (4)
as one can neglect scattering in the cavity between point contacts. The quantity
the
0 := Y + Y’,
(5)
is the phase an electron acquires when travelling once back and forth in the cavity without the phase shift due to the reflection at both barriers. The latter phase shift is determined by the phases of the reflection amplitudes of the barriers: 0, := arg (rirg).
which relates the incoming to the outgoing current amplitudes, where r denotes the reflection and t the transmission amplitude for scattering from left to right; r’ and t’ denote the corresponding amplitudes for scattering from right to left. Equivalently, the scattering can be described by a two-dimensional symplectic transfer matrix which connects the current amplitudes to the right of the scattering region with those to the left of it (see Fig. 1):
t-’
describe two quantum point contacts - labelled by A and B - individually, the transfer matrix for the system of the two coupled barriers has the form:
amplitudes
-_t-‘f -rt-‘r’
T, denote
(3)
+
the
transfer
matrices
which 215
(6)
Using the relation (3) between the transfer matrix and the transmission and reflection amplitudes, one can calculate the transmission coefficient of the doublepoint contact device:* T=
lAIB
1 -
2Jm
cos (0
+ 0,)
+ R,R,’
(7)
For deriving equation (7), the fact has been used that even in the absence of time reversal symmetry the reflection coefficients for transmission in both direction are the same (due to current conservation) as long as the number of transport channels is the same in both directions. * Kouwenhoven formula under metric barriers holds in general symmetry.
et al. [I] deduced a comparable the additional assumption of symwhich gives t = t’ and r = r’; this only in the presence of time reversal
RESONANT
216
Fig. 1. Transmission tacts (schematically).
through
TRANSMISSION
two quantum
point con-
If the two barriers are equal, expression (7) for the transmission coefficient of the two-barrier system reduces to: T=
C 1 -
2R, cos (0
+ 0,)
=
2nm
(mEZ)
a
T=l.
MAGNETIC
(10) is very well fulfilled in the conductance measurements for both the direction perpendicular and parallel to the boundary of the constriction where w, denotes the
=
- u,xZ + u,yl
+ 41;
(11)
U, and U, determine the curvature of the potential near the barrier and V, sets the effective zero of the energy scale. If one defines the reduced energy
(9)
That a bound state could exist in the resonant cavity formed between the two quantum point contacts is therefore a necessary condition for resonant transmission to occur at a given energy. In direct generalization, an even number of coupled equal barriers is transparent whenever the total phase shift for an electron travelling once back and forth between two adjacent barriers is an integer times 27~. Equation (8) predicts oscillations in the transmission coefficient and via the Landauer formula also in the conductance of a double-point contact device. For a comparison with experimental data it is necessary to relate the total phase shift in the cavity 0 + 0, to the microscopic structure of the device. Although the observed oscillations bear a similarity to the Aharonov-Bohm effect, they cannot be explained by invoking this effect because the motion of the electron is not geometrically confined to a one-dimensional path and the electron moves in a simply connected region. As a direct consequence the gauge transformation necessary to establish the Aharonov-Bohm effect is not well defined in this geometry. If the electron moves in a sufficiently smooth potential Vand high magnetic field B, it is nevertheless possible to relate rigorously the phase shift an electron picks up to the number of flux quanta enclosed by the path of the electron. Experimental data suggest that the high-field condition
Vol. 78, No. 3
FIELD
cyclotron frequency and I, the magnetic length [4]. In the high-field regime the motion of an electron can be semiclassically described by decoupled cyclotron and center coordinates [5] and by applying a WKB-type approximation for the center coordinates first proposed by Fertig [6]. In high magnetic fields the center coordinates follow the equipotential lines of the electrostatic potential in the device except near saddle points where the two-dimensional transfer matrix can be calculated: in the vicinity of a saddle point, the potential can be expanded - after an appropriate isometric transformation of the coordinate system - in the form l&(x, y)
+ R:,
Incoming waves are fully transmitted if the total phase shift in the cavity is an integer times 271: O+O,
IN HIGH
E - ho,.(n ?/ :=
+ 3) -
h
JiTyql,’
(12)
)
where n stands for the Landau level index, the explicit form of the transfer matrix for scattering at the saddle point can be calculated:? e’~~‘Jl + exp (- rc~)
i exp (- 74ON
T= (
-i
ee’*“)
exp ( - r@/2))
1 + exp (-ny)
1’ (13)
where @ = m,(y) is defined
as -
k+i:
O(y) := ! + arg I (
>
?!ln!!2!_
(14)
By moving along an equipotential line the electron picks up an additional phase equal to the number of flux quanta enclosed between this equipotential line and the x-axis multiplied by 2~ (see equation (2.6) in [6]). Each turning point - where the partial derivative of the potential with respect to y changes its sign and the partial derivative with respect to x is non-zero contributes an additional term 7c/2 (see equation (2.7) in [6].$ Using these approximate results, it is now possible to calculate the transmission coefficient of the twobarrier system from the microscopic shape of the potential. As a barrier corresponds in the high-field limit to a region where the potential is saddle-shaped, + See equation (2.11) in [6]; note the slightly different definition of the transfer matrix. 1 As one usually considers closed equipotential lines (i.e. for the calculation of eigenenergies), the dependence of the connection formulae on the choice of the coordinate system drops out of the calculations.
Vol. 78, No. 3
RESONANT
one obtains for the transmission two-barrier system (7): T-’
1 + e-PA + e-VS +
=
+
x cos
coefficient
of the
‘&-~(?A+%)
Jm
2e-‘h+~sV2
TRANSMISSION
(271%/v-@ - @(YA) -
Jm
Q)(?JB)).
(15)
yA, yB denote the reduced energy with respect to the barrier A and B, respectively, and X@ is the number of flux quanta enclosed in the equipotential line in the cavity - or rather more precisely between the classical turning points of the two saddle points at the energy in question. In this approximation, the phase (5) an electron picks up when propagating along the equipotential line in the cavity is proportional to the number of enclosed flux quanta: 0
=
27cJlr*.
(16)
Not surprisingly, the Bohm-Aharonov-result is recovered for this part of the phase shift as the strong magnetic field forces the electron moving in a smooth potential to follow one-dimensional paths - the equipotential lines; it thus allows for defining a path the electron moves along in the classical sense of the word. The additional phase shift (6) due to the reflection at the barriers is 0,
=
-(D(yA)
-
@(ys) + 72.
2Q(y,)
=
2a(m + 3)
(m E Z).
(18)
In the limit of low transmittancy of the single barriers, i.e. ya -+ - co, resonant transmission occurs whenever a half-integer number of flux quanta is enclosed by the path in the cavity: Jlr’
=
m + 3
(m E No),
MAGNETIC
FIELD
217
special case the general rule that resonant transmission occurs whenever the cavity between the two quantum point contacts allows for a bound state. A comparable condition for resonant transmission was deduced by Glazman and Jonson [8] in the limit of a hard wall potential. They studied the conductance of a double-point contact device under the assumption of an infinitely steep confining potential and allowed for a small curvature of the boundary, but no electrostatic potential in the sample itself. Therefore, the Aharonov-Bohm phase has been established in two opposite limits as the essential part for the oscillations in the conductance of double-point contact devices. Summarizing the results, in the limit of a smooth potential and high magnetic field the condition for resonant transmission in a two-barrier system has been given in terms of the microscopic structure of the potential in the device. Experiments suggest that the condition of a slowly varying potential is well fulfilled in the conductance measurements for both the direction perpendicular and parallel to the boundary of the constriction; the observed oscillations in the conductance of double-point contact devices have thereby found a microscopic explanation. The language which has been used here should be suitable for describing different experiments in constrained semiconductor heterostructures.
(17)
The condition (9) for resonant transmission in a system formed by two equal barriers is therefore 27cM” -
IN HIGH
Acknowledgements - I am greatly indebted to Jgnos Hajdu and Martin JanBen for many long and fruitful discussions. I also appreciate that I was able to discuss with Bart van Wees, Frank Hekking and K. Harmans their experiments.
REFERENCES 1.
(19)
where it has been used that lim,,,,,Q(y) = 0. As equation (19) is the eigenenergy condition for isolated equipotential lines* calculated by the BohrSommerfeld quantization rule [7], one recovers in this * “Isolated” means that any saddle point is sufficiently far away from the considered equipotential line so that scattering does not need to be taken into account. Therefore the equipotential line cannot couple to another equipotential line of the same energy.
2. 3. 4. 5. 6. 7. 8.
L.P. Kouwenhoven et al., in: (Edited by K. von Klitzing) Localization and ConJinement of Electrons in Semiconductors, Springer Series in Solid State Physics, in press. L.P. Kouwenhoven et al., Surface Science 229, 290 (1990). L.P. Kouwenhoven et al., Phys. Rev. B40, 8083 (1989). B. van Wees, private communication. R. Kubo, S.J. Miyake & N. Hashitsume, Solid State Physics 17, 269, Academic Press (1965). H.A. Fertig, Phys. Rev. B38, 996 (1988). M. Tsukada, J. Phys. Sot. Jpn. 41, 1466 (1976). L.I. Glazman & M. Jonson, Phys. Rev. B41, 10686 (1990).
RESONANT
Vol. 78, No. 3, pp. 215-217,
TRANSMISSION
0038-1098/91 $3.00 + .OO Pergamon Press plc
1991.
THROUGH A DOUBLE-POINT IN HIGH MAGNETIC FIELD
CONTACT
DEVICE
Lydia Jaeger Institut
fur Theoretische
Physik,
Ziilpicher
StraBe 77, 5000 Koln 41, Germany
(Received 4 January 1991 by B. Miihlschlegel) Recently, oscillations have been observed in the conductance of doublepoint contact devices. They are explained microscopically in the experimentally relevant limit of a slowly varying potential and high magnetic field using a WKB-type approximation. The condition for resonant transmission is given; in the limit of low transmittancy of the single barriers, resonant transmission occurs whenever a half-integer number of flux quanta is enclosed in the path of the electron between the two quantum point contacts. RECENTLY, oscillations have been observed in the conductance of coupled quantum point contacts in the interface of semiconductor heterostructures which are similar in structure to Aharonov-Bohm oscillations in mesoscopic quasi-one-dimensional metallic rings. The oscillations occur in magnetic field ranges where one transport channel is neither fully transmitting nor fully reflecting, i.e. in the transition region between two plateaus in the conductance [l, 21. As the transport in the considered experiments is ballistic and adiabatic [3], the scattering is conveniently described in the transition region by a two-dimensional Unitarian scattering matrix s
=
r it
t’ (1)
r’ 1 ’
(2) In terms of the reflection and transmission the transfer matrix takes the form T
=
If T,
rt-’
and
/,-iy’
n \ (4)
as one can neglect scattering in the cavity between point contacts. The quantity
the
0 := Y + Y’,
(5)
is the phase an electron acquires when travelling once back and forth in the cavity without the phase shift due to the reflection at both barriers. The latter phase shift is determined by the phases of the reflection amplitudes of the barriers: 0, := arg (rirg).
which relates the incoming to the outgoing current amplitudes, where r denotes the reflection and t the transmission amplitude for scattering from left to right; r’ and t’ denote the corresponding amplitudes for scattering from right to left. Equivalently, the scattering can be described by a two-dimensional symplectic transfer matrix which connects the current amplitudes to the right of the scattering region with those to the left of it (see Fig. 1):
t-’
describe two quantum point contacts - labelled by A and B - individually, the transfer matrix for the system of the two coupled barriers has the form:
amplitudes
-_t-‘f -rt-‘r’
T, denote
(3)
+
the
transfer
matrices
which 215
(6)
Using the relation (3) between the transfer matrix and the transmission and reflection amplitudes, one can calculate the transmission coefficient of the doublepoint contact device:* T=
lAIB
1 -
2Jm
cos (0
+ 0,)
+ R,R,’
(7)
For deriving equation (7), the fact has been used that even in the absence of time reversal symmetry the reflection coefficients for transmission in both direction are the same (due to current conservation) as long as the number of transport channels is the same in both directions. * Kouwenhoven formula under metric barriers holds in general symmetry.
et al. [I] deduced a comparable the additional assumption of symwhich gives t = t’ and r = r’; this only in the presence of time reversal
RESONANT
216
Fig. 1. Transmission tacts (schematically).
through
TRANSMISSION
two quantum
point con-
If the two barriers are equal, expression (7) for the transmission coefficient of the two-barrier system reduces to: T=
C 1 -
2R, cos (0
+ 0,)
=
2nm
(mEZ)
a
T=l.
MAGNETIC
(10) is very well fulfilled in the conductance measurements for both the direction perpendicular and parallel to the boundary of the constriction where w, denotes the
=
- u,xZ + u,yl
+ 41;
(11)
U, and U, determine the curvature of the potential near the barrier and V, sets the effective zero of the energy scale. If one defines the reduced energy
(9)
That a bound state could exist in the resonant cavity formed between the two quantum point contacts is therefore a necessary condition for resonant transmission to occur at a given energy. In direct generalization, an even number of coupled equal barriers is transparent whenever the total phase shift for an electron travelling once back and forth between two adjacent barriers is an integer times 27~. Equation (8) predicts oscillations in the transmission coefficient and via the Landauer formula also in the conductance of a double-point contact device. For a comparison with experimental data it is necessary to relate the total phase shift in the cavity 0 + 0, to the microscopic structure of the device. Although the observed oscillations bear a similarity to the Aharonov-Bohm effect, they cannot be explained by invoking this effect because the motion of the electron is not geometrically confined to a one-dimensional path and the electron moves in a simply connected region. As a direct consequence the gauge transformation necessary to establish the Aharonov-Bohm effect is not well defined in this geometry. If the electron moves in a sufficiently smooth potential Vand high magnetic field B, it is nevertheless possible to relate rigorously the phase shift an electron picks up to the number of flux quanta enclosed by the path of the electron. Experimental data suggest that the high-field condition
Vol. 78, No. 3
FIELD
cyclotron frequency and I, the magnetic length [4]. In the high-field regime the motion of an electron can be semiclassically described by decoupled cyclotron and center coordinates [5] and by applying a WKB-type approximation for the center coordinates first proposed by Fertig [6]. In high magnetic fields the center coordinates follow the equipotential lines of the electrostatic potential in the device except near saddle points where the two-dimensional transfer matrix can be calculated: in the vicinity of a saddle point, the potential can be expanded - after an appropriate isometric transformation of the coordinate system - in the form l&(x, y)
+ R:,
Incoming waves are fully transmitted if the total phase shift in the cavity is an integer times 271: O+O,
IN HIGH
E - ho,.(n ?/ :=
+ 3) -
h
JiTyql,’
(12)
)
where n stands for the Landau level index, the explicit form of the transfer matrix for scattering at the saddle point can be calculated:? e’~~‘Jl + exp (- rc~)
i exp (- 74ON
T= (
-i
ee’*“)
exp ( - r@/2))
1 + exp (-ny)
1’ (13)
where @ = m,(y) is defined
as -
k+i:
O(y) := ! + arg I (
>
?!ln!!2!_
(14)
By moving along an equipotential line the electron picks up an additional phase equal to the number of flux quanta enclosed between this equipotential line and the x-axis multiplied by 2~ (see equation (2.6) in [6]). Each turning point - where the partial derivative of the potential with respect to y changes its sign and the partial derivative with respect to x is non-zero contributes an additional term 7c/2 (see equation (2.7) in [6].$ Using these approximate results, it is now possible to calculate the transmission coefficient of the twobarrier system from the microscopic shape of the potential. As a barrier corresponds in the high-field limit to a region where the potential is saddle-shaped, + See equation (2.11) in [6]; note the slightly different definition of the transfer matrix. 1 As one usually considers closed equipotential lines (i.e. for the calculation of eigenenergies), the dependence of the connection formulae on the choice of the coordinate system drops out of the calculations.
Vol. 78, No. 3
RESONANT
one obtains for the transmission two-barrier system (7): T-’
1 + e-PA + e-VS +
=
+
x cos
coefficient
of the
‘&-~(?A+%)
Jm
2e-‘h+~sV2
TRANSMISSION
(271%/v-@ - @(YA) -
Jm
Q)(?JB)).
(15)
yA, yB denote the reduced energy with respect to the barrier A and B, respectively, and X@ is the number of flux quanta enclosed in the equipotential line in the cavity - or rather more precisely between the classical turning points of the two saddle points at the energy in question. In this approximation, the phase (5) an electron picks up when propagating along the equipotential line in the cavity is proportional to the number of enclosed flux quanta: 0
=
27cJlr*.
(16)
Not surprisingly, the Bohm-Aharonov-result is recovered for this part of the phase shift as the strong magnetic field forces the electron moving in a smooth potential to follow one-dimensional paths - the equipotential lines; it thus allows for defining a path the electron moves along in the classical sense of the word. The additional phase shift (6) due to the reflection at the barriers is 0,
=
-(D(yA)
-
@(ys) + 72.
2Q(y,)
=
2a(m + 3)
(m E Z).
(18)
In the limit of low transmittancy of the single barriers, i.e. ya -+ - co, resonant transmission occurs whenever a half-integer number of flux quanta is enclosed by the path in the cavity: Jlr’
=
m + 3
(m E No),
MAGNETIC
FIELD
217
special case the general rule that resonant transmission occurs whenever the cavity between the two quantum point contacts allows for a bound state. A comparable condition for resonant transmission was deduced by Glazman and Jonson [8] in the limit of a hard wall potential. They studied the conductance of a double-point contact device under the assumption of an infinitely steep confining potential and allowed for a small curvature of the boundary, but no electrostatic potential in the sample itself. Therefore, the Aharonov-Bohm phase has been established in two opposite limits as the essential part for the oscillations in the conductance of double-point contact devices. Summarizing the results, in the limit of a smooth potential and high magnetic field the condition for resonant transmission in a two-barrier system has been given in terms of the microscopic structure of the potential in the device. Experiments suggest that the condition of a slowly varying potential is well fulfilled in the conductance measurements for both the direction perpendicular and parallel to the boundary of the constriction; the observed oscillations in the conductance of double-point contact devices have thereby found a microscopic explanation. The language which has been used here should be suitable for describing different experiments in constrained semiconductor heterostructures.
(17)
The condition (9) for resonant transmission in a system formed by two equal barriers is therefore 27cM” -
IN HIGH
Acknowledgements - I am greatly indebted to Jgnos Hajdu and Martin JanBen for many long and fruitful discussions. I also appreciate that I was able to discuss with Bart van Wees, Frank Hekking and K. Harmans their experiments.
REFERENCES 1.
(19)
where it has been used that lim,,,,,Q(y) = 0. As equation (19) is the eigenenergy condition for isolated equipotential lines* calculated by the BohrSommerfeld quantization rule [7], one recovers in this * “Isolated” means that any saddle point is sufficiently far away from the considered equipotential line so that scattering does not need to be taken into account. Therefore the equipotential line cannot couple to another equipotential line of the same energy.
2. 3. 4. 5. 6. 7. 8.
L.P. Kouwenhoven et al., in: (Edited by K. von Klitzing) Localization and ConJinement of Electrons in Semiconductors, Springer Series in Solid State Physics, in press. L.P. Kouwenhoven et al., Surface Science 229, 290 (1990). L.P. Kouwenhoven et al., Phys. Rev. B40, 8083 (1989). B. van Wees, private communication. R. Kubo, S.J. Miyake & N. Hashitsume, Solid State Physics 17, 269, Academic Press (1965). H.A. Fertig, Phys. Rev. B38, 996 (1988). M. Tsukada, J. Phys. Sot. Jpn. 41, 1466 (1976). L.I. Glazman & M. Jonson, Phys. Rev. B41, 10686 (1990).