- Email: [email protected]
Continuous phase shift in the forced interference of edge states in the quantum Hall regime
Solid State Communications 122 (2002) 489–492 www.elsevier.com/locate/ssc
Continuous phase shift in the forced interference of edge states in the quantum Hall regime Mincheol Shin* School of Engineering, Information and Communications University, Daejeon 305-714, South Korea Received 17 March 2002; accepted 1 April 2002 by H. Akai
Abstract The edge states in the quantum Hall regime are forced to interfere on one side of an Aharonov – Bohm ring by a potential barrier. The net effect of the interference caused by the potential barrier is found to change the electron’s phase by D in the final transmittance, and the phase shift D is continuous as the barrier potential is continuously changed. This result is in sharp contrast with the earlier study that D ¼ 2np; where n is an integer, regardless of the barrier potential. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 72.20.My; 73.20.Dx; 73.40.Hm; 72.15.Gd Keywords: E. Mesoscopic system; C. Edge state; E. Ballistic transport; A. Quantum wire
In his paper, [1] Mu¨ller considered the situation where the edge states in the quantum Hall regime [2] are forced to interfere on one side of an Aharonov– Bohm ring by a potential barrier attached on one arm of the ring (Fig. 1(a)). He argued that the tunneled part and the scattered part of the edge states always interfere constructively, because the area enclosed between them are always an integer multiple of F0 =B; where F0 ¼ hc=e is the magnetic flux quantum and B is the magnetic field perpendicular to the ring. He demonstrated it by solving the time-dependent Schro¨dinger equation for the system, especially focusing on the situation where the change of the inner ring size would lead the phase shift between the tunneled and the scattered waves. We reexamined the problem and suggested that the constructive interference is not a necessary condition [3]. In this paper, we will give detailed and augmented arguments that support our claim. We consider the case where the phase of the tunneled wave varies to the change in the strength (height) of the barrier, while the size of the inner circle is fixed. This can be experimentally possible by attaching a gate on one arm of a ring and changing the gate potential [4]. According to the Mu¨ller’s argument, the change in the gate potential would * Tel.: þ82-42-866-6904; fax: þ 82-42-866-6923. E-mail address: [email protected] (M. Shin).
not lead to the phase shift of the scattered and the tunneled waves, since the scattered wave will always adjust its course so as to meet the constructive interference condition BS ¼ nF0
ð1Þ
where S is the area enclosed by the two waves. We will argue later that the gate potential change does lead to the phase shift between them. Our argument is based on it that the constructive condition of Eq. (1) is not necessary. Let us suppose that an electron wave of unit flux I in the edge state is incident from the left of the ring region (see Fig. 1(a)). As it propagates along the edge channel, it scatters with the barrier in the lower arm of the ring. The barrier is assumed to interact with an incident wave in such a way that the transmission and the reflection amplitudes are tG and rG ; respectively, when it is incident from the left of the barrier, and ~tG and r~G when incident from the right of the barrier. After the initial scattering, part of the incident wave transmits through the barrier and continues to propagate along the edge channel on the other side of the barrier. The reflected wave propagates around the inner circle, adding fP to its phase in the meanwhile, and hits the barrier again. The reflected wave at this second scattering propagates along the edge channel to the right, but the transmitted wave still remains in the ring region and propagates around the inner circle once again. The same scattering process continues
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 0 9 8 ( 0 2 ) 0 0 1 3 5 - 7
490
M. Shin / Solid State Communications 122 (2002) 489–492
fC ¼ fP þ f~G ; it can be readily obtained that T ¼ expðiDÞ;
ð4Þ
where
D ; fG þ fC þ d
ð5Þ
and tanðdÞ ¼ n 2ltg lsin fC ð1 2 ltg lcos fC Þ= ð1 2 ltg lcos fC Þ2
ð6Þ
o
2 ðltg lsin fC Þ2 : In obtaining Eqs. (4) and (5), we have used the fact that rG ¼ 2~rG and tG ~tG 2 rG r~G ¼ expðiðfG þ f~G ÞÞ: Eq. (4) indicates that the net effect of the scattering by the barrier is to change the electron’s phase by D in the transmission amplitude T. The transmission probability is always one, lTl2 ¼ 1; which should be so because there is no path for the electron’s backscattering to the other side of the wire. A main result in this study is that the phase D changes to the change in the barrier potential through fG ; fC and d in Eqs. (5) and (6). To explicitly show their relationship with the barrier potential, let us consider a symmetric system, where the ring has the central radius of RC and the barrier spans an angle of uG (Fig. 1(b)). Then
fG < ðBS=F0 ÞuL =2p þ kG RC uL
ð7Þ
and
fC < 2ðBS=F0 Þ þ kRC ð2p 2 uL Þ þ kG RC uL
Fig. 1. Ring geometries with a barrier in the lower arm.
until all the flux initially incident from the ‘source’ lead exits to the ‘drain’ lead. The summation of the initially transmitted wave and all the reflected waves by the subsequent scatterings yields the eventual transmission amplitude T, and it can be easily calculated as follows by supposing the steady state of the electron waves in the system. In Fig. 1(a), the amplitude of the wave leaving the barrier to the left along the edge of the inner circle is represented as A1, that of the wave incident on the right of the barrier along the edge of the inner circle as A2, and that of the wave transmitted as T. In the steady state, the amplitudes satisfy A1 ¼ rG I þ ~tG A2 ;
A2 ¼ PA1 ;
T ¼ tG I þ r~G A2 : ð2Þ
where P is the propagator along the edge of the inner circle, P ¼ expðifP Þ: If we write tG ¼ ltG lexpðifG Þ;
~tG ¼ ltG lexpðif~G Þ;
ð3Þ
and denote as fC the phase gain as the electron makes one complete turn along the edge of the inner circle clockwise;
ð8Þ
where k is the wave vector of the electron in the leads and in the ring annulus excluding the barrier and kG is the wave vector beneath the barrier. If we model the barrier as a simple potential rise of constant height VG =E0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kG ¼ kG ðVG Þ < k2 2 ðVG =E0 Þ; ð9Þ where E0 ; "2 =2mp w2 ; where mp is the effective mass of electron and w is the width of the wire. If we apply a constant magnetic field to the system and vary the barrier potential, fG and fC change through change in the phase kG ðVG Þ of electron beneath the barrier, and d changes through both change in the phase and change in the transmission probability ltG l: Since kG and ltG l are continuous functions of the barrier height VG =E0 ; we can readily see via Eqs. (4)– (8) that a continuous change in the barrier potential leads to the continuous change in the phase D of T. This result of ours disagrees with the conclusion which can be drawn from Mu¨ller’s argument that the phase D cannot be continuously varied no matter how we change the relative phase of the tunneled and the scattered waves, via the barrier potential change for an instance. Both of Mu¨ller’s and our results however yield lTl2 ¼ 1 for the system in Fig. 1(a) or (b), so the disagreement cannot be experimentally clarified with such a set-up. We therefore
M. Shin / Solid State Communications 122 (2002) 489–492
Fig. 2. A three-dimensional plot of lT 0 l2 for a very short barrier case.
values of ltG l2 ; D0 of Eq. (10) can become a multiple of 2p and lT 0 l2 reaches the maximum of 1 at the points. However, when fC happens to be 2np at the same time, d ¼ 0 (mod 2p) and lT 0 l2 becomes nearly independent of ltG l2 : This feature results in high asymmetry with respect to the points fC ¼ 2np in a three-dimensional plot of lT 0 l2 : In Fig. 2, peaks corresponding to the maximum value of 1 run in the plane of fC and ltG l2 for fC , 2p and terminate at fC ¼ 2p; but for fC . 2p; lT 0 l2 becomes almost constant. For a long central barrier, on the other hand, the change in the electron’s phase under the barrier, rather than the transmission probability as in the case of the very short barrier, dominantly determines the dependence of D0 on the barrier potential. Namely, when we apply a constant magnetic field and vary the barrier potential,
D0 ðVG Þ < 2RC uL kG ðVG Þ þ const; suggest to consider the system in Fig. 1(c), where we attach second barriers at the end of the two leads. These barriers reflect the electron with the reflection probability lr 0 l2 so that the transmission probability of the whole system becomes now (omitting uninteresting phase factors) lT 0 l2 ¼ lt0 l4 =ð1 þ lr 0 l4 2 2lr 0 l2 cos D0 Þ 0 2
ð10Þ
0 2
where lt l ¼ 1 2 lr l ;and
D0 ; D þ ðBS0 =F0 þ kL0 Þ
ð11Þ
where S0 is the area of the ring region enclosed by the second barriers and the top and bottom wire walls, and L0 is its perimeter. The edge state condition should be fulfilled. With a sufficiently high reflection probability lr 0 l2 of the second barriers, the transmission probability lT 0 l2 ; which is directly related to the conductance of the system actually measured in real experiments, is a good measure of the change in the phase D, if any. Note that if we follow Mu¨ller’s argument, D ¼ 2np; where n is an integer, and so lT 0 l2 of Eq. (10) will be constant with respect to the center barrier (gate) potential. In the following, we will provide in detail the consequences of our claim that the phase D of Eqs. (4) and (11) is continuously varied, when applied to the system in Fig. 1(c). First, we note that if fC ¼ 2np; i.e. if the constructive interference condition is satisfied when the electron makes one complete turn around the inner circle, D ¼ fG (mod 2p) independently of the transmission probability ltG l2 through the barrier. And when fC ¼ ð2n þ 1Þp; i.e. for the destructive condition, D ¼ fG þ p (mod 2p) independently of ltG l2 : This implies that for the special cases, the barrier acts as if it is an ideal one-dimensional phase modulator which only changes the electron’s phase without any backscattering, even though it actually scatters the electron with the probability lrG l2 where lrG l2 can assume any value from 0 to 1. We next discuss the case of a very short central barrier. fG and fC are then almost constant with respect to the barrier potential VG ; and d responds to change in VG through the transmission probability ltG l2 of the barrier. For certain
491
ð12Þ
where kG ðVG Þ may be given by Eq. (9). To make it certain that the earlier relation holds for the long barrier case, we have actually solved the time-independent Schro¨dinger equation for the system in Fig. 1(c). The Hamiltonian for an electron of charge e and effective mass mp moving ballistically in the structure is; "2 eA 2 2i7 þ þVðx; yÞ ð13Þ H¼ "c 2mp where A is the vector potential of the perpendicular magnetic field B ¼ ð0; 0; BÞ and the potential Vðx; yÞ is given by 8 1 at the wire walls; > > > > > < VG at the barrier in the ring region; Vðx; yÞ ¼ ð14Þ > > VL at the lead barriers; > > > : 0 otherwise: The barrier in the ring region is modeled as a simple potential rise of constant height VG =E0 and the lead barriers as those of constant height VL =E0 : We then numerically solve the resultant Schro¨dinger equations and obtain eigenvalues (eigenmomentum) and eigenfunctions of the straight part and the bent parts. We next write down the wave functions of all different regions by expanding them in terms of corresponding eigenfunctions. The wave functions and their derivatives are matched at all boundaries of different regions and, resultant linear equations are solved to yield the transmission amplitudes. Detailed description of the mode-matching technique can be found in Refs. [5,6]. Note that the Hamiltonian of Eq. (13) is essentially the same as the one used by Mu¨ller. The only difference is that he solved the time-dependent Schro¨dinger equation while we solved the time-independent one. In Fig. 3(a), we show the result of solving the timeindependent Schro¨dinger equation in the long barrier case. The parameters used in the calculation are: RC ¼ 3:5w; uG ¼ p=3; EF =E0 ¼ ð2pÞ2 ; B=B0 ¼ 15; VL =E0 ¼ 2:1 and
492
M. Shin / Solid State Communications 122 (2002) 489–492
~ 0 ¼ 0 (heavy solid line), 0.15 (dotted line), 0.30 (dashed line), and 0.45 (light solid Fig. 3. (a)lT 0 l2 versus VG =E0 for a long barrier case at kB T=E ~ 0 Þ: line). (b) The oscillation amplitudes of lT 0 l2 with respect to temperature ðkB T=E
~ 0 ¼ 0; 0.15, 0.30, 0.45, where B0 ¼ "c=ew2 ; EF is the kB T=E Fermi energy and T~ is temperature. At the parameters, only one propagating channel is formed in the system, and due to the sufficiently high magnetic field, it is in the edge state in all regions in the structure. The lead barriers back-scatter 11.2 % of incident electrons at the lead potential. As we increase the potential of the barrier in the ring region from zero until it is equal to the Fermi energy, we observe oscillations whose periods are in good agreement with Eq. (12). The transmission probability follows Eq. (10) very well at zero temperature, but the amplitudes of the oscillations decay almost exponentially with increase in temperature (Fig. 3(b)). The terms containing k in Eqs. (7), (8), and (11) are responsible for the rapid decay. Namely, at the temperature where the fluctuation of k is allowed to be as large as
the Mu¨ller’s that D ¼ 2np; where n is an integer, regardless of the barrier potential. We have proposed an experimentally possible set-up where the phase shift D with respect to the barrier potential can be directly detected, and made detailed prediction in cases of very short and long barriers. In the long barrier case, the temperature dependence of the transmission probability is also provided.
DkðL0 þ 2pRC Þ < 2p;
References
ð15Þ
the amplitude of the oscillation becomes almost zero. In conclusion, we have re-examined the situation considered earlier by Mu¨ller where the edge states are forced to interfere on one side of an Aharonov– Bohm ring by a potential barrier. We have found that the net effect of the interference induced by the potential barrier results in the electron’s phase shift D in the final transmittance, and the phase shift D is continuous as the barrier potential is continuously changed. This result is in sharp contrast with
Acknowledgments This work has been supported by the Ministry of Informations and Communications, Korea.
J.E. Mu¨ller, Phys. Rev. Lett. 72 (1994) 2616. M. Bu¨ttiker, Phys. Rev. B 38 (1988) 9875. M. Shin, Phys. Rev. Lett. 77 (1996) 5146. P.G.N. de Vegvar, G. Timp, P.M. Mankiewich, R. Behringer, J. Cunningham, Phys. Rev. B 40 (1989) 3491. [5] R.L. Shult, H.W. Wyld, D.G. Ravenhall, Phys. Rev. B 41 (1990) 12760. [6] M. Shin, K.W. Park, S. Lee, E.-H. Lee, Phys. Rev. B 53 (1996) 1014.
[1] [2] [3] [4]
Continuous phase shift in the forced interference of edge states in the quantum Hall regime Mincheol Shin* School of Engineering, Information and Communications University, Daejeon 305-714, South Korea Received 17 March 2002; accepted 1 April 2002 by H. Akai
Abstract The edge states in the quantum Hall regime are forced to interfere on one side of an Aharonov – Bohm ring by a potential barrier. The net effect of the interference caused by the potential barrier is found to change the electron’s phase by D in the final transmittance, and the phase shift D is continuous as the barrier potential is continuously changed. This result is in sharp contrast with the earlier study that D ¼ 2np; where n is an integer, regardless of the barrier potential. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 72.20.My; 73.20.Dx; 73.40.Hm; 72.15.Gd Keywords: E. Mesoscopic system; C. Edge state; E. Ballistic transport; A. Quantum wire
In his paper, [1] Mu¨ller considered the situation where the edge states in the quantum Hall regime [2] are forced to interfere on one side of an Aharonov– Bohm ring by a potential barrier attached on one arm of the ring (Fig. 1(a)). He argued that the tunneled part and the scattered part of the edge states always interfere constructively, because the area enclosed between them are always an integer multiple of F0 =B; where F0 ¼ hc=e is the magnetic flux quantum and B is the magnetic field perpendicular to the ring. He demonstrated it by solving the time-dependent Schro¨dinger equation for the system, especially focusing on the situation where the change of the inner ring size would lead the phase shift between the tunneled and the scattered waves. We reexamined the problem and suggested that the constructive interference is not a necessary condition [3]. In this paper, we will give detailed and augmented arguments that support our claim. We consider the case where the phase of the tunneled wave varies to the change in the strength (height) of the barrier, while the size of the inner circle is fixed. This can be experimentally possible by attaching a gate on one arm of a ring and changing the gate potential [4]. According to the Mu¨ller’s argument, the change in the gate potential would * Tel.: þ82-42-866-6904; fax: þ 82-42-866-6923. E-mail address: [email protected] (M. Shin).
not lead to the phase shift of the scattered and the tunneled waves, since the scattered wave will always adjust its course so as to meet the constructive interference condition BS ¼ nF0
ð1Þ
where S is the area enclosed by the two waves. We will argue later that the gate potential change does lead to the phase shift between them. Our argument is based on it that the constructive condition of Eq. (1) is not necessary. Let us suppose that an electron wave of unit flux I in the edge state is incident from the left of the ring region (see Fig. 1(a)). As it propagates along the edge channel, it scatters with the barrier in the lower arm of the ring. The barrier is assumed to interact with an incident wave in such a way that the transmission and the reflection amplitudes are tG and rG ; respectively, when it is incident from the left of the barrier, and ~tG and r~G when incident from the right of the barrier. After the initial scattering, part of the incident wave transmits through the barrier and continues to propagate along the edge channel on the other side of the barrier. The reflected wave propagates around the inner circle, adding fP to its phase in the meanwhile, and hits the barrier again. The reflected wave at this second scattering propagates along the edge channel to the right, but the transmitted wave still remains in the ring region and propagates around the inner circle once again. The same scattering process continues
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 0 9 8 ( 0 2 ) 0 0 1 3 5 - 7
490
M. Shin / Solid State Communications 122 (2002) 489–492
fC ¼ fP þ f~G ; it can be readily obtained that T ¼ expðiDÞ;
ð4Þ
where
D ; fG þ fC þ d
ð5Þ
and tanðdÞ ¼ n 2ltg lsin fC ð1 2 ltg lcos fC Þ= ð1 2 ltg lcos fC Þ2
ð6Þ
o
2 ðltg lsin fC Þ2 : In obtaining Eqs. (4) and (5), we have used the fact that rG ¼ 2~rG and tG ~tG 2 rG r~G ¼ expðiðfG þ f~G ÞÞ: Eq. (4) indicates that the net effect of the scattering by the barrier is to change the electron’s phase by D in the transmission amplitude T. The transmission probability is always one, lTl2 ¼ 1; which should be so because there is no path for the electron’s backscattering to the other side of the wire. A main result in this study is that the phase D changes to the change in the barrier potential through fG ; fC and d in Eqs. (5) and (6). To explicitly show their relationship with the barrier potential, let us consider a symmetric system, where the ring has the central radius of RC and the barrier spans an angle of uG (Fig. 1(b)). Then
fG < ðBS=F0 ÞuL =2p þ kG RC uL
ð7Þ
and
fC < 2ðBS=F0 Þ þ kRC ð2p 2 uL Þ þ kG RC uL
Fig. 1. Ring geometries with a barrier in the lower arm.
until all the flux initially incident from the ‘source’ lead exits to the ‘drain’ lead. The summation of the initially transmitted wave and all the reflected waves by the subsequent scatterings yields the eventual transmission amplitude T, and it can be easily calculated as follows by supposing the steady state of the electron waves in the system. In Fig. 1(a), the amplitude of the wave leaving the barrier to the left along the edge of the inner circle is represented as A1, that of the wave incident on the right of the barrier along the edge of the inner circle as A2, and that of the wave transmitted as T. In the steady state, the amplitudes satisfy A1 ¼ rG I þ ~tG A2 ;
A2 ¼ PA1 ;
T ¼ tG I þ r~G A2 : ð2Þ
where P is the propagator along the edge of the inner circle, P ¼ expðifP Þ: If we write tG ¼ ltG lexpðifG Þ;
~tG ¼ ltG lexpðif~G Þ;
ð3Þ
and denote as fC the phase gain as the electron makes one complete turn along the edge of the inner circle clockwise;
ð8Þ
where k is the wave vector of the electron in the leads and in the ring annulus excluding the barrier and kG is the wave vector beneath the barrier. If we model the barrier as a simple potential rise of constant height VG =E0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kG ¼ kG ðVG Þ < k2 2 ðVG =E0 Þ; ð9Þ where E0 ; "2 =2mp w2 ; where mp is the effective mass of electron and w is the width of the wire. If we apply a constant magnetic field to the system and vary the barrier potential, fG and fC change through change in the phase kG ðVG Þ of electron beneath the barrier, and d changes through both change in the phase and change in the transmission probability ltG l: Since kG and ltG l are continuous functions of the barrier height VG =E0 ; we can readily see via Eqs. (4)– (8) that a continuous change in the barrier potential leads to the continuous change in the phase D of T. This result of ours disagrees with the conclusion which can be drawn from Mu¨ller’s argument that the phase D cannot be continuously varied no matter how we change the relative phase of the tunneled and the scattered waves, via the barrier potential change for an instance. Both of Mu¨ller’s and our results however yield lTl2 ¼ 1 for the system in Fig. 1(a) or (b), so the disagreement cannot be experimentally clarified with such a set-up. We therefore
M. Shin / Solid State Communications 122 (2002) 489–492
Fig. 2. A three-dimensional plot of lT 0 l2 for a very short barrier case.
values of ltG l2 ; D0 of Eq. (10) can become a multiple of 2p and lT 0 l2 reaches the maximum of 1 at the points. However, when fC happens to be 2np at the same time, d ¼ 0 (mod 2p) and lT 0 l2 becomes nearly independent of ltG l2 : This feature results in high asymmetry with respect to the points fC ¼ 2np in a three-dimensional plot of lT 0 l2 : In Fig. 2, peaks corresponding to the maximum value of 1 run in the plane of fC and ltG l2 for fC , 2p and terminate at fC ¼ 2p; but for fC . 2p; lT 0 l2 becomes almost constant. For a long central barrier, on the other hand, the change in the electron’s phase under the barrier, rather than the transmission probability as in the case of the very short barrier, dominantly determines the dependence of D0 on the barrier potential. Namely, when we apply a constant magnetic field and vary the barrier potential,
D0 ðVG Þ < 2RC uL kG ðVG Þ þ const; suggest to consider the system in Fig. 1(c), where we attach second barriers at the end of the two leads. These barriers reflect the electron with the reflection probability lr 0 l2 so that the transmission probability of the whole system becomes now (omitting uninteresting phase factors) lT 0 l2 ¼ lt0 l4 =ð1 þ lr 0 l4 2 2lr 0 l2 cos D0 Þ 0 2
ð10Þ
0 2
where lt l ¼ 1 2 lr l ;and
D0 ; D þ ðBS0 =F0 þ kL0 Þ
ð11Þ
where S0 is the area of the ring region enclosed by the second barriers and the top and bottom wire walls, and L0 is its perimeter. The edge state condition should be fulfilled. With a sufficiently high reflection probability lr 0 l2 of the second barriers, the transmission probability lT 0 l2 ; which is directly related to the conductance of the system actually measured in real experiments, is a good measure of the change in the phase D, if any. Note that if we follow Mu¨ller’s argument, D ¼ 2np; where n is an integer, and so lT 0 l2 of Eq. (10) will be constant with respect to the center barrier (gate) potential. In the following, we will provide in detail the consequences of our claim that the phase D of Eqs. (4) and (11) is continuously varied, when applied to the system in Fig. 1(c). First, we note that if fC ¼ 2np; i.e. if the constructive interference condition is satisfied when the electron makes one complete turn around the inner circle, D ¼ fG (mod 2p) independently of the transmission probability ltG l2 through the barrier. And when fC ¼ ð2n þ 1Þp; i.e. for the destructive condition, D ¼ fG þ p (mod 2p) independently of ltG l2 : This implies that for the special cases, the barrier acts as if it is an ideal one-dimensional phase modulator which only changes the electron’s phase without any backscattering, even though it actually scatters the electron with the probability lrG l2 where lrG l2 can assume any value from 0 to 1. We next discuss the case of a very short central barrier. fG and fC are then almost constant with respect to the barrier potential VG ; and d responds to change in VG through the transmission probability ltG l2 of the barrier. For certain
491
ð12Þ
where kG ðVG Þ may be given by Eq. (9). To make it certain that the earlier relation holds for the long barrier case, we have actually solved the time-independent Schro¨dinger equation for the system in Fig. 1(c). The Hamiltonian for an electron of charge e and effective mass mp moving ballistically in the structure is; "2 eA 2 2i7 þ þVðx; yÞ ð13Þ H¼ "c 2mp where A is the vector potential of the perpendicular magnetic field B ¼ ð0; 0; BÞ and the potential Vðx; yÞ is given by 8 1 at the wire walls; > > > > > < VG at the barrier in the ring region; Vðx; yÞ ¼ ð14Þ > > VL at the lead barriers; > > > : 0 otherwise: The barrier in the ring region is modeled as a simple potential rise of constant height VG =E0 and the lead barriers as those of constant height VL =E0 : We then numerically solve the resultant Schro¨dinger equations and obtain eigenvalues (eigenmomentum) and eigenfunctions of the straight part and the bent parts. We next write down the wave functions of all different regions by expanding them in terms of corresponding eigenfunctions. The wave functions and their derivatives are matched at all boundaries of different regions and, resultant linear equations are solved to yield the transmission amplitudes. Detailed description of the mode-matching technique can be found in Refs. [5,6]. Note that the Hamiltonian of Eq. (13) is essentially the same as the one used by Mu¨ller. The only difference is that he solved the time-dependent Schro¨dinger equation while we solved the time-independent one. In Fig. 3(a), we show the result of solving the timeindependent Schro¨dinger equation in the long barrier case. The parameters used in the calculation are: RC ¼ 3:5w; uG ¼ p=3; EF =E0 ¼ ð2pÞ2 ; B=B0 ¼ 15; VL =E0 ¼ 2:1 and
492
M. Shin / Solid State Communications 122 (2002) 489–492
~ 0 ¼ 0 (heavy solid line), 0.15 (dotted line), 0.30 (dashed line), and 0.45 (light solid Fig. 3. (a)lT 0 l2 versus VG =E0 for a long barrier case at kB T=E ~ 0 Þ: line). (b) The oscillation amplitudes of lT 0 l2 with respect to temperature ðkB T=E
~ 0 ¼ 0; 0.15, 0.30, 0.45, where B0 ¼ "c=ew2 ; EF is the kB T=E Fermi energy and T~ is temperature. At the parameters, only one propagating channel is formed in the system, and due to the sufficiently high magnetic field, it is in the edge state in all regions in the structure. The lead barriers back-scatter 11.2 % of incident electrons at the lead potential. As we increase the potential of the barrier in the ring region from zero until it is equal to the Fermi energy, we observe oscillations whose periods are in good agreement with Eq. (12). The transmission probability follows Eq. (10) very well at zero temperature, but the amplitudes of the oscillations decay almost exponentially with increase in temperature (Fig. 3(b)). The terms containing k in Eqs. (7), (8), and (11) are responsible for the rapid decay. Namely, at the temperature where the fluctuation of k is allowed to be as large as
the Mu¨ller’s that D ¼ 2np; where n is an integer, regardless of the barrier potential. We have proposed an experimentally possible set-up where the phase shift D with respect to the barrier potential can be directly detected, and made detailed prediction in cases of very short and long barriers. In the long barrier case, the temperature dependence of the transmission probability is also provided.
DkðL0 þ 2pRC Þ < 2p;
References
ð15Þ
the amplitude of the oscillation becomes almost zero. In conclusion, we have re-examined the situation considered earlier by Mu¨ller where the edge states are forced to interfere on one side of an Aharonov– Bohm ring by a potential barrier. We have found that the net effect of the interference induced by the potential barrier results in the electron’s phase shift D in the final transmittance, and the phase shift D is continuous as the barrier potential is continuously changed. This result is in sharp contrast with
Acknowledgments This work has been supported by the Ministry of Informations and Communications, Korea.
J.E. Mu¨ller, Phys. Rev. Lett. 72 (1994) 2616. M. Bu¨ttiker, Phys. Rev. B 38 (1988) 9875. M. Shin, Phys. Rev. Lett. 77 (1996) 5146. P.G.N. de Vegvar, G. Timp, P.M. Mankiewich, R. Behringer, J. Cunningham, Phys. Rev. B 40 (1989) 3491. [5] R.L. Shult, H.W. Wyld, D.G. Ravenhall, Phys. Rev. B 41 (1990) 12760. [6] M. Shin, K.W. Park, S. Lee, E.-H. Lee, Phys. Rev. B 53 (1996) 1014.
[1] [2] [3] [4]