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Correlations between electronic energy bands spin splitting and magnetic profile symmetry of magnetic superlattice
ARTICLE IN PRESS
Physica E 40 (2008) 2959–2964 www.elsevier.com/locate/physe
Correlations between electronic energy bands spin splitting and magnetic profile symmetry of magnetic superlattice Huaizhe Xua,b, a
Department of Physics, Beihang University, P.O. Box 912, Beijing 100083, PR China Key Laboratory of Semiconductor Materials Science, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, P. R. China
b
Received 6 February 2008; accepted 21 February 2008 Available online 12 March 2008
Abstract We have theoretically investigated the energy band structures of two typical magnetic superlattices formed by perpendicular or parallel magnetization ferromagnetic stripes periodically deposited on a two-dimensional electron gas (2DEG), where the magnetic profile in the perpendicular magnetization is of inversion anti-symmetry, but of inversion symmetry in parallel magnetization, respectively. We have shown that the energy bands of perpendicular magnetization display the spin-splitting and transverse wave-vector symmetry, while the energy bands of the parallel magnetization exhibit spin degeneration and transverse wave-vector asymmetry. These distinguishing spindependent and transverse wave-vector asymmetry features are essential for future spintronics devices applications. r 2008 Elsevier B.V. All rights reserved. PACS: 73.21.Cd; 72.25.Dc; 85.30.De Keywords: Energy dispersion relation; Magnetic superlattice; Spin splitting; Electronic band; Inversion anti-symmetry
1. Introduction The breakthroughs over the last few years in understanding and controlling the spin-related effects have set the stage for a period of explosive growth in research and technological exploitation of spintronics [1–3], where both spin and charge of carriers (electrons or holes) are exploited to provide new functionalities for microelectronic devices. The expected advantages of spin devices include nonvolatility, higher integration densities, lower power operation and higher switching speeds. However, it is found that the injection of spin-polarized electrons from ferromagnetic metals to semiconductor was significantly hindered due to the large resistance mismatch [2,3], which has stimulated the strong interest in exploring highefficiency spin injection by using hybrid magnetic metal/ semiconductor devices [4–7], where the strong interaction of the intrinsic spin with magnetic field may lead to spin Tel./fax: +86 010 82317935.
E-mail address: [email protected] 1386-9477/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.02.016
filtering via the giant Zeeman effect. Theoretical efforts on 2DEG in an inhomogeneous external magnetic field modulation, such as momentum-dependent tunneling through a magnetic barrier or electron transport in a weakly spatially modulated magnetic field, have been carried out [8–13]. The spin filtering effect has been predicted in devices realized by depositing magnetic stripes on the two-dimensional electron gas (2DEG). The hybrid structures consisting of patterned ferromagnetic metals on the top of semiconductor have appeared as a possible route for future information-processing devices. The growing interest in the magneto-conductance properties of the 2DEG in spatially periodic lateral magnetic fields has been further stimulated by the recent experimental availability of such systems [14–21]. The spin splitting in magnetic superlattice attracts our attention not only because of its importance for understanding of the fundamental properties of hybrid ferromagnetic metal/2DEG systems, but also because of its possible applications in the field of spintronics [14–21]. Motivated by the wealth of new and interesting behavior of
ARTICLE IN PRESS 2960
H. Xu / Physica E 40 (2008) 2959–2964
a 2DEG in a spatially periodic lateral magnetic field, we have investigated the electronic band structure in two typical magnetic superlattices, those created by perpendicular or parallel magnetization ferromagnetic stripes periodically deposited on a 2DEG heterojunction. The spin splitting or degeneracy of energy bands has been revealed and correlated to the spatial anti-inversion symmetry in perpendicular magnetization or to the inversion symmetry magnetic-field modulation in parallel magnetization. Finally, we have pointed out that the dispersion relation for spin electrons in the magnetic superlattice of parallel magnetization given in Ref. [12] is incorrect, which can be traced back to a calculation mistake. The striking spin-dependent features are indispensable for the operations of spin-related devices. Thus, we believe the results here will be useful for future designing and developing of spin-filters or spin-dependent tunneling electron devices.
2. Theoretical calculations We consider an electron moving in a two-dimensional (2D) plane (x,y) in the presence of perpendicular magnetic field (along the z-direction). Following the procedure of Refs. [10,11], the magnetic field is taken homogeneous along the y-axis and varied along the x-axis. Such a ~ is described in the Landau gauge ~¼ rA magnetic field B ~ ¼ ½0; AðxÞ; 0. In the singleby the vector potential A particle approximation, the Hamiltonian describing such a system can be written as [21]
pffiffiffiffiffiffiffiffiffiffiffiffiffi mass, and the magnetic length l B ¼ _=eB0 , B0 being some typical magnetic field. We express all the relevant quantities in dimensionless units: (1) the magnetic field ~ ! ~z ðxÞ ! B0 B ~z ðxÞ, (2) the vector potential AðxÞ B ~ (3) the coordinate ~ r ! l B~ r, and (4) the energy B0 l B AðxÞ, E-_ocE. For GaAs and an estimated B0 ¼ 0.1 T, we have lB ¼ 813 A˚ and _oc ¼ 0.17 meV. Now the Schro¨dinger equation is of the dimensionless form 2 d þ 2E 2V ðx; k ; sÞ cðxÞ ¼ 0 (4) y dx2 with V ðx; ky ; sÞ ¼ ½AðxÞ þ ky 2 =2 þ mn gn sBz ðxÞ=4m0
The V(x,ky,s) term is normally interpreted as effective potential. It depends not only on the electron wave vector for motion in the y-direction, but also on the magnetic configuration, as well as on the interaction between the electron spin and the non-homogeneous magnetic field. When assuming the B(x) profiles to be periodic in x-axis, and the average magnetic-field strength of the magneticfield modulation to be zero, i.e., /B(x)S ¼ 0, the resulting effective potential will be periodic with period l, i.e., V(x,ky) ¼ V(x+nl,ky). Therefore, we need only to find out the solution in one unit period, then use Bloch’s theorem to propagate this solution throughout the lattice. The total wave function for the 2D electrons, up to a normalization factor, is written as ~
n 1 ~ ~ 2 þ eg s_ Bz ðxÞ H¼ ½P þ eAðxÞ n 2m 2m0 2
Cn;k ð~ rÞ ¼ eik~r cn;k ðxÞ (1)
where m* is the effective mass of electron, m0 is the free~ is the momentum of electron, electron mass in vacuum, P g* is the effective g-factor of electron in a real 2DEG, s ¼ +1/1 for spin-up/down electrons. Since the y-component of the free-electron momentum operator commutes with the Hamiltonian, i.e., [H,Py] ¼ 0, the problem is translational invariant along the y-direction and the corresponding wave vector ky is a conserved quantity. Therefore, the wave function can be written as a product as Cðx; yÞ ¼ eiky y cðxÞ
(2)
where _ky is the expectation value of py in y-direction. The wave function c(x) satisfies the one-dimensional (1D) Schro¨dinger equation: 2 he i2 mn gn sB ðxÞ 2mn E d z AðxÞ þ k þ 2 cðxÞ ¼ 0 y _ 2_m0 dx2 _ (3) For convenience, we introduce the electron cyclonic frequency oc ¼ eB0 =mn , where mn is the electron effective
(5)
(6)
where ~ r ¼ ðx; yÞ, k~ ¼ ðk; ky Þ, cn,k(x) is the Bloch function, which can be expressed in units of lB. It is noted that increasing the period l at constant B0 is equivalent to increasing the strength of the magnetic field B0 for a fixed period l.
3. Results and discussion 3.1. Perpendicular magnetization We consider a periodic magnetic field realized by means of metallic ferromagnetic stripes deposited on the top of a 2DEG heterostructure. First, we assume the ferromagnetic strips are perpendicularly magnetized; thus we obtain a magnetic superlattice created by the inhomogeneous magnetic field in the 2DEG [11,19–21]; its magnetic profile is schematically illustrated as in Fig. 1. The magnetic field has been simplified as a square barrier or well as in Refs. [19–21]; the dotted line represents the magnetic vector potential. It is seen that the magnetic field modulation is of inversion anti-symmetry feature, i.e., the magnetic field is symmetric but the vector potential is anti-symmetric at the middle points of each magnetic barrier or well.
ARTICLE IN PRESS H. Xu / Physica E 40 (2008) 2959–2964
In each region the xi0 and li are expressed as 8 ð0pxpb=4Þ ky =B > < i x0 ¼ b=2 ky =B ðb=4pxp3b=4Þ > : b þ k =B ð3b=4pxpbÞ
z b/2
2961
b/2
(12)
y
B
b/4
w
x
Fig. 1. Schematic illustration of the ferromagnetic metal stripes/semiconductor 2DEG structure, the model magnetic field profile B(x) (solid line) and vector potential A(x) (dotted line) of the magnetic superlattice realized by perpendicular magnetization.
The magnetic field experienced by the 2DEG is given in Refs. [5,16]: 8 B ð0pxob=4Þ > > > > < þB ðb=4xp3b=4Þ (7) BðxÞ ¼ B ð3b=4oxpbÞ > > > > :0 ðboxpb þ wÞ with 8 Bx > > > > < Bb=2 þ bx AðxÞ ¼ Bb Bx > > > > :0
ð0oxpb=4Þ ðb=4oxp3b=4Þ ð3b=4oxpbÞ ð0oxpb þ wÞ
(8)
ci ðxÞ ¼ C i cU 0 ðxÞ þ Di cU 1 ðxÞ With cU 0 ðxÞ ¼ expðx2i =2ÞU 0 ðxi Þ (9)
Here Ci and Di are constants to be determined from the boundary conditions. U0(xi) and U1(xi) are the two Hermitian functions [22,23]: U 0 ðxi Þ ¼ 1 þ
2 li 3 ð2 li Þð6 li Þ 5 xi þ xi þ 5! 3! ð2 li Þð6 li Þ ð4n 2 li Þ 2nþ1 xi þ þ ð2n þ 1Þ!
By imposing the continuity conditions of the wave function and its first derivative, as well as the Bloch periodicity on the wave function at the edges of the unit cells, we can obtain the dispersion relation between the electron energy E and its wave vector k~ ¼ ðk; kx Þ; it can be written in a very concise form using the transfer matrix method as [24] CosðklÞ ¼ Tr S=2
(14)
Here l is the period length, Tr S is the trace of the total 2 2 matrix S, which only needs to be calculated in one period. It has the following form: ðx ¼ 3b=4ÞS ðx ¼ 3b=4ÞS 1 ðx ¼ bÞS w "
The Schro¨dinger equation (4) in the magnetic barrier region (B(x)6¼0) can be solved analytically, and the wave function ci(x) can be written as [22,23]
li 2 li ð4 li Þ 4 xi þ x þ 4! 2! i li ð4 li Þ ð4n 4 li Þ 2n xi þ þ ð2nÞ!
(13)
1 S ¼ S ðx ¼ 0ÞS 1 ðx ¼ b=4ÞS þ ðx ¼ b=4ÞS þ
Within one period, the wave functions in the magnetic well region (B(x)=0) are the free-electron wave functions; it can be written as cw ðx; yÞ ¼ eiky y ðC w eikw x þ Dw eikw x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with kw ¼ 2E k2y .
cU 1 ðxÞ ¼ expðx2i =2ÞU 1 ðxi Þ pffiffiffiffi xi ¼ Bðx xi0 Þ
and 8 n n ð0pxpb=4Þ > < 2E=B þ m g s=2m0 1 n n li ¼ 2E=B m g s=2m0 1 ðb=4pxp3b=4Þ > : 2E=B þ mn gn s=2m 1 ð3b=4pxpbÞ 0
(10)
U 1 ðxi Þ ¼ xi þ
(11)
Sw ¼
Sin kðkw w wÞ Cos ðkw wÞ kw Sinðkw wÞ Cos ðkw wÞ 2
cU 0 ðxi Þ
cU 1 ðxi Þ
dx
dcU 1 ðxi Þ dx
S ðxi Þ ¼ 4 dcU 0 ðxi Þ
(15)
# (16)
3 5
(17)
The subscript signs () and (+) are referred to the B regions or the +B regions, respectively. Fig. 2 shows the energy minibands of a model magnetic superlattice created by perpendicular magnetization ferromagnetic stripes, which are periodically deposited on an InSb 2DEG. We have assumed g* ¼ 51, m* ¼ 0.014m0, b ¼ 2, B ¼ 10, w ¼ 1 in numerical calculations. The spin splitting resulting from the interaction between the spin and the magnetic field is clearly seen, the electronic minibands between the spin-up and spin-down do not superimpose due to the periodic magnetic field modulation. Fig. 3 shows the electron energy E as a function of wave vector ky. The widths of the spin-dependent bands and the energy gaps strongly depend on the transverse wave vector ky. It is seen that all the minibands become narrow as |ky| is increasing. The strong wave-vector filtering feature and the spin-dependent energy of this magnetic superlattice make it a good candidate for the spintronics applications. For example, a spin filter of 100% based on spin-dependent tunneling has already been demonstrated in structures consisting of single or double magnetic barriers [19–21]. It is noted that the energy bands are symmetric for ky-ky, which is a result of the symmetrical effective potential
ARTICLE IN PRESS H. Xu / Physica E 40 (2008) 2959–2964
2962
12
z
10 c
b
c
x
Energy (E/E0)
8 -B
6
Fig. 4. Schematic illustration of the ferromagnetic metal stripes/semiconductor 2DEG structure, and the model magnetic field profile B(x) (solid line) and vector potential A(x) (dotted line) of the magnetic superlattice realized by parallel magnetization.
4
2
0 -1.0
-0.5
0.0 k (1/lB)
0.5
1.0
Fig. 2. The energy dispersion curves for the InSb system with w ¼ 1; solid lines are for spin-up electrons, and dashed lines are for spin-down electrons, respectively.
d-functions with alternating sign arranged along the x-axis [14,15]; the dotted line represents the magnetic vector potential. The magnetic field modulation is of inversion symmetry, i.e., the magnetic field is anti-symmetric and the vector potential is symmetric at the middle of each magnetic barrier or well: BzðxÞ ¼ Bl B
þ1 X
½dðx c þ anÞ dðx þ anÞ
(18)
1
12
For such a magnetic field the vector potential is a step ~ ¼ ð0; Bl B ðxÞ; 0Þ, where function and can be taken as A ( 1=2 ð0oxocÞ ðxÞ ¼ (19) 1=2 ðcoxoaÞ
10
Energy (E/E0)
8
This magnetic profile is similar to the frequently used magnetic Kronig–Penney model, except that we also assumed c6¼b here for generality and for convenience to compare with Ref. [12]. Obviously, the effective potential of this model is step like, so that the solutions of the Schro¨dinger equation for this potential are plane waves in both the well and the barrier regions. Similarly, the dispersion relation can be written as [24]
6
4
2
CosðkaÞ ¼ Tr S=2
0 -4
-2
0 ky
2
4 -4
-2
0 ky
2
4
Fig. 3. The dependence of band energy on the transverse wave vector ky; (a) for spin-up and (b) for spin-down, respectively.
V(ky,x). This result implies that there is no in-plane conductance (or Hall current along the y-direction) when electrons transversally go through this superlattice. This structure cannot be used for the aim of exploring the spin Hall Effect in 2DEG.
Here a is the period length, Tr S is the trace of the total 2 2 matrix S, which only needs to be calculated in one period. Here the matrix S becomes S ¼ S þd Sc S d S b with d ¼
c¼
3.2. Parallel magnetization b¼ Second, we consider the magnetic superlattice created by parallel magnetization of ferromagnetic stripes, which are periodically deposited on a 2DEG heterojunction [9–18]. The magnetic profile is schematically illustrated in Fig. 4. The magnetic field profile has been simplified by a series of
(20)
a¼
1
0
g
1
(21)
!
cos ðacÞ a sin ðacÞ
1a sin ðacÞ cos ðacÞ
!
CoshðbbÞ
b1 SinhðbbÞ
b SinhðbbÞ
CoshðbbÞ
(22) ! (23)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2E ðkx B=2Þ2 ; and b ¼ ðkx þ B=2Þ2 2E ,
g ¼ g sBm =2m0 .
(24)
ARTICLE IN PRESS H. Xu / Physica E 40 (2008) 2959–2964
After some straightforward calculations, we obtained the analytical dispersion relation as a 2 þ g 2 b2 cosðkaÞ ¼ cos hðbbÞ cos ðacÞ sin hðbbÞ sinðacÞ 2ab (25) It can be seen clearly from Eq. (25) that the energy band of the magnetic superlattice realized by parallel magnetization is spin-degeneracy. Physically, it can be attributed to the inversion symmetry of the magnetic field profile. This spin-degeneracy feature explains why there is no spin polarization as the tunneling electron goes through such a kind of magnetic superlattice of finite periods [15]. It is not difficult to find out the difference between Eq. (25) and Eq. (13) of Ref. [12], such as the sign in front of g2 at the numerator and the two extra terms in the right side of Eq. (13); thus, we consider that Eq. (13) of Ref. [12] is incorrect. It can be easily demonstrated that Eq. (13) of Ref. [12] was deduced P from a mistaken magnetic profile of form zðxÞ ¼ Bl B þ1 1 ½dðx c þ anÞ þ dðx þ anÞ, or zðxÞ ¼ Bl B Sþ1 1 ½dðx c þ anÞ þ dðx þ anÞ, and the vector potential of form Eq. (19). It is really surprising that similar mistakes had also been found in Refs. [17,18], while the authors were calculating the electron transmission probability in the simplest magnetic profile consisting of one piece of magnetic strip, and thus only a pair of positive and negative d-functions magnetic field. Fig. 5 shows the electron energy E as a function of the wave vector ky numerically calculated for a model of B ¼ 1, c ¼ 3, b ¼ 6 and g* ¼ 2.0, m* ¼ m0. We have used the same parameters as that of Ref. [12]. It is seen that the energy band bottoms shift away from ky ¼ 0 due to including the spin–magnetic field interaction and assuming c6¼b, which are much different from the case for c ¼ b, as 4
Energy
3
2
2963
well as for spinless electrons [9,10,12]. Furthermore, the widths of the electron energy bands and the band gaps depend strongly on ky, and are asymmetric for ky-ky when c6¼b. In particular, the electrons with positive ky are found in a higher-energy state than those with ky (negative) within the same energy miniband. It means electrons moving in opposite y-directions no longer have the same energy. As |ky| increases, the bands become narrower and consequently the corresponding states are more localized in the y-direction. We noted that the dispersion relation for E versus ky for c=b is symmetric for ky-ky, and there is no band gap at ky=0. Therefore, we attributed this asymmetry to the broken symmetry of the effective potential V(ky,x) with respect to ky for c6¼b, stemming from the magnetic profile being no longer invariant under spatial inversion at each d-function potential. The ky-ky asymmetry will result in an inplane conductance (or Hall current along the y-direction) when electrons transport through these kinds of superlattices. 4. Conclusion We have calculated analytically and numerically the energy bands for electron in two-magnetic superlattices realized by perpendicular or parallel magnetization ferromagnetic stripes periodically deposited on a 2DEG heterojunction. We have shown that the spin splitting of the energy bands appears in magnetic superlattice realized by perpendicular magnetization ferromagnetic stripes, while the magnetic superlattice realized by parallel magnetization shows only the spin degeneration. These distinguishing differences are related to the inversion antisymmetry in the perpendicular magnetization and the inversion symmetry in parallel magnetization. Finally, we have pointed out that the dispersion relation for spin electrons in the magnetic superlattice realized by parallel magnetization given in Ref. [12] is incorrect, which can be traced back to a calculation mistake. We believe that the structural symmetry-related spin splitting features in these two magnetic superlattices are useful for designing and developing spin filters or spin-dependent tunneling electron devices. Finally, our rigorous calculation is based on the transfer matrix approach, which can be easily extended to more complex structures. Acknowledgement The author would like to thank the financial support from National Nature Science Foundation of China under Grant No. 60776067.
1
0
References -2
-1
0 ky
1
2
Fig. 5. The band energy dependence on the transverse wave vector ky.
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[2] H.J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H.-P. Scho¨nherr, K.H. Ploog, Phys. Rev. Lett. 87 (2001) 016601. [3] J. De Boeck, W. Van Roy, J. Das, V. Motsnyi, Z. Liu, L. Lagae, H. Boeve, K. Dessein, G. Borghs, Semicond. Sci. Technol. 17 (2002) 342. [4] P.D. Ye, D. Weiss, R.R. Gerhardts, M. Seeger, K. von Klitzing, K. Eberl, H. Nickel, Phys. Rev. Lett. 74 (1995) 3013. [5] W. Van Roy, J. De Boeck, G. Borghs, Appl. Phys. Lett. 61 (1992) 3056. [6] H. Schomig, A. Forchel, S. Halm, G. Bacher, J. Puls, F. Henneberger, Appl. Phys. Lett. 84 (2004) 2826. [7] J. Kossut, I. Yamakawa, A. Nakamura, G. Cywinski, K. Fronc, M. Czeczott, J. Wrobel, F. Kyrychenko, T. Wojtowicz, S. Takeyama, Appl. Phys. Lett. 79 (2001) 1789. [8] J.E. Muller, Phys. Rev. Lett. 68 (1992) 385. [9] I.S. Ibrahim, F.M. Peeters, Am. J. Phys. 63 (1995) 171. [10] I.S. Ibrahim, F.M. Peeters, Phys. Rev. B 52 (1995) 17321. [11] A. Matulis, F.M. Peeters, Phys. Rev. Lett. 72 (1994) 1518. [12] A. Krakovsky, Phys. Rev. B 53 (1996) 8469.
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Physica E 40 (2008) 2959–2964 www.elsevier.com/locate/physe
Correlations between electronic energy bands spin splitting and magnetic profile symmetry of magnetic superlattice Huaizhe Xua,b, a
Department of Physics, Beihang University, P.O. Box 912, Beijing 100083, PR China Key Laboratory of Semiconductor Materials Science, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, P. R. China
b
Received 6 February 2008; accepted 21 February 2008 Available online 12 March 2008
Abstract We have theoretically investigated the energy band structures of two typical magnetic superlattices formed by perpendicular or parallel magnetization ferromagnetic stripes periodically deposited on a two-dimensional electron gas (2DEG), where the magnetic profile in the perpendicular magnetization is of inversion anti-symmetry, but of inversion symmetry in parallel magnetization, respectively. We have shown that the energy bands of perpendicular magnetization display the spin-splitting and transverse wave-vector symmetry, while the energy bands of the parallel magnetization exhibit spin degeneration and transverse wave-vector asymmetry. These distinguishing spindependent and transverse wave-vector asymmetry features are essential for future spintronics devices applications. r 2008 Elsevier B.V. All rights reserved. PACS: 73.21.Cd; 72.25.Dc; 85.30.De Keywords: Energy dispersion relation; Magnetic superlattice; Spin splitting; Electronic band; Inversion anti-symmetry
1. Introduction The breakthroughs over the last few years in understanding and controlling the spin-related effects have set the stage for a period of explosive growth in research and technological exploitation of spintronics [1–3], where both spin and charge of carriers (electrons or holes) are exploited to provide new functionalities for microelectronic devices. The expected advantages of spin devices include nonvolatility, higher integration densities, lower power operation and higher switching speeds. However, it is found that the injection of spin-polarized electrons from ferromagnetic metals to semiconductor was significantly hindered due to the large resistance mismatch [2,3], which has stimulated the strong interest in exploring highefficiency spin injection by using hybrid magnetic metal/ semiconductor devices [4–7], where the strong interaction of the intrinsic spin with magnetic field may lead to spin Tel./fax: +86 010 82317935.
E-mail address: [email protected] 1386-9477/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.02.016
filtering via the giant Zeeman effect. Theoretical efforts on 2DEG in an inhomogeneous external magnetic field modulation, such as momentum-dependent tunneling through a magnetic barrier or electron transport in a weakly spatially modulated magnetic field, have been carried out [8–13]. The spin filtering effect has been predicted in devices realized by depositing magnetic stripes on the two-dimensional electron gas (2DEG). The hybrid structures consisting of patterned ferromagnetic metals on the top of semiconductor have appeared as a possible route for future information-processing devices. The growing interest in the magneto-conductance properties of the 2DEG in spatially periodic lateral magnetic fields has been further stimulated by the recent experimental availability of such systems [14–21]. The spin splitting in magnetic superlattice attracts our attention not only because of its importance for understanding of the fundamental properties of hybrid ferromagnetic metal/2DEG systems, but also because of its possible applications in the field of spintronics [14–21]. Motivated by the wealth of new and interesting behavior of
ARTICLE IN PRESS 2960
H. Xu / Physica E 40 (2008) 2959–2964
a 2DEG in a spatially periodic lateral magnetic field, we have investigated the electronic band structure in two typical magnetic superlattices, those created by perpendicular or parallel magnetization ferromagnetic stripes periodically deposited on a 2DEG heterojunction. The spin splitting or degeneracy of energy bands has been revealed and correlated to the spatial anti-inversion symmetry in perpendicular magnetization or to the inversion symmetry magnetic-field modulation in parallel magnetization. Finally, we have pointed out that the dispersion relation for spin electrons in the magnetic superlattice of parallel magnetization given in Ref. [12] is incorrect, which can be traced back to a calculation mistake. The striking spin-dependent features are indispensable for the operations of spin-related devices. Thus, we believe the results here will be useful for future designing and developing of spin-filters or spin-dependent tunneling electron devices.
2. Theoretical calculations We consider an electron moving in a two-dimensional (2D) plane (x,y) in the presence of perpendicular magnetic field (along the z-direction). Following the procedure of Refs. [10,11], the magnetic field is taken homogeneous along the y-axis and varied along the x-axis. Such a ~ is described in the Landau gauge ~¼ rA magnetic field B ~ ¼ ½0; AðxÞ; 0. In the singleby the vector potential A particle approximation, the Hamiltonian describing such a system can be written as [21]
pffiffiffiffiffiffiffiffiffiffiffiffiffi mass, and the magnetic length l B ¼ _=eB0 , B0 being some typical magnetic field. We express all the relevant quantities in dimensionless units: (1) the magnetic field ~ ! ~z ðxÞ ! B0 B ~z ðxÞ, (2) the vector potential AðxÞ B ~ (3) the coordinate ~ r ! l B~ r, and (4) the energy B0 l B AðxÞ, E-_ocE. For GaAs and an estimated B0 ¼ 0.1 T, we have lB ¼ 813 A˚ and _oc ¼ 0.17 meV. Now the Schro¨dinger equation is of the dimensionless form 2 d þ 2E 2V ðx; k ; sÞ cðxÞ ¼ 0 (4) y dx2 with V ðx; ky ; sÞ ¼ ½AðxÞ þ ky 2 =2 þ mn gn sBz ðxÞ=4m0
The V(x,ky,s) term is normally interpreted as effective potential. It depends not only on the electron wave vector for motion in the y-direction, but also on the magnetic configuration, as well as on the interaction between the electron spin and the non-homogeneous magnetic field. When assuming the B(x) profiles to be periodic in x-axis, and the average magnetic-field strength of the magneticfield modulation to be zero, i.e., /B(x)S ¼ 0, the resulting effective potential will be periodic with period l, i.e., V(x,ky) ¼ V(x+nl,ky). Therefore, we need only to find out the solution in one unit period, then use Bloch’s theorem to propagate this solution throughout the lattice. The total wave function for the 2D electrons, up to a normalization factor, is written as ~
n 1 ~ ~ 2 þ eg s_ Bz ðxÞ H¼ ½P þ eAðxÞ n 2m 2m0 2
Cn;k ð~ rÞ ¼ eik~r cn;k ðxÞ (1)
where m* is the effective mass of electron, m0 is the free~ is the momentum of electron, electron mass in vacuum, P g* is the effective g-factor of electron in a real 2DEG, s ¼ +1/1 for spin-up/down electrons. Since the y-component of the free-electron momentum operator commutes with the Hamiltonian, i.e., [H,Py] ¼ 0, the problem is translational invariant along the y-direction and the corresponding wave vector ky is a conserved quantity. Therefore, the wave function can be written as a product as Cðx; yÞ ¼ eiky y cðxÞ
(2)
where _ky is the expectation value of py in y-direction. The wave function c(x) satisfies the one-dimensional (1D) Schro¨dinger equation: 2 he i2 mn gn sB ðxÞ 2mn E d z AðxÞ þ k þ 2 cðxÞ ¼ 0 y _ 2_m0 dx2 _ (3) For convenience, we introduce the electron cyclonic frequency oc ¼ eB0 =mn , where mn is the electron effective
(5)
(6)
where ~ r ¼ ðx; yÞ, k~ ¼ ðk; ky Þ, cn,k(x) is the Bloch function, which can be expressed in units of lB. It is noted that increasing the period l at constant B0 is equivalent to increasing the strength of the magnetic field B0 for a fixed period l.
3. Results and discussion 3.1. Perpendicular magnetization We consider a periodic magnetic field realized by means of metallic ferromagnetic stripes deposited on the top of a 2DEG heterostructure. First, we assume the ferromagnetic strips are perpendicularly magnetized; thus we obtain a magnetic superlattice created by the inhomogeneous magnetic field in the 2DEG [11,19–21]; its magnetic profile is schematically illustrated as in Fig. 1. The magnetic field has been simplified as a square barrier or well as in Refs. [19–21]; the dotted line represents the magnetic vector potential. It is seen that the magnetic field modulation is of inversion anti-symmetry feature, i.e., the magnetic field is symmetric but the vector potential is anti-symmetric at the middle points of each magnetic barrier or well.
ARTICLE IN PRESS H. Xu / Physica E 40 (2008) 2959–2964
In each region the xi0 and li are expressed as 8 ð0pxpb=4Þ ky =B > < i x0 ¼ b=2 ky =B ðb=4pxp3b=4Þ > : b þ k =B ð3b=4pxpbÞ
z b/2
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b/2
(12)
y
B
b/4
w
x
Fig. 1. Schematic illustration of the ferromagnetic metal stripes/semiconductor 2DEG structure, the model magnetic field profile B(x) (solid line) and vector potential A(x) (dotted line) of the magnetic superlattice realized by perpendicular magnetization.
The magnetic field experienced by the 2DEG is given in Refs. [5,16]: 8 B ð0pxob=4Þ > > > > < þB ðb=4xp3b=4Þ (7) BðxÞ ¼ B ð3b=4oxpbÞ > > > > :0 ðboxpb þ wÞ with 8 Bx > > > > < Bb=2 þ bx AðxÞ ¼ Bb Bx > > > > :0
ð0oxpb=4Þ ðb=4oxp3b=4Þ ð3b=4oxpbÞ ð0oxpb þ wÞ
(8)
ci ðxÞ ¼ C i cU 0 ðxÞ þ Di cU 1 ðxÞ With cU 0 ðxÞ ¼ expðx2i =2ÞU 0 ðxi Þ (9)
Here Ci and Di are constants to be determined from the boundary conditions. U0(xi) and U1(xi) are the two Hermitian functions [22,23]: U 0 ðxi Þ ¼ 1 þ
2 li 3 ð2 li Þð6 li Þ 5 xi þ xi þ 5! 3! ð2 li Þð6 li Þ ð4n 2 li Þ 2nþ1 xi þ þ ð2n þ 1Þ!
By imposing the continuity conditions of the wave function and its first derivative, as well as the Bloch periodicity on the wave function at the edges of the unit cells, we can obtain the dispersion relation between the electron energy E and its wave vector k~ ¼ ðk; kx Þ; it can be written in a very concise form using the transfer matrix method as [24] CosðklÞ ¼ Tr S=2
(14)
Here l is the period length, Tr S is the trace of the total 2 2 matrix S, which only needs to be calculated in one period. It has the following form: ðx ¼ 3b=4ÞS ðx ¼ 3b=4ÞS 1 ðx ¼ bÞS w "
The Schro¨dinger equation (4) in the magnetic barrier region (B(x)6¼0) can be solved analytically, and the wave function ci(x) can be written as [22,23]
li 2 li ð4 li Þ 4 xi þ x þ 4! 2! i li ð4 li Þ ð4n 4 li Þ 2n xi þ þ ð2nÞ!
(13)
1 S ¼ S ðx ¼ 0ÞS 1 ðx ¼ b=4ÞS þ ðx ¼ b=4ÞS þ
Within one period, the wave functions in the magnetic well region (B(x)=0) are the free-electron wave functions; it can be written as cw ðx; yÞ ¼ eiky y ðC w eikw x þ Dw eikw x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with kw ¼ 2E k2y .
cU 1 ðxÞ ¼ expðx2i =2ÞU 1 ðxi Þ pffiffiffiffi xi ¼ Bðx xi0 Þ
and 8 n n ð0pxpb=4Þ > < 2E=B þ m g s=2m0 1 n n li ¼ 2E=B m g s=2m0 1 ðb=4pxp3b=4Þ > : 2E=B þ mn gn s=2m 1 ð3b=4pxpbÞ 0
(10)
U 1 ðxi Þ ¼ xi þ
(11)
Sw ¼
Sin kðkw w wÞ Cos ðkw wÞ kw Sinðkw wÞ Cos ðkw wÞ 2
cU 0 ðxi Þ
cU 1 ðxi Þ
dx
dcU 1 ðxi Þ dx
S ðxi Þ ¼ 4 dcU 0 ðxi Þ
(15)
# (16)
3 5
(17)
The subscript signs () and (+) are referred to the B regions or the +B regions, respectively. Fig. 2 shows the energy minibands of a model magnetic superlattice created by perpendicular magnetization ferromagnetic stripes, which are periodically deposited on an InSb 2DEG. We have assumed g* ¼ 51, m* ¼ 0.014m0, b ¼ 2, B ¼ 10, w ¼ 1 in numerical calculations. The spin splitting resulting from the interaction between the spin and the magnetic field is clearly seen, the electronic minibands between the spin-up and spin-down do not superimpose due to the periodic magnetic field modulation. Fig. 3 shows the electron energy E as a function of wave vector ky. The widths of the spin-dependent bands and the energy gaps strongly depend on the transverse wave vector ky. It is seen that all the minibands become narrow as |ky| is increasing. The strong wave-vector filtering feature and the spin-dependent energy of this magnetic superlattice make it a good candidate for the spintronics applications. For example, a spin filter of 100% based on spin-dependent tunneling has already been demonstrated in structures consisting of single or double magnetic barriers [19–21]. It is noted that the energy bands are symmetric for ky-ky, which is a result of the symmetrical effective potential
ARTICLE IN PRESS H. Xu / Physica E 40 (2008) 2959–2964
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12
z
10 c
b
c
x
Energy (E/E0)
8 -B
6
Fig. 4. Schematic illustration of the ferromagnetic metal stripes/semiconductor 2DEG structure, and the model magnetic field profile B(x) (solid line) and vector potential A(x) (dotted line) of the magnetic superlattice realized by parallel magnetization.
4
2
0 -1.0
-0.5
0.0 k (1/lB)
0.5
1.0
Fig. 2. The energy dispersion curves for the InSb system with w ¼ 1; solid lines are for spin-up electrons, and dashed lines are for spin-down electrons, respectively.
d-functions with alternating sign arranged along the x-axis [14,15]; the dotted line represents the magnetic vector potential. The magnetic field modulation is of inversion symmetry, i.e., the magnetic field is anti-symmetric and the vector potential is symmetric at the middle of each magnetic barrier or well: BzðxÞ ¼ Bl B
þ1 X
½dðx c þ anÞ dðx þ anÞ
(18)
1
12
For such a magnetic field the vector potential is a step ~ ¼ ð0; Bl B ðxÞ; 0Þ, where function and can be taken as A ( 1=2 ð0oxocÞ ðxÞ ¼ (19) 1=2 ðcoxoaÞ
10
Energy (E/E0)
8
This magnetic profile is similar to the frequently used magnetic Kronig–Penney model, except that we also assumed c6¼b here for generality and for convenience to compare with Ref. [12]. Obviously, the effective potential of this model is step like, so that the solutions of the Schro¨dinger equation for this potential are plane waves in both the well and the barrier regions. Similarly, the dispersion relation can be written as [24]
6
4
2
CosðkaÞ ¼ Tr S=2
0 -4
-2
0 ky
2
4 -4
-2
0 ky
2
4
Fig. 3. The dependence of band energy on the transverse wave vector ky; (a) for spin-up and (b) for spin-down, respectively.
V(ky,x). This result implies that there is no in-plane conductance (or Hall current along the y-direction) when electrons transversally go through this superlattice. This structure cannot be used for the aim of exploring the spin Hall Effect in 2DEG.
Here a is the period length, Tr S is the trace of the total 2 2 matrix S, which only needs to be calculated in one period. Here the matrix S becomes S ¼ S þd Sc S d S b with d ¼
c¼
3.2. Parallel magnetization b¼ Second, we consider the magnetic superlattice created by parallel magnetization of ferromagnetic stripes, which are periodically deposited on a 2DEG heterojunction [9–18]. The magnetic profile is schematically illustrated in Fig. 4. The magnetic field profile has been simplified by a series of
(20)
a¼
1
0
g
1
(21)
!
cos ðacÞ a sin ðacÞ
1a sin ðacÞ cos ðacÞ
!
CoshðbbÞ
b1 SinhðbbÞ
b SinhðbbÞ
CoshðbbÞ
(22) ! (23)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2E ðkx B=2Þ2 ; and b ¼ ðkx þ B=2Þ2 2E ,
g ¼ g sBm =2m0 .
(24)
ARTICLE IN PRESS H. Xu / Physica E 40 (2008) 2959–2964
After some straightforward calculations, we obtained the analytical dispersion relation as a 2 þ g 2 b2 cosðkaÞ ¼ cos hðbbÞ cos ðacÞ sin hðbbÞ sinðacÞ 2ab (25) It can be seen clearly from Eq. (25) that the energy band of the magnetic superlattice realized by parallel magnetization is spin-degeneracy. Physically, it can be attributed to the inversion symmetry of the magnetic field profile. This spin-degeneracy feature explains why there is no spin polarization as the tunneling electron goes through such a kind of magnetic superlattice of finite periods [15]. It is not difficult to find out the difference between Eq. (25) and Eq. (13) of Ref. [12], such as the sign in front of g2 at the numerator and the two extra terms in the right side of Eq. (13); thus, we consider that Eq. (13) of Ref. [12] is incorrect. It can be easily demonstrated that Eq. (13) of Ref. [12] was deduced P from a mistaken magnetic profile of form zðxÞ ¼ Bl B þ1 1 ½dðx c þ anÞ þ dðx þ anÞ, or zðxÞ ¼ Bl B Sþ1 1 ½dðx c þ anÞ þ dðx þ anÞ, and the vector potential of form Eq. (19). It is really surprising that similar mistakes had also been found in Refs. [17,18], while the authors were calculating the electron transmission probability in the simplest magnetic profile consisting of one piece of magnetic strip, and thus only a pair of positive and negative d-functions magnetic field. Fig. 5 shows the electron energy E as a function of the wave vector ky numerically calculated for a model of B ¼ 1, c ¼ 3, b ¼ 6 and g* ¼ 2.0, m* ¼ m0. We have used the same parameters as that of Ref. [12]. It is seen that the energy band bottoms shift away from ky ¼ 0 due to including the spin–magnetic field interaction and assuming c6¼b, which are much different from the case for c ¼ b, as 4
Energy
3
2
2963
well as for spinless electrons [9,10,12]. Furthermore, the widths of the electron energy bands and the band gaps depend strongly on ky, and are asymmetric for ky-ky when c6¼b. In particular, the electrons with positive ky are found in a higher-energy state than those with ky (negative) within the same energy miniband. It means electrons moving in opposite y-directions no longer have the same energy. As |ky| increases, the bands become narrower and consequently the corresponding states are more localized in the y-direction. We noted that the dispersion relation for E versus ky for c=b is symmetric for ky-ky, and there is no band gap at ky=0. Therefore, we attributed this asymmetry to the broken symmetry of the effective potential V(ky,x) with respect to ky for c6¼b, stemming from the magnetic profile being no longer invariant under spatial inversion at each d-function potential. The ky-ky asymmetry will result in an inplane conductance (or Hall current along the y-direction) when electrons transport through these kinds of superlattices. 4. Conclusion We have calculated analytically and numerically the energy bands for electron in two-magnetic superlattices realized by perpendicular or parallel magnetization ferromagnetic stripes periodically deposited on a 2DEG heterojunction. We have shown that the spin splitting of the energy bands appears in magnetic superlattice realized by perpendicular magnetization ferromagnetic stripes, while the magnetic superlattice realized by parallel magnetization shows only the spin degeneration. These distinguishing differences are related to the inversion antisymmetry in the perpendicular magnetization and the inversion symmetry in parallel magnetization. Finally, we have pointed out that the dispersion relation for spin electrons in the magnetic superlattice realized by parallel magnetization given in Ref. [12] is incorrect, which can be traced back to a calculation mistake. We believe that the structural symmetry-related spin splitting features in these two magnetic superlattices are useful for designing and developing spin filters or spin-dependent tunneling electron devices. Finally, our rigorous calculation is based on the transfer matrix approach, which can be easily extended to more complex structures. Acknowledgement The author would like to thank the financial support from National Nature Science Foundation of China under Grant No. 60776067.
1
0
References -2
-1
0 ky
1
2
Fig. 5. The band energy dependence on the transverse wave vector ky.
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