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Spin–orbit Berry phase in a quantum loop
ARTICLE IN PRESS
Physica E 34 (2006) 397–400 www.elsevier.com/locate/physe
Spin–orbit Berry phase in a quantum loop Maxim P. Trushin, Alexander L. Chudnovskiy 1. Institut fu¨r Theoretische Physik, Universita¨t Hamburg, Jungiusstr 9, D-20355 Hamburg, Germany Available online 19 April 2006
Abstract We have found a manifestation of spin–orbit Berry phase in the conductance of a mesoscopic loop with Rashba spin–orbit coupling placed in the external magnetic field which is perpendicular to the loop plain. The transmission probabilities at different radiuses of the loop are calculated. In addition, the nonadiabatic regime is investigated. r 2006 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 05.60.Gg; 03.65.Vf Keywords: Ballistic transport; Spin–orbit coupling; Berry phase
The beauty of topological Berry phase concept [1] inspires much theoretical and experimental activity aimed to find its manifestations in different areas of modern physics [2]. Berry describes a quantal system in an eigenstate, slowly transported round a circuit by varying parameters in its Hamiltonian. According to the adiabatic theorem, if the Hamiltonian is returned to its original form the system will return to its original state, apart from a phase factor. In addition to the familiar dynamical phase, such a state can acquire a geometrical, circuit-dependent phase factor, which is the result of the adiabatic variation of the external parameters. (See also a fundamental generalization of this idea for nonadiabatic evolution [3].) A possible candidate for the role of such a parameter in solid state physics is the external magnetic field B that interacts with the electron spin via the Hamiltonian H Z ¼ gmB ðr BÞ=2, where r ¼ fsx ; sy ; sz g are Pauli matrices, and mB , g are Bohr magneton and g-factor, respectively. When the value of the magnetic field is constant and its direction follows adiabatically a closed trajectory the spin wave function acquires the topological phase factor which is proportional to the solid angle subtended in a space by the magnetic field [1]. This proposition is the central issue explored in the pioneering and recent papers [4]. In detail, the authors consider the Corresponding author.
E-mail address: [email protected] (M.P. Trushin). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.090
adiabatic as well as nonadiabatic motion of electrons through a mesoscopic ring in the presence of a static, inhomogeneous magnetic field. As was noted above, in order to observe the geometric phase in an electronic system with spin, the application of an orientationally inhomogeneous (e.g. radial) magnetic field is necessary [4]. However, the manner in which the magnetic field is varied leads to rather difficult experiments. Fortunately, the desired magnetic field texture can be experimentally implemented via fabricating the ring from a material with spin–orbit (SO) interaction of Rashba type [5]. Indeed, Rashba Hamiltonian H R ¼ a½r kz (here a is the SO coupling constant, k is the wave vector) gives rise to a radial built-in Zeeman-like magnetic field Bin ¼ 2ak=ðgmB Þ. Therefore, when electrons adiabatically encircle a ring in presence of the additional, external magnetic field B perpendicular to the ring plane (jBj ¼ Bz ), electron spin, influenced by the total effective magnetic field Beff ¼ Bin þ B, subtends a cone-shaped trajectory in parameter space, and the particle acquires the spin geometric phase which is given by the formula 0 1 Bz B C fB ¼ p@1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 Bz þ Bin
(1)
Moreover, since the particle carries a charge as well, it picks up an Aharonov–Bohm phase fAB ¼ 2pF=F0 , where
ARTICLE IN PRESS M.P. Trushin, A.L. Chudnovskiy / Physica E 34 (2006) 397–400
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F0 ¼ 2p _c=e is the flux quantum, and F ¼ pR2 Bz is the magnetic flux enclosed by the ring. It is important to emphasize, that the mentioned magnetic field Bin does not relate to the real external magnetic field Bz , but stems from the internal properties of the substance. Most important, however, the external Bz and in-plane Bin components form the desired inhomogeneous magnetic field texture and in that way can provide the geometric phase indications in the interference conductance pattern. This pretty idea is attracting both theoretical [6–9] and experimental [10,11] attention. Experimental schemes aiming to observe the Berry phase in materials with SO coupling were based on measurements of Aharonov–Bohm oscillations [10,11]. In the present work, we find Berry phase manifestation in the conductance oscillations that stem from the interference between two spin states with different dynamical phases. We concentrate on the theoretical investigation of the setup similar to Ref. [11] (see Fig. 1). In our system, however, possibility to bypass the ring is assumed to be negligible, so that the Aharonov–Bohm effect does not occur, and the system actually represents a loop connected to the input and output leads. Furthermore, we also consider the influence of the gate voltage applied to the loop. In order to find the transmission probability through such a system we need the solution of the corresponding Schro¨dinger equation. To this end we divide the system in three parts: input channel, the loop and output channel. The Hamiltonians describing the propagation of electron in the input/output channels read 0 1 2 ia k^x _2 k^x =ð2m Þ þ Z A H wire ¼ @ (2) 2 ia k^x _2 k^x =ð2m Þ Z whereas the propagation through the loop of radius R is governed by the Hamiltonian [12] 1 0 0 q^ 2j þ Z þ V aeij q^ j 12 =R B C (3) H loop ¼ @ A. 2 ij ^ 1 0 q^ j Z þ V ae qj þ 2 =R Here k^x ¼ iðq=qxÞ ðF=F0 Þð1=RÞ, q^ j ¼ iq=qj F=F0 are momentum and angular momentum operators, respectively, 0 ¼ _2 =ð2m R2 Þ is the size confinement energy with the effective electron mass m , Z ¼ g mB Bz =2 is the Zeeman energy, and V denotes an energy shift determined by the gate voltage applied to the loop. We adopt the
Vgate
y
B 1D loop R
x
0 input
−R
output
Fig. 1. Geometry of the system. Note, that in contrast to Ref. [11] the electron beam does not split while it enters the loop.
vector potential A that is tangential to the direction of the current. In the loop we choose Aj ðjÞ ¼ F=2pR. The vector potential in the leads is determined by the continuity condition at the junction point with the loop, hence we have Ax ¼ F=2pR. We denote the wave functions for each part as C loop ðjÞ for the loop, C ðxÞ and C ðxÞ for input and output in out channels, respectively. In order to find the wave functions of the whole system, we impose boundary conditions that warrant the continuity of the wave function and its first derivative on the boundaries between the loop and input/ output channels. Solution of Schro¨dinger equations for Hamiltonians (2), (3) give us the desired spinor wave functions. For the input channel we have ! þ þ cos gþ ðeik x þ Aþ eik x Þ þ ðiF=F0 RÞx Cin ðxÞ ¼ e , (4) þ þ i sin gþ ðeik x Aþ eik x Þ ! ik x ik x i sin g ðe A e Þ ðiF=F0 RÞx C , (5) in ðxÞ ¼ e cos g ðeik x þ A eik x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where tan g ¼ Z =k a þ 1 þ ðZ =k aÞ2 , and k are the Fermi wave vectors that satisfy the dispersion relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 E F ¼ ð_2 k =2m Þ ðak Þ2 þ 2Z , E F being the Fermi energy. The coefficients A are the reflection amplitudes that have to be found imposing the boundary conditions. For the output channel the reflection amplitudes are assumed to be zero, and the corresponding spinors read ! þ Dþ cos gþ eiðk þðF=F0 RÞÞx þ Cout ðxÞ ¼ , (6) þ iDþ sin gþ eiðk þðF=F0 RÞÞx
C out ðxÞ
¼
iD sin g eiðk D cos g eiðk
þðF=F0 RÞÞx
þðF=F0 RÞÞx
! .
(7)
Here D are the transmission amplitudes. The eigenfunctions of the Hamiltonian (3) are defined as linear combinations of the right- and left-propagating spinors. The Fermi angular momentum q R satisfies the relation ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 E F ¼ V þ 0 =4 þ 0 q ðq R R a=RÞ þ ðqR 0 Z Þ . The relation for q L differs only by the sign before Z . Imposing the abovementioned boundary conditions on the wave functions, we obtain a solution of the Schro¨dinger equation for the whole system. At this point it is pertinent to turn to the current densities calculations. Using foregoing results one can easily find the input, reflected and transmitted current densities. Each current density is given as a sum of its two spin components j ¼ j þ þ j , whereas 2 2 j in ¼ ð_=m Þ½k ðam =_ Þ sinð2g Þ, j refl ¼ jA j j in , and 2 j out ¼ jD j j in . The transmission probability is defined as T ¼ j out =j in , and the reflection one R ¼ j refl =j in . The plots TðBz Þ are shown in Fig. 2 (solid lines) for different radius of the loop. The influence of the Berry phase can be seen in
ARTICLE IN PRESS M.P. Trushin, A.L. Chudnovskiy / Physica E 34 (2006) 397–400
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0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8 (c)
Fig. 2. Transmission probability for the loop of radiuses R (solid lines: (a) R ¼ 5 105 cm, (b) R ¼ 105 cm, (c) R ¼ 5 106 cm) and for corresponding straight wire of length L ¼ 2pR (dotted lines) vs. magnetic field. The barrier height V is taken equal to 18.75 meV. The other parameters are taken relevant for InAs: a ¼ 2 1011 eVm, m ¼ 0:033me , g ¼ 12, E F ¼ 30 meV.
the difference of the transmission of the loop with that of a straight wire with the same length L ¼ 2pR (dotted line in the same figures).
399
Fig. 2 exhibits the following characteristic features. First, the transmission probability oscillates as a function of the external magnetic field Bz . The oscillations have a natural explanation if one follows the evolution of the wave function as a particle propagates through the system. Namely, after entering the loop, the component of the input wave function Cþ in propagates as a linear combination þ of the modes Cþ and C loop with the wave vectors qR and loop þ qR (or qL and qL for reflected electrons), respectively. The same is true for the propagation of the state C in . Due to the interference between propagated states at the input and output of the loop, the oscillating factors appear in the transmission probability, which shows up in Fig. 2. Second, there is a strong difference between transmission probabilities for the loop and straight wire at certain intermediate values of the magnetic field (see Fig. 2), while at higher values and at Bz ¼ 0 the both curves just coincide. Moreover, the transmission probability for the loop at that special range of magnetic fields is usually smaller than for the corresponding straight wire. This is a particular manifestation of the Berry phase that we explain in what follows. First of all, note that the Berry phase is always zero in the straight wire. In contrast to that simple case, an additional Berry phase dependent interference factor sin fB occurs while an electron wave function propagates through the loop. The Berry phase (1) is negligible at Bz bBin and equal to p at Bz ¼ 0. Therefore, the factor sin fB does not show up in these cases. At certain intermediate values of Bz the difference between straight wire and loop geometry is essential. In particular, at certain special Bz Berry phase is equal to p=2 and the difference between transmission probabilities for the loop and straight wireis maximal. We find it necessary to estimate such magnetic field using the quasiclassical formula (1) and assuming parameters relevant for InAs: a ¼ 2 1011 eVm, g ¼ 12, k ¼ 106 cm1 . Then,pthe ffiffiffi Berry phase value p=2 corresponds to Bz ¼ jBin j= 3 or, numerically, 2:9 T that is in a good agreement with the plots in Fig. 2a,b. At this point one begins to wonder what happens in Fig. 2c. It is clearly seen, that the maximum of the difference between transmission probabilities of the loop and straight wire is shifted to higher magnetic fields. We can explain the effect if we remember, that the formula (1) (and Berry concept as well) is valid only for the adiabatic motion. The latter means, that am R=_2 must be larger than one, so that the electron spin precesses a few times while it is moving through the loop. This is not the case depicted in Fig. 2c, where am R=_2 0:5 and the motion is definitely not adiabatic. Note, that our general approach is valid for both adiabatic and nonadiabatic cases, because we use direct solution of Schro¨dinger equation. Therefore, we are able to see the Aharonov– Anadan geometric phase [3] effects in Fig. 2c. This kind of geometrical phase is the nonadiabatic generalization of Berry’s and in our case this is nothing else as ftop ¼ p½1 ðq L qR Þ. Of course, this expression has its limit (1) in the appropriate regime.
ARTICLE IN PRESS 400
M.P. Trushin, A.L. Chudnovskiy / Physica E 34 (2006) 397–400
In conclusion, we have studied the quantum transport in a mesoscopic loop with Rashba coupling and Zeeman splitting. Here we have found that the Berry phase should give a well-pronounced interference effect as the substantial deviation of the transmission probability (compared to the straight wire of the same length L ¼ 2pR) at some specific value of the external magnetic field. Moreover, we have investigated our system under the nonadiabatic regime and found, that the characteristic magnetic fields, which provide the deviation, are shifted to higher values. Estimations based on typical parameters for InAs show that the effect predicted is accessible for current experimental techniques. The authors acknowledge financial support by DFG from Graduiertenkolleg ‘‘Physik nanostrukturierter Festko¨rper’’ (M.T.) and Sonderforschungsbereich 508 (A.C.). References [1] M.V. Berry, Proc. Roy. Soc. London A 392 (1984) 45. [2] A. Shapere, F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore, 1989. [3] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593.
[4] D. Loss, P.M. Goldbart, A.V. Balatsky, Phys. Rev. Lett. 65 (1990) 1655; D. Loss, P. Goldbart, Phys. Rev. B 45 (1990) 13544; A. Stern, Phys. Rev. Lett. 68 (1992) 1022; H.-A. Engel, D. Loss, Phys. Rev. B 62 (2000) 10238; D. Frustaglia, M. Hentschel, K. Richter, Phys. Rev. Lett. 87 (2001) 256602; D. Frustaglia, K. Richter, Phys. Rev. B 69 (2004) 235310; M. Popp, D. Frustaglia, K. Richter, Phys. Rev. B 68 (2003) 041303; M. Hentschel, H. Schomerus, D. Frustaglia, K. Richter, Phys. Rev. B 69 (2004) 155326; M. Hentschel, D. Frustaglia, K. Richter, Phys. Rev. B 69 (2004) 155327. [5] Yu.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039. [6] A.G. Aronov, Y.B. Lyanda-Geller, Phys. Rev. Lett. 70 (1993) 343. [7] Y. Lyanda-Geller, Phys. Rev. Lett. 71 (1993) 657. [8] T.-Z. Qian, Z.-B. Su, Phys. Rev. Lett. 72 (1994) 2311. [9] A.G. Mal’shukov, V.V. Shlyapin, K.A. Chao, Phys. Rev. B 60 (1999) R2161. [10] A.F. Morpurgo, J.P. Heida, T.M. Klapwijk, B.J. van Wees, G. Borghs, Phys. Rev. Lett. 80 (1998) 1050; J. Nitta, H. Takayanagi, S. Calvet, Microelectron. Eng. 47 (1999) 85; J.-B. Yau, E.P. De Poortere, M. Shayegan, Phys. Rev. Lett. 88 (2002) 146801. [11] M.J. Yang, C.H. Yang, Y.B. Lyanda-Geller, Europhys. Lett. 66 (2004) 826. [12] F.E. Meijer, A.F. Morpurgo, T.M. Klapwijk, Phys. Rev. B 66 (2002) 033107.
Physica E 34 (2006) 397–400 www.elsevier.com/locate/physe
Spin–orbit Berry phase in a quantum loop Maxim P. Trushin, Alexander L. Chudnovskiy 1. Institut fu¨r Theoretische Physik, Universita¨t Hamburg, Jungiusstr 9, D-20355 Hamburg, Germany Available online 19 April 2006
Abstract We have found a manifestation of spin–orbit Berry phase in the conductance of a mesoscopic loop with Rashba spin–orbit coupling placed in the external magnetic field which is perpendicular to the loop plain. The transmission probabilities at different radiuses of the loop are calculated. In addition, the nonadiabatic regime is investigated. r 2006 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 05.60.Gg; 03.65.Vf Keywords: Ballistic transport; Spin–orbit coupling; Berry phase
The beauty of topological Berry phase concept [1] inspires much theoretical and experimental activity aimed to find its manifestations in different areas of modern physics [2]. Berry describes a quantal system in an eigenstate, slowly transported round a circuit by varying parameters in its Hamiltonian. According to the adiabatic theorem, if the Hamiltonian is returned to its original form the system will return to its original state, apart from a phase factor. In addition to the familiar dynamical phase, such a state can acquire a geometrical, circuit-dependent phase factor, which is the result of the adiabatic variation of the external parameters. (See also a fundamental generalization of this idea for nonadiabatic evolution [3].) A possible candidate for the role of such a parameter in solid state physics is the external magnetic field B that interacts with the electron spin via the Hamiltonian H Z ¼ gmB ðr BÞ=2, where r ¼ fsx ; sy ; sz g are Pauli matrices, and mB , g are Bohr magneton and g-factor, respectively. When the value of the magnetic field is constant and its direction follows adiabatically a closed trajectory the spin wave function acquires the topological phase factor which is proportional to the solid angle subtended in a space by the magnetic field [1]. This proposition is the central issue explored in the pioneering and recent papers [4]. In detail, the authors consider the Corresponding author.
E-mail address: [email protected] (M.P. Trushin). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.090
adiabatic as well as nonadiabatic motion of electrons through a mesoscopic ring in the presence of a static, inhomogeneous magnetic field. As was noted above, in order to observe the geometric phase in an electronic system with spin, the application of an orientationally inhomogeneous (e.g. radial) magnetic field is necessary [4]. However, the manner in which the magnetic field is varied leads to rather difficult experiments. Fortunately, the desired magnetic field texture can be experimentally implemented via fabricating the ring from a material with spin–orbit (SO) interaction of Rashba type [5]. Indeed, Rashba Hamiltonian H R ¼ a½r kz (here a is the SO coupling constant, k is the wave vector) gives rise to a radial built-in Zeeman-like magnetic field Bin ¼ 2ak=ðgmB Þ. Therefore, when electrons adiabatically encircle a ring in presence of the additional, external magnetic field B perpendicular to the ring plane (jBj ¼ Bz ), electron spin, influenced by the total effective magnetic field Beff ¼ Bin þ B, subtends a cone-shaped trajectory in parameter space, and the particle acquires the spin geometric phase which is given by the formula 0 1 Bz B C fB ¼ p@1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 Bz þ Bin
(1)
Moreover, since the particle carries a charge as well, it picks up an Aharonov–Bohm phase fAB ¼ 2pF=F0 , where
ARTICLE IN PRESS M.P. Trushin, A.L. Chudnovskiy / Physica E 34 (2006) 397–400
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F0 ¼ 2p _c=e is the flux quantum, and F ¼ pR2 Bz is the magnetic flux enclosed by the ring. It is important to emphasize, that the mentioned magnetic field Bin does not relate to the real external magnetic field Bz , but stems from the internal properties of the substance. Most important, however, the external Bz and in-plane Bin components form the desired inhomogeneous magnetic field texture and in that way can provide the geometric phase indications in the interference conductance pattern. This pretty idea is attracting both theoretical [6–9] and experimental [10,11] attention. Experimental schemes aiming to observe the Berry phase in materials with SO coupling were based on measurements of Aharonov–Bohm oscillations [10,11]. In the present work, we find Berry phase manifestation in the conductance oscillations that stem from the interference between two spin states with different dynamical phases. We concentrate on the theoretical investigation of the setup similar to Ref. [11] (see Fig. 1). In our system, however, possibility to bypass the ring is assumed to be negligible, so that the Aharonov–Bohm effect does not occur, and the system actually represents a loop connected to the input and output leads. Furthermore, we also consider the influence of the gate voltage applied to the loop. In order to find the transmission probability through such a system we need the solution of the corresponding Schro¨dinger equation. To this end we divide the system in three parts: input channel, the loop and output channel. The Hamiltonians describing the propagation of electron in the input/output channels read 0 1 2 ia k^x _2 k^x =ð2m Þ þ Z A H wire ¼ @ (2) 2 ia k^x _2 k^x =ð2m Þ Z whereas the propagation through the loop of radius R is governed by the Hamiltonian [12] 1 0 0 q^ 2j þ Z þ V aeij q^ j 12 =R B C (3) H loop ¼ @ A. 2 ij ^ 1 0 q^ j Z þ V ae qj þ 2 =R Here k^x ¼ iðq=qxÞ ðF=F0 Þð1=RÞ, q^ j ¼ iq=qj F=F0 are momentum and angular momentum operators, respectively, 0 ¼ _2 =ð2m R2 Þ is the size confinement energy with the effective electron mass m , Z ¼ g mB Bz =2 is the Zeeman energy, and V denotes an energy shift determined by the gate voltage applied to the loop. We adopt the
Vgate
y
B 1D loop R
x
0 input
−R
output
Fig. 1. Geometry of the system. Note, that in contrast to Ref. [11] the electron beam does not split while it enters the loop.
vector potential A that is tangential to the direction of the current. In the loop we choose Aj ðjÞ ¼ F=2pR. The vector potential in the leads is determined by the continuity condition at the junction point with the loop, hence we have Ax ¼ F=2pR. We denote the wave functions for each part as C loop ðjÞ for the loop, C ðxÞ and C ðxÞ for input and output in out channels, respectively. In order to find the wave functions of the whole system, we impose boundary conditions that warrant the continuity of the wave function and its first derivative on the boundaries between the loop and input/ output channels. Solution of Schro¨dinger equations for Hamiltonians (2), (3) give us the desired spinor wave functions. For the input channel we have ! þ þ cos gþ ðeik x þ Aþ eik x Þ þ ðiF=F0 RÞx Cin ðxÞ ¼ e , (4) þ þ i sin gþ ðeik x Aþ eik x Þ ! ik x ik x i sin g ðe A e Þ ðiF=F0 RÞx C , (5) in ðxÞ ¼ e cos g ðeik x þ A eik x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where tan g ¼ Z =k a þ 1 þ ðZ =k aÞ2 , and k are the Fermi wave vectors that satisfy the dispersion relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 E F ¼ ð_2 k =2m Þ ðak Þ2 þ 2Z , E F being the Fermi energy. The coefficients A are the reflection amplitudes that have to be found imposing the boundary conditions. For the output channel the reflection amplitudes are assumed to be zero, and the corresponding spinors read ! þ Dþ cos gþ eiðk þðF=F0 RÞÞx þ Cout ðxÞ ¼ , (6) þ iDþ sin gþ eiðk þðF=F0 RÞÞx
C out ðxÞ
¼
iD sin g eiðk D cos g eiðk
þðF=F0 RÞÞx
þðF=F0 RÞÞx
! .
(7)
Here D are the transmission amplitudes. The eigenfunctions of the Hamiltonian (3) are defined as linear combinations of the right- and left-propagating spinors. The Fermi angular momentum q R satisfies the relation ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 E F ¼ V þ 0 =4 þ 0 q ðq R R a=RÞ þ ðqR 0 Z Þ . The relation for q L differs only by the sign before Z . Imposing the abovementioned boundary conditions on the wave functions, we obtain a solution of the Schro¨dinger equation for the whole system. At this point it is pertinent to turn to the current densities calculations. Using foregoing results one can easily find the input, reflected and transmitted current densities. Each current density is given as a sum of its two spin components j ¼ j þ þ j , whereas 2 2 j in ¼ ð_=m Þ½k ðam =_ Þ sinð2g Þ, j refl ¼ jA j j in , and 2 j out ¼ jD j j in . The transmission probability is defined as T ¼ j out =j in , and the reflection one R ¼ j refl =j in . The plots TðBz Þ are shown in Fig. 2 (solid lines) for different radius of the loop. The influence of the Berry phase can be seen in
ARTICLE IN PRESS M.P. Trushin, A.L. Chudnovskiy / Physica E 34 (2006) 397–400
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Fig. 2. Transmission probability for the loop of radiuses R (solid lines: (a) R ¼ 5 105 cm, (b) R ¼ 105 cm, (c) R ¼ 5 106 cm) and for corresponding straight wire of length L ¼ 2pR (dotted lines) vs. magnetic field. The barrier height V is taken equal to 18.75 meV. The other parameters are taken relevant for InAs: a ¼ 2 1011 eVm, m ¼ 0:033me , g ¼ 12, E F ¼ 30 meV.
the difference of the transmission of the loop with that of a straight wire with the same length L ¼ 2pR (dotted line in the same figures).
399
Fig. 2 exhibits the following characteristic features. First, the transmission probability oscillates as a function of the external magnetic field Bz . The oscillations have a natural explanation if one follows the evolution of the wave function as a particle propagates through the system. Namely, after entering the loop, the component of the input wave function Cþ in propagates as a linear combination þ of the modes Cþ and C loop with the wave vectors qR and loop þ qR (or qL and qL for reflected electrons), respectively. The same is true for the propagation of the state C in . Due to the interference between propagated states at the input and output of the loop, the oscillating factors appear in the transmission probability, which shows up in Fig. 2. Second, there is a strong difference between transmission probabilities for the loop and straight wire at certain intermediate values of the magnetic field (see Fig. 2), while at higher values and at Bz ¼ 0 the both curves just coincide. Moreover, the transmission probability for the loop at that special range of magnetic fields is usually smaller than for the corresponding straight wire. This is a particular manifestation of the Berry phase that we explain in what follows. First of all, note that the Berry phase is always zero in the straight wire. In contrast to that simple case, an additional Berry phase dependent interference factor sin fB occurs while an electron wave function propagates through the loop. The Berry phase (1) is negligible at Bz bBin and equal to p at Bz ¼ 0. Therefore, the factor sin fB does not show up in these cases. At certain intermediate values of Bz the difference between straight wire and loop geometry is essential. In particular, at certain special Bz Berry phase is equal to p=2 and the difference between transmission probabilities for the loop and straight wireis maximal. We find it necessary to estimate such magnetic field using the quasiclassical formula (1) and assuming parameters relevant for InAs: a ¼ 2 1011 eVm, g ¼ 12, k ¼ 106 cm1 . Then,pthe ffiffiffi Berry phase value p=2 corresponds to Bz ¼ jBin j= 3 or, numerically, 2:9 T that is in a good agreement with the plots in Fig. 2a,b. At this point one begins to wonder what happens in Fig. 2c. It is clearly seen, that the maximum of the difference between transmission probabilities of the loop and straight wire is shifted to higher magnetic fields. We can explain the effect if we remember, that the formula (1) (and Berry concept as well) is valid only for the adiabatic motion. The latter means, that am R=_2 must be larger than one, so that the electron spin precesses a few times while it is moving through the loop. This is not the case depicted in Fig. 2c, where am R=_2 0:5 and the motion is definitely not adiabatic. Note, that our general approach is valid for both adiabatic and nonadiabatic cases, because we use direct solution of Schro¨dinger equation. Therefore, we are able to see the Aharonov– Anadan geometric phase [3] effects in Fig. 2c. This kind of geometrical phase is the nonadiabatic generalization of Berry’s and in our case this is nothing else as ftop ¼ p½1 ðq L qR Þ. Of course, this expression has its limit (1) in the appropriate regime.
ARTICLE IN PRESS 400
M.P. Trushin, A.L. Chudnovskiy / Physica E 34 (2006) 397–400
In conclusion, we have studied the quantum transport in a mesoscopic loop with Rashba coupling and Zeeman splitting. Here we have found that the Berry phase should give a well-pronounced interference effect as the substantial deviation of the transmission probability (compared to the straight wire of the same length L ¼ 2pR) at some specific value of the external magnetic field. Moreover, we have investigated our system under the nonadiabatic regime and found, that the characteristic magnetic fields, which provide the deviation, are shifted to higher values. Estimations based on typical parameters for InAs show that the effect predicted is accessible for current experimental techniques. The authors acknowledge financial support by DFG from Graduiertenkolleg ‘‘Physik nanostrukturierter Festko¨rper’’ (M.T.) and Sonderforschungsbereich 508 (A.C.). References [1] M.V. Berry, Proc. Roy. Soc. London A 392 (1984) 45. [2] A. Shapere, F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore, 1989. [3] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593.
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