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Transfer-matrix approach to electron transport in inhomogeneous magnetic fields
Computer Physics Communications 142 (2001) 429–435 www.elsevier.com/locate/cpc
Transfer-matrix approach to electron transport in inhomogeneous magnetic fields K. Yakubo Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Abstract A transfer-matrix method to study quantum transport of two-dimensional (2D) electron systems subject to inhomogeneous magnetic fields has been developed. This method enables us to calculate conductance within the same order of a computing time with those of one-dimensional transfer-matrix calculations. This is because the present method does not involve any diagonalization procedure, while a conventional transfer-matrix technique for 2D systems requires to diagonalize the Hamiltonian for thin strips in the system, which consumes a long computing time. Using this method, we calculate conductances of 2D electrons in periodically modulated and random magnetic fields. 2001 Elsevier Science B.V. All rights reserved. PACS: 73.50.Jt; 73.23.Ps; 72.15.Rn; 02.70.-c
1. Introduction Spatially modulated magnetic fields lead novel quantum transport of two-dimensional (2D) electrons. For example, the magnetoconductance of 2D electrons in a periodically modulated magnetic field shows oscillatory behavior [1]. Recently, various alternative magnetic structures have been proposed on 2D electron systems, such as magnetic quantum dots or magnetic antidots [2,3] and random magnetic fields [4–9]. In particular, transport properties of 2D electrons in a random magnetic field (RMF) are crucial for understanding magnetotransport around a half-filled Landau level [10]. It has been extensively investigated whether extended states exist in the 2D system subject to the RMF [6–9]. However, very little numerical work on transport properties was reported so far. Numerical studies of 2D electrons are very few also for other inhomogeneous magnetic fields. This is due to the lack of a numerical method efficient for investigating quantum transport of 2D electrons in inhomogeneous magnetic fields. In this paper, we show that a transfer-matrix method is quite effective and useful to calculate conductances of 2D noninteracting electron systems in the presence of inhomogeneous magnetic fields. In a conventional transfermatrix method for a 2D system, we have to diagonalize numerically Hamiltonians for thin strips into which the system is divided in the transport direction. This procedure usually takes a long computing time. Therefore, the transfer-matrix method is not suitable for two- or three-dimensional problems. In the present transfer-matrix method, however, analytical expressions for eigenvectors of Hamiltonians are used to construct transfer matrices, E-mail address: [email protected] (K. Yakubo). 0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 1 ) 0 0 3 8 3 - 6
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K. Yakubo / Computer Physics Communications 142 (2001) 429–435
and it is not necessary to diagonalize Hamiltonians for thin strips numerically. As a consequence, we can calculate conductances within the same order of a computing time with those of one-dimensional transfer-matrix calculations, and magnetotransport of 2D electrons in complex magnetic structures can be treated numerically. Furthermore, current distributions in inhomogeneous magnetic fields are also calculated by using the transfer matrix. The paper is organized as follows: In Section 2 the transfer-matrix method for 2D inhomogeneous magnetic field systems is formulated. Some applications are presented in Section 3. Section 4 is devoted to the conclusions.
2. Transfer-matrix formalism Let us consider a 2D electron system in the presence of an inhomogeneous magnetic field B(r) = (0, 0, B(x, y)). There exist no other electrostatic potentials in the system. In the Landau gauge, the vector potential is given by Ax = Az = 0 and x Ay (x, y) =
B(x , y) dx .
(1)
0
Hereafter, we define the x direction as the transport direction. In the transfer-matrix method, the system is divided into M thin strips in the x direction. The width of strips is so small that Ay (x, y) can be regarded as a constant with respect to x in each strip. The Schödinger equation for the lth strip is then given by 2 2 1 ∂ 2 ∂ − eAy (xl , y) + − ih¯ (2) − h¯ ψl (x, y) = Eψl (x, y), 2m∗ ∂x 2 ∂y where m∗ is the electron effective mass and E is the electron incident energy. We find the solution of Eq. (2) in the form of ψl (x, y) = e±ikx x ϕl (y).
(3)
The function ϕl (y) should satisfy the equation: 2 1 ∂ − eAy (xl , y) + ε ϕl (y) = 0, −ih¯ 2m∗ ∂y where ε = E − (h¯ 2 kx2 /2m∗ ). Eq. (4) can be solved analytically for arbitrary Ay (x, y) as y iky y ie −iky y exp ϕl (y) = αe + βe Ay (xl , y ) dy , h¯
(4)
(5)
0
√ where ky = 2m∗ ε/h¯ . Assuming the periodic boundary conditions in the y-direction, that is, ϕl (y + Ly ) = ϕl (y) and B(x, y + Ly ) = B(x, Ly ) with the system size Ly in the y-direction, we have y ν y ie ikyl ϕl (y) = γ e (6) exp Ay (xl , y ) dy , h¯ 0
where Ly e 1 ν kyl = 2νπ − Ay (xl , y) dy , Ly h¯ 0
(7)
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ν = 0, ±1, ±2, . . . , and γ is a constant. Therefore, from Eqs. (1) and (3), the general solution ψl (x, y) is written as
ν ν ν ν ψl (x, y) = fl (y) Aνl ei(kxl x+kyl y) + Blν e−i(kxl x−kyl y) , (8) ν = where kxl
ν ν )2 /h and 2m∗ E − (h¯ kyl ¯
ie fl (y) = exp h¯
yxl
B(x , y ) dx dy .
(9)
0 0
ν In order to relate the coefficients Aνl and Blν in Eq. (8) with Aνl+1 and Bl+1 , let us consider the matching conditions ψl (xl+1 , y) = ψl+1 (xl+1 , y) and ∂x ψl (xl+1 , y) = ∂x ψl+1 (xl+1 , y). Substituting Eq. (8) into the first ∗ (y)e matching condition, multiplying fl+1
Aνl+1 e
ν x ikxl+1 l+1
ν + Bl+1 e
ν x −ikxl+1 l+1
=
ν −ikyl+1 y
, and integrating over y within the range of [0, Ly ], we have
ν ν Clνν Aνl eikxl xl+1 + Blν e−ikxl xl+1 ,
(10)
dy.
(11)
ν
where νν
Cl
1 = Ly
Ly
∗ fl+1 (y)fl (y)e
ν −k ν i(kyl yl+1 )y
0 Clνν
The quantity represents the channel mixing between the νth and ν th channels. From the relation νν 2 νν 2 ν |Cl | = 1, |Cl | can be regarded as the probability to switch the channel ν to ν at the lth strip. In the similar way, from the second matching condition, one obtains ν ik ν x ν νν ν ik ν x ν x ν ν ν −ikxl+1 l+1 Al+1 e xl+1 l+1 − Bl+1 = (12) e kxl Cl Al e xl l+1 − Blν e−ikxl xl+1 . kxl+1 ν
Solving the coupled equations [Eqs. (10) and (12)] and cutting off higher irrelevant channels (|ν| > νc ), we finally obtain the matrix form expression: D l+1 = T(l)D l ,
(13)
c −1 if integer λ where D l is the 2Nc -dimensional coefficient vector whose components are given by Dlλ = Aλ−ν l λ−Nc −νc −1 λ is bounded by 1 λ Nc and Dl = Bl if Nc + 1 λ 2Nc , where Nc = 2νc + 1 is the number of (l) has its components given by channels. The 2Nc × 2Nc matrix T
T
λλ
µ
(kxl+1 + mm kxl ) µ
(l) =
µ 2kxl+1
µ µ µµ i(mkxl+1 −m kxl )xl+1
Cl
e
,
(14)
where µ = {(λ − 1) mod Nc } + 1, m = 2[(λ − 1)/Nc ] − 1, and [x] denotes the largest integer less than or equal µ µ µµ ν , k ν , and C νν with ν = µ − ν − 1, respectively. From to x. Quantities kxl , kyl , and Cl are the same with kxl c yl l Eq. (13), we can calculate D M from D 0 by TM = M T (l). The conductance G is calculated by the Landauer l formula [11]. We can obtain the current distribution from the calculated transmission and reflection coefficients AνM and B0ν . Using again Eq. (13), one obtains coefficient vector D l for all l, and calculates ψl (x, y) from Eq. (8). From the definition of the quantum mechanical current density, j = (ih¯ /2m∗ )[ψl (x, y)∇ψl∗ (x, y) − ψl∗ (x, y)∇ψl (x, y)] − (e/m∗ )A|ψl (x, y)|2 , the spatial distribution of the current density can be computed.
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3. Applications We have applied the present transfer-matrix method to two kinds of 2D electron systems in inhomogeneous magnetic fields. As the first application, we compute the conductance of a 2D system subject to a periodically modulated magnetic field (the modulation is only in the transport direction), namely, B(x, y) = BU + BW cos(qx). Yagi and Iye [1] have found that the magnetoconductance (conductance as a function of BU ) shows an oscillatory behavior with conductance maxima at √ 2h¯ 2πne BU = (n = 1, 2, . . .), (15) ed(n − 1/4) where ne is the electron density, and d is the period of the modulated magnetic field (d = 2π/q). In order to check our numerical method, we confirm this magnetoconductance behavior. The amplitude and the period of the oscillating magnetic field are set to be BW = 0.5 T and d = 40 nm, respectively. The Fermi energy EF of the electron system is fixed at 6.0 meV and the electron effective mass m∗ is chosen as 0.067me , where me is the electron bare mass. The system size Ly is 0.5 µm for which the number of channels Nc for EF = 6.0 meV is 17. The length Lx of the system size in the x-direction varies with BU . If Lx is fixed, the conductance rapidly decreases as increasing BU , because the effective one-dimensional potential barrier (Veff = e2 BU2 x 2 /2m∗ ) due to the uniform magnetic field becomes higher. In order to concentrate the oscillatory behavior of the conductance, the value of Lx is determined so that the transmission probability for Veff becomes constant for any BU . The calculated conductance at low temperature (T = 0.2 K) is shown in Fig. 1. One can find the clear oscillatory behavior of the conductance. Peak positions at lower uniform fields agree well with the theoretical prediction Eq. (15). For BU 0.6 T, peak positions are slightly shifted from the theoretical values. These shifts are caused by the finite Lx in which the number of periods of the oscillating magnetic field is restricted (only 4 periods are included in the system for BU = 1.0 T). It should be noted that peaks labeled by dashed arrows do not correspond to peaks given by Eq. (15), but the Shubnikov–de Haas peaks given by BU = π hn ¯ e /e(n + 1/2). These results show that the transfer-matrix method works well. The second application is quantum transport of 2D electrons in a long-range random magnetic field (RMF). As mentioned in Section 1, 2D transport in the RMF are attracting strong interest in recent years. We examine
Fig. 1. Magnetoconductance of 2D electrons in a periodically modulated magnetic field.
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Fig. 2. Incident electron energy dependences of the conductance of 2D electrons in RMF’s.
the conductance of this system by using the transfer-matrix method. The magnetic field is given by B(x, y) = BU + BR (x, y), where BR (x, y) =
NB
i=1
(x − xi )2 + (y − yi )2 , Bi exp − ξ2
(16)
ξ is 20 nm, Bi is uniformly distributed within [−BR0 , BR0 ], and NB = 200. Centers of Gaussian magnetic fields (xi , yi ) are randomly chosen in the system. The magnitude of fluctuations in BR (x, y) is characterized by 2 BR ≡ BR (x, y) dx dy/S, where S is the area of the system. In this case, both sizes of the system in the x and y directions are fixed at 200 nm. Fig. 2 shows incident energy dependences of conductances at zero temperature. The uniform magnetic field BU is equal to zero. Values of BR are 0.5 and 5.0 T for solid and dotted lines, respectively. The conductance for the weak RMF behaves as in the case of the quantum point contact system in which the conductance is quantized due to varying the number of transport channels. This means that electrons are weakly scattered from the RMF. In this case, the conductance of this system is larger than 2e2 / h, which implies that single channel transport can be described by a semiclassical theory such as the Drude theory. On the contrary, for the strong RMF, the conductance is much smaller than 2e2/ h because of the weak localization effect. In such case, the quantum interference effect is crucial for electron transport. In fact, the conductance exhibits many resonant peaks by the quantum interference. We also calculate magnetoconductance for the BR = 5.0 T case. Fig. 3 shows the conductance change as a function of BU . The electron incident energy is fixed at 6.0 meV. The magnetoconductance is positive for the strong RMF system, while it is negative for the weak RMF (BR = 0.5 T) one. The positive magnetoconductance originates in the enlargement of the localization length due to breaking the global time reversal symmetry. The negative magnetoconductance for the weak RMF results from the shrinking of the wavefunction. Finally, we show current distributions both in the weak RMF (BR = 0.1 T) and the strong RMF (BR = 5.0 T) in Fig. 4. In the case of the weak RMF, electrons are not scattered so much by the RMF, while for the strong RMF the current flows along the BR (x, y) = 0 contour with strong scattering from the magnetic field.
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Fig. 3. Magnetoconductance of 2D electrons in RMF’s. The ordinate represents the conductance change 0G ≡ G(B) − G(B = 0) rescaled by G(B = 0).
Fig. 4. Current density distributions for weak and strong RMF’s. For both figures, the incident electron energy is set to be 6.0 meV and BU = 0. Gray scale backgrounds represent B 2 (x, y) (dark portions correspond to strong fields) and arrows indicate current flows.
4. Conclusions We have shown that the transfer-matrix method is quite efficient for calculating conductances of 2D electron systems in the presence of inhomogeneous magnetic fields. The computational effort is essentially the same with those for one-dimensional transfer-matrix calculations, because the present method does not involve any diagonalization procedure. Current distributions in inhomogeneous magnetic fields are also computed by utilizing the transfer matrix. We applied this method to a 2D electron system subject to a periodically modulated magnetic field. We have obtained the oscillating magnetoconductance which agrees well with the theoretical prediction. It has been also found that there exist two types of electron transport in RMF’s. For a weak RMF, the current is weakly scattered from the RMF, which leads the Drude type conductance, the conductance larger than 2e2/ h, and the negative magnetoconductance. On the contrary, a strong RMF leads the weak-localization effect, the conductance much smaller than 2e2 / h, and the positive magnetoconductance.
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Finally, we mention computing times of present calculations. To obtain Fig. 1, it takes 9.5 hours by using Hitach SR8000 supercomputer. In this calculation, the finite-temperature conductance at a given uniform magnetic field requires zero-temperature conductances at 100 energy points around EF . The computing time for Fig. 2 with 500 energy points is 131 s by the same computer. To obtain a single line (with 100 data points) in Fig. 3, the computing time is 43 min. The current distribution (Fig. 4) is calculated by Sun Workstation (Ultra 80/300 MHz) within 224 s. As seen from the above computing times, we can make very fast calculation of the conductance of 2D electron system in magnetic fields with complex structures.
Acknowledgement The author is grateful for valuable discussion with T. Nakayama. Numerical calculations were partially performed on the facilities of Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
References [1] R. Yagi, Y. Iye, J. Phys. Soc. Jpn. 62 (1993) 1279. [2] L. Solimany, B. Kramer, Solid State Commun. 96 (1995) 471; H.-S. Sim, K.-H. Ahn, K.J. Chang, G. Ihm, N. Kim, S.J. Lee, Phys. Rev. Lett. 80 (1998) 1501. [3] A.K. Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov, F.M. Peeters, Nature 390 (1997) 259; J. Reijniers, F.M. Peeters, A. Matulis, Phys. Rev. B 59 (1999) 2817. [4] K. Geim, S.J. Bending, I.V. Grigorieva, Phys. Rev. Lett. 69 (1992) 2252. [5] F.B. Mancoff, R.M. Clarke, C.M. Marcus, S.C. Zhang, K. Campman, A.C. Gossard, Phys. Rev. B 51 (1995) 13 269. [6] T. Sugiyama, N. Nagaosa, Phys. Rev. Lett. 70 (1993) 1980. [7] K. Yakubo, Y. Goto, Phys. Rev. B 54 (1996) 13 432. [8] C. Xie, X.R. Wang, D.Z. Liu, Phys. Rev. Lett. 80 (1998) 3563. [9] D. Taras-Semchuk, K.B. Efetov, Phys. Rev. Lett. 85 (2000) 1060. [10] B.I. Halperin, P.A. Lee, N. Read, Phys. Rev. B 47 (1993) 7312. [11] M. Büttiker, Y. Imry, R. Landauer, S. Pinhas, Phys. Rev. B. 31 (1985) 6207.
Transfer-matrix approach to electron transport in inhomogeneous magnetic fields K. Yakubo Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Abstract A transfer-matrix method to study quantum transport of two-dimensional (2D) electron systems subject to inhomogeneous magnetic fields has been developed. This method enables us to calculate conductance within the same order of a computing time with those of one-dimensional transfer-matrix calculations. This is because the present method does not involve any diagonalization procedure, while a conventional transfer-matrix technique for 2D systems requires to diagonalize the Hamiltonian for thin strips in the system, which consumes a long computing time. Using this method, we calculate conductances of 2D electrons in periodically modulated and random magnetic fields. 2001 Elsevier Science B.V. All rights reserved. PACS: 73.50.Jt; 73.23.Ps; 72.15.Rn; 02.70.-c
1. Introduction Spatially modulated magnetic fields lead novel quantum transport of two-dimensional (2D) electrons. For example, the magnetoconductance of 2D electrons in a periodically modulated magnetic field shows oscillatory behavior [1]. Recently, various alternative magnetic structures have been proposed on 2D electron systems, such as magnetic quantum dots or magnetic antidots [2,3] and random magnetic fields [4–9]. In particular, transport properties of 2D electrons in a random magnetic field (RMF) are crucial for understanding magnetotransport around a half-filled Landau level [10]. It has been extensively investigated whether extended states exist in the 2D system subject to the RMF [6–9]. However, very little numerical work on transport properties was reported so far. Numerical studies of 2D electrons are very few also for other inhomogeneous magnetic fields. This is due to the lack of a numerical method efficient for investigating quantum transport of 2D electrons in inhomogeneous magnetic fields. In this paper, we show that a transfer-matrix method is quite effective and useful to calculate conductances of 2D noninteracting electron systems in the presence of inhomogeneous magnetic fields. In a conventional transfermatrix method for a 2D system, we have to diagonalize numerically Hamiltonians for thin strips into which the system is divided in the transport direction. This procedure usually takes a long computing time. Therefore, the transfer-matrix method is not suitable for two- or three-dimensional problems. In the present transfer-matrix method, however, analytical expressions for eigenvectors of Hamiltonians are used to construct transfer matrices, E-mail address: [email protected] (K. Yakubo). 0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 1 ) 0 0 3 8 3 - 6
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K. Yakubo / Computer Physics Communications 142 (2001) 429–435
and it is not necessary to diagonalize Hamiltonians for thin strips numerically. As a consequence, we can calculate conductances within the same order of a computing time with those of one-dimensional transfer-matrix calculations, and magnetotransport of 2D electrons in complex magnetic structures can be treated numerically. Furthermore, current distributions in inhomogeneous magnetic fields are also calculated by using the transfer matrix. The paper is organized as follows: In Section 2 the transfer-matrix method for 2D inhomogeneous magnetic field systems is formulated. Some applications are presented in Section 3. Section 4 is devoted to the conclusions.
2. Transfer-matrix formalism Let us consider a 2D electron system in the presence of an inhomogeneous magnetic field B(r) = (0, 0, B(x, y)). There exist no other electrostatic potentials in the system. In the Landau gauge, the vector potential is given by Ax = Az = 0 and x Ay (x, y) =
B(x , y) dx .
(1)
0
Hereafter, we define the x direction as the transport direction. In the transfer-matrix method, the system is divided into M thin strips in the x direction. The width of strips is so small that Ay (x, y) can be regarded as a constant with respect to x in each strip. The Schödinger equation for the lth strip is then given by 2 2 1 ∂ 2 ∂ − eAy (xl , y) + − ih¯ (2) − h¯ ψl (x, y) = Eψl (x, y), 2m∗ ∂x 2 ∂y where m∗ is the electron effective mass and E is the electron incident energy. We find the solution of Eq. (2) in the form of ψl (x, y) = e±ikx x ϕl (y).
(3)
The function ϕl (y) should satisfy the equation: 2 1 ∂ − eAy (xl , y) + ε ϕl (y) = 0, −ih¯ 2m∗ ∂y where ε = E − (h¯ 2 kx2 /2m∗ ). Eq. (4) can be solved analytically for arbitrary Ay (x, y) as y iky y ie −iky y exp ϕl (y) = αe + βe Ay (xl , y ) dy , h¯
(4)
(5)
0
√ where ky = 2m∗ ε/h¯ . Assuming the periodic boundary conditions in the y-direction, that is, ϕl (y + Ly ) = ϕl (y) and B(x, y + Ly ) = B(x, Ly ) with the system size Ly in the y-direction, we have y ν y ie ikyl ϕl (y) = γ e (6) exp Ay (xl , y ) dy , h¯ 0
where Ly e 1 ν kyl = 2νπ − Ay (xl , y) dy , Ly h¯ 0
(7)
K. Yakubo / Computer Physics Communications 142 (2001) 429–435
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ν = 0, ±1, ±2, . . . , and γ is a constant. Therefore, from Eqs. (1) and (3), the general solution ψl (x, y) is written as
ν ν ν ν ψl (x, y) = fl (y) Aνl ei(kxl x+kyl y) + Blν e−i(kxl x−kyl y) , (8) ν = where kxl
ν ν )2 /h and 2m∗ E − (h¯ kyl ¯
ie fl (y) = exp h¯
yxl
B(x , y ) dx dy .
(9)
0 0
ν In order to relate the coefficients Aνl and Blν in Eq. (8) with Aνl+1 and Bl+1 , let us consider the matching conditions ψl (xl+1 , y) = ψl+1 (xl+1 , y) and ∂x ψl (xl+1 , y) = ∂x ψl+1 (xl+1 , y). Substituting Eq. (8) into the first ∗ (y)e matching condition, multiplying fl+1
Aνl+1 e
ν x ikxl+1 l+1
ν + Bl+1 e
ν x −ikxl+1 l+1
=
ν −ikyl+1 y
, and integrating over y within the range of [0, Ly ], we have
ν ν Clνν Aνl eikxl xl+1 + Blν e−ikxl xl+1 ,
(10)
dy.
(11)
ν
where νν
Cl
1 = Ly
Ly
∗ fl+1 (y)fl (y)e
ν −k ν i(kyl yl+1 )y
0 Clνν
The quantity represents the channel mixing between the νth and ν th channels. From the relation νν 2 νν 2 ν |Cl | = 1, |Cl | can be regarded as the probability to switch the channel ν to ν at the lth strip. In the similar way, from the second matching condition, one obtains ν ik ν x ν νν ν ik ν x ν x ν ν ν −ikxl+1 l+1 Al+1 e xl+1 l+1 − Bl+1 = (12) e kxl Cl Al e xl l+1 − Blν e−ikxl xl+1 . kxl+1 ν
Solving the coupled equations [Eqs. (10) and (12)] and cutting off higher irrelevant channels (|ν| > νc ), we finally obtain the matrix form expression: D l+1 = T(l)D l ,
(13)
c −1 if integer λ where D l is the 2Nc -dimensional coefficient vector whose components are given by Dlλ = Aλ−ν l λ−Nc −νc −1 λ is bounded by 1 λ Nc and Dl = Bl if Nc + 1 λ 2Nc , where Nc = 2νc + 1 is the number of (l) has its components given by channels. The 2Nc × 2Nc matrix T
T
λλ
µ
(kxl+1 + mm kxl ) µ
(l) =
µ 2kxl+1
µ µ µµ i(mkxl+1 −m kxl )xl+1
Cl
e
,
(14)
where µ = {(λ − 1) mod Nc } + 1, m = 2[(λ − 1)/Nc ] − 1, and [x] denotes the largest integer less than or equal µ µ µµ ν , k ν , and C νν with ν = µ − ν − 1, respectively. From to x. Quantities kxl , kyl , and Cl are the same with kxl c yl l Eq. (13), we can calculate D M from D 0 by TM = M T (l). The conductance G is calculated by the Landauer l formula [11]. We can obtain the current distribution from the calculated transmission and reflection coefficients AνM and B0ν . Using again Eq. (13), one obtains coefficient vector D l for all l, and calculates ψl (x, y) from Eq. (8). From the definition of the quantum mechanical current density, j = (ih¯ /2m∗ )[ψl (x, y)∇ψl∗ (x, y) − ψl∗ (x, y)∇ψl (x, y)] − (e/m∗ )A|ψl (x, y)|2 , the spatial distribution of the current density can be computed.
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3. Applications We have applied the present transfer-matrix method to two kinds of 2D electron systems in inhomogeneous magnetic fields. As the first application, we compute the conductance of a 2D system subject to a periodically modulated magnetic field (the modulation is only in the transport direction), namely, B(x, y) = BU + BW cos(qx). Yagi and Iye [1] have found that the magnetoconductance (conductance as a function of BU ) shows an oscillatory behavior with conductance maxima at √ 2h¯ 2πne BU = (n = 1, 2, . . .), (15) ed(n − 1/4) where ne is the electron density, and d is the period of the modulated magnetic field (d = 2π/q). In order to check our numerical method, we confirm this magnetoconductance behavior. The amplitude and the period of the oscillating magnetic field are set to be BW = 0.5 T and d = 40 nm, respectively. The Fermi energy EF of the electron system is fixed at 6.0 meV and the electron effective mass m∗ is chosen as 0.067me , where me is the electron bare mass. The system size Ly is 0.5 µm for which the number of channels Nc for EF = 6.0 meV is 17. The length Lx of the system size in the x-direction varies with BU . If Lx is fixed, the conductance rapidly decreases as increasing BU , because the effective one-dimensional potential barrier (Veff = e2 BU2 x 2 /2m∗ ) due to the uniform magnetic field becomes higher. In order to concentrate the oscillatory behavior of the conductance, the value of Lx is determined so that the transmission probability for Veff becomes constant for any BU . The calculated conductance at low temperature (T = 0.2 K) is shown in Fig. 1. One can find the clear oscillatory behavior of the conductance. Peak positions at lower uniform fields agree well with the theoretical prediction Eq. (15). For BU 0.6 T, peak positions are slightly shifted from the theoretical values. These shifts are caused by the finite Lx in which the number of periods of the oscillating magnetic field is restricted (only 4 periods are included in the system for BU = 1.0 T). It should be noted that peaks labeled by dashed arrows do not correspond to peaks given by Eq. (15), but the Shubnikov–de Haas peaks given by BU = π hn ¯ e /e(n + 1/2). These results show that the transfer-matrix method works well. The second application is quantum transport of 2D electrons in a long-range random magnetic field (RMF). As mentioned in Section 1, 2D transport in the RMF are attracting strong interest in recent years. We examine
Fig. 1. Magnetoconductance of 2D electrons in a periodically modulated magnetic field.
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Fig. 2. Incident electron energy dependences of the conductance of 2D electrons in RMF’s.
the conductance of this system by using the transfer-matrix method. The magnetic field is given by B(x, y) = BU + BR (x, y), where BR (x, y) =
NB
i=1
(x − xi )2 + (y − yi )2 , Bi exp − ξ2
(16)
ξ is 20 nm, Bi is uniformly distributed within [−BR0 , BR0 ], and NB = 200. Centers of Gaussian magnetic fields (xi , yi ) are randomly chosen in the system. The magnitude of fluctuations in BR (x, y) is characterized by 2 BR ≡ BR (x, y) dx dy/S, where S is the area of the system. In this case, both sizes of the system in the x and y directions are fixed at 200 nm. Fig. 2 shows incident energy dependences of conductances at zero temperature. The uniform magnetic field BU is equal to zero. Values of BR are 0.5 and 5.0 T for solid and dotted lines, respectively. The conductance for the weak RMF behaves as in the case of the quantum point contact system in which the conductance is quantized due to varying the number of transport channels. This means that electrons are weakly scattered from the RMF. In this case, the conductance of this system is larger than 2e2 / h, which implies that single channel transport can be described by a semiclassical theory such as the Drude theory. On the contrary, for the strong RMF, the conductance is much smaller than 2e2/ h because of the weak localization effect. In such case, the quantum interference effect is crucial for electron transport. In fact, the conductance exhibits many resonant peaks by the quantum interference. We also calculate magnetoconductance for the BR = 5.0 T case. Fig. 3 shows the conductance change as a function of BU . The electron incident energy is fixed at 6.0 meV. The magnetoconductance is positive for the strong RMF system, while it is negative for the weak RMF (BR = 0.5 T) one. The positive magnetoconductance originates in the enlargement of the localization length due to breaking the global time reversal symmetry. The negative magnetoconductance for the weak RMF results from the shrinking of the wavefunction. Finally, we show current distributions both in the weak RMF (BR = 0.1 T) and the strong RMF (BR = 5.0 T) in Fig. 4. In the case of the weak RMF, electrons are not scattered so much by the RMF, while for the strong RMF the current flows along the BR (x, y) = 0 contour with strong scattering from the magnetic field.
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Fig. 3. Magnetoconductance of 2D electrons in RMF’s. The ordinate represents the conductance change 0G ≡ G(B) − G(B = 0) rescaled by G(B = 0).
Fig. 4. Current density distributions for weak and strong RMF’s. For both figures, the incident electron energy is set to be 6.0 meV and BU = 0. Gray scale backgrounds represent B 2 (x, y) (dark portions correspond to strong fields) and arrows indicate current flows.
4. Conclusions We have shown that the transfer-matrix method is quite efficient for calculating conductances of 2D electron systems in the presence of inhomogeneous magnetic fields. The computational effort is essentially the same with those for one-dimensional transfer-matrix calculations, because the present method does not involve any diagonalization procedure. Current distributions in inhomogeneous magnetic fields are also computed by utilizing the transfer matrix. We applied this method to a 2D electron system subject to a periodically modulated magnetic field. We have obtained the oscillating magnetoconductance which agrees well with the theoretical prediction. It has been also found that there exist two types of electron transport in RMF’s. For a weak RMF, the current is weakly scattered from the RMF, which leads the Drude type conductance, the conductance larger than 2e2/ h, and the negative magnetoconductance. On the contrary, a strong RMF leads the weak-localization effect, the conductance much smaller than 2e2 / h, and the positive magnetoconductance.
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Finally, we mention computing times of present calculations. To obtain Fig. 1, it takes 9.5 hours by using Hitach SR8000 supercomputer. In this calculation, the finite-temperature conductance at a given uniform magnetic field requires zero-temperature conductances at 100 energy points around EF . The computing time for Fig. 2 with 500 energy points is 131 s by the same computer. To obtain a single line (with 100 data points) in Fig. 3, the computing time is 43 min. The current distribution (Fig. 4) is calculated by Sun Workstation (Ultra 80/300 MHz) within 224 s. As seen from the above computing times, we can make very fast calculation of the conductance of 2D electron system in magnetic fields with complex structures.
Acknowledgement The author is grateful for valuable discussion with T. Nakayama. Numerical calculations were partially performed on the facilities of Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
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