Electron tunneling in a non-magnetic heterostructure in presence of both Dresselhaus and Rashba spin–orbit terms

Physica E 43 (2010) 142–145

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Physica E journal homepage: www.elsevier.com/locate/physe

Electron tunneling in a non-magnetic heterostructure in presence of both Dresselhaus and Rashba spin–orbit terms Jian-Duo Lu a,b, a b

Department of Applied Physics, Wuhan University of Science and Technology, Wuhan 430081, China Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430081, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 19 May 2010 Accepted 29 June 2010

In this paper, the electron tunneling in a non-magnetic heterostructure is theoretically investigated in presence of both Dresselhaus and Rashba spin–orbit terms. We find that the considerable spin polarization can be achieved in such a non-magnetic structure, and in the different electron energy regions or for the different barrier heights, the Dresselhaus and Rashba spin–orbit couplings play a different role in the spin polarization. We also find that the transmission probability and the spin polarization both show a periodic profile with the increase of the well width. & 2010 Elsevier B.V. All rights reserved.

1. Introduction In the past 10 years, electron spin in semiconductors has received a rapidly growing interest [1–6] due to its potential application in spintronics devices. To successfully incorporate spin into existing semiconductor technology, one has to overcome many technical difficulties, the most important one of which is to advance the spin-injection efficiency from ferromagnetic metal (FM) into semiconductor (SC). Although the use of FM–SC junctions as spin polarization devices has been widely considered [7,8], the spin-injection efficiency is very poor [9] due to the conductance mismatch between the FM and SC materials [10]. Therefore, the new method of spin injection should be investigated to advance the spin-injection efficiency. Recently, due to the poor spin-injection efficiency from FM into SC, many attentions [11–14] have been given to study the electron transport properties through a non-magnetic semiconductor, in which the electron–spin polarization stems from the spin–orbit couplings including the Dresselhaus effect [15] due to the inversion asymmetry of the bulk material and the Rashba effect [16] caused by the inversion asymmetry of the confining potential. Spin-dependent transports in a non-magnetic semiconductor have shown much interesting prospect of technological applications such as light emitting diodes and lasers, tunneling spin valves and spin filter [17]. In this research, the spin-dependent transport through a nonmagnetic semiconductor has been theoretically studied, in which the effects of both Dresselhaus and Rashba spin–orbit interactions  Correspondence address: Department of Applied Physics, Wuhan University of

Science and Technology, Wuhan 430081, China. E-mail address: [email protected] 1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.06.035

are taken into account. The numerical results show that the considerable spin polarization can be achieved due to Dresselhaus and Rashba spin–orbit effects, although the magnetic field is zero in such a structure. They also show that the transmission probability and the spin polarization both periodically change with the increase of the well width. These features could be helpful for the fabrication of tunable spin-dependent electric devices based on non-magnetic semiconductors.

2. Theoretical method and formulas We consider the transmission of a single electron with the initial wave vector k ¼ ðkJ ,kz Þ through an asymmetrical double-barrier structure, AlAs/GaSb/AlAs/GaSb/AlAs, along zJ½0 0 1 direction as plotted in Fig. 1. Here symbols a and c denote the barrier thickness, symbol b gives the well width, kJ is the wave vector in the plane of the barrier, and kz denotes the wave vector component normal to the barrier and pointing in the direction of tunneling. The relevant parameters for AlAs [18] are the electron effective mass m1 ¼ 0:15me , the material constant g ¼ 3:6 eV˚A 3 , and for GaSb [19] the electron effective mass m2 ¼ 0:041me , the material constants g ¼ 187 eV˚A 3 and a ¼ 0:3 eV˚A. We assume the low enough temperature so that the electron– phonon interaction can be ignored. Therefore, the electron motion in each layer of the structure can be described by the Hamiltonian including both Dresselhaus and Rashba spin–orbit terms [20], 2

2

2 2 ^ ¼  ‘ @ þ ‘ kJ þ Vb þ H ^ D þH ^R H 2 2mi @z 2mi

ði ¼ 1,2Þ:

ð1Þ

^ D is the spin-dependent Here Vb is the heterostructure potential, H ^ R is in-plane Rashba term. Assuming k3 Dresselhaus term, and H

J.-D. Lu / Physica E 43 (2010) 142–145

143

the kinetic energy of incident electron is much smaller than the barrier height Vb, the Dresselhaus term may be expressed as [19,21]

With the diagonalized Dresselhaus and in-plane Rashba spin– orbit coupling terms, the spin-dependent tunneling through double barriers is described by

2 ^ D ¼ gðs^ x kx s^ y ky Þ @ , H @z2

d2 us ðzÞ

ð2Þ

dz

and in the asymmetrical quantum wells in-plane Rashba term exists with non-zero a due to heteropotential asymmetry [22] ^ R ¼ aðs^ x ky s^ y kx Þ, H

ð3Þ

where s^ x and s^ y are the Pauli matrices, and the coordinate axes x, y, z are assumed to be parallel to the cubic crystallographic axes [1 0 0], [0 1 0], [0 0 1], respectively. For the fixed in-plane wave vector kJ , the wave functions of the electron are of the form ~Þ, cs ð~ r Þ ¼ ws us ðzÞexpði~ kJ  r

ð4Þ

!

where r ¼ ðx,yÞ is an in-plane coordinate of the barrier and ws are spinors. The electron propagating through barriers in the jth region can be described as ujs ðzÞ ¼ Ajs eikz z þBjs eikz z , j ¼1, 3 and 5, and ujs ðzÞ ¼ Ajs eqs z þBjs eqs z , j ¼2 and 4, where qs are the wave vectors through the barrier including both Dresselhaus and inplane Rashba spin–orbit couplings. The Dresselhaus and in-plane Rashba terms can be diagonalized by the spinors pffiffiffi ws ¼ 1= 2ð1,seif Þ, ð5Þ which describe the electron spin states, where f is the polar angle of the wave vector k in the x–y plane and k ¼ ðkJ cosf,kJ sinf,kz Þ. The eigen-spin states with s ¼ 7 1 propagate through the barrier with conservation of the spin orientation.

(kII,kz) 1 AlAs 0

2

Vb

2

3

GaSb

AlAs a

4

5

GaSb a+b

AlAs a+b+c

z II [001]

Fig. 1. Electron transmission through [0 0 1]-grown asymmetrical double-barrier ˚ and heterostructure. Here we choose kJ ¼ 2  108 m1 , the system size a¼ 50 A. ˚ c ¼40 A.

1.0

‘

ð6Þ

ð7Þ

m1

m1

Taking into account the boundary conditions, a system of linear equation for Ajs and Bjs can be derived. Hence the transmission coefficient Ts can be calculated by transfer-matrix method and the spin polarization P that determines the difference of transparency for the spin states jþ S and jS through the barrier is P ¼ ðT þ T Þ=ðT þ þ T Þ.

3. Numerical results and discussions We first investigate the transmission probability and the spin polarization as a function of electron energy in Fig. 2. From Fig. 2(a)–(c), we can clearly see that the considerable spin splitting occurs in the transmission curves of spin-down and spin-up electrons. For the energy, about E o0:2 eV, the Dresselhaus spin–orbit interaction plays a greater role on the spin splitting than the Rashba spin–orbit term does, while for the energy, E4 0:2 eV, the spin splitting is mainly due to the Rashba spin–orbit effect. This point can be more clearly seen from the spin polarization curves as shown in Fig. 2(d). It is clearly shown that the large degree of spin polarization can be achieved in such a non-magnetic nanostructure due to both Dresselhaus and Rashba spin–orbit terms, but it is the Dresselhaus spin–orbit interaction playing a greater role on the large spin polarization for Eo 0:2 eV, while it is the Rashba spin–orbit effect for E4 0:2 eV.

Tup Tdown

0.8

with both Dresselhaus and Rashba spin−orbit terms

0.6 0.4

Tup Tdown

0.2

0.2

0.0 0.8

with Rashba spin−orbit term

0.8

Polarization

Transmission

us ðzÞ ¼ 0:

Here q0 is the reciprocal length of decay of the wave function in the barrier when the spin–orbit coupling interactions are omitted as given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2m2 Vb m m2 k2z 2 k2J 1 : ð8Þ q0 ¼ 2

Transmission

0.6

0.6 0.4

1 þ 2sgm2 kJ =‘

2

2

1.0

0.8

0.0

q20 2sam2 kJ =‘

Therefore the wave vectors qs are given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uq2 2sam kJ =‘2 2 : qs ¼ t 0 2 1 þ 2sgm2 kJ =‘

with Dresselhaus spin−orbit term

0.4



Tup Tdown

0.4 0.0 −0.4

0.2

with both Dresselhaus and Rashba spin−orbit terms

with Dresselhaus spin−orbit term

with Rashba spin−orbit term

−0.8

0.0 0.0

0.1 0.2 Electron energy in eV

0.3

0.0

0.1 0.2 Electron energy in eV

0.3

˚ Fig. 2. The transmission probability and the spin polarization P as a function of electron energy at fixed the barrier height, Vb ¼0.32 eV and the well width, b ¼ 30 A.

144

J.-D. Lu / Physica E 43 (2010) 142–145

the Dresselhaus spin–orbit interaction. As a result of this character, the large spin polarization can be obviously seen in such a non-magnetic nanostructure as shown in Fig. 3(d). It can reach about 90% for about the well width, b¼30 or 71 A˚ when Dresselhaus and Rashba spin–orbit terms are both taken into account. Here, it is interesting to note that the transmission and the spin polarization curves both show a periodic profile with the increase of the well width. This can be ascribed to the reasons as illustrated in Ref. [23]. Finally, we examine the dependence of the transmission probabilities and the spin polarization on the barrier height as given in Fig. 4. From the figure, it can be obviously seen that, with the increase of the barrier height the transmission first shows two resonant peaks and then decreases to zero, and the left peak is

This character is different from the electron transmission in the non-magnetic heterostructure investigated in Ref. [23]. It is also shown that the polarization efficiency is different through the whole energy region and reaches highest at resonant energies, it reaches about 90% with both Dresselhaus and Rashba spin–orbit terms. We next investigate the electron transmission probabilities and the spin polarization as a function of the well width in Fig. 3. Apparently, two resonant peaks appear in the transmitted curves of both spin-up and spin-down electrons, and the transmitted peak for spin-up electron is a little higher than that of spin-down electron. It is also found that the transmitted peaks for spin-down electron clearly shift rightwards when both Dresselhaus and Rashba spin–orbit terms are taken into account, but mainly due to

1.0

1.0 with Dresselhaus spin−orbit term

0.8 Transmission

0.8 0.6 Tup Tdown

0.2 0.0

0.6

Tup Tdown

0.4 0.2 0.0 0.8

with Rashba spin−orbit term

with both Dresselhaus and Rashba spin−orbit terms

0.8 0.6

Polarization

Transmission

0.4

with both Dresselhaus and Rashba spin−orbit terms

Tup Tdown

0.4

0.4 0.0 with Rashba spin−orbit term

−0.4

0.2

with Dresselhaus spin−orbit term

−0.8

0.0 20

30

40

50

60

70

80

20

the well width (b) in 10−10m

30

40

50

60

70

80

the well width (b) in 10−10m

Fig. 3. The transmission probability and the spin polarization P as a function of the well width. Here we choose the barrier height, Vb ¼ 0.32 eV and the electron energy, E¼ 160 meV.

1.0

1.0 with Dresselhaus spin−orbit term

0.6

Tup Tdown

0.2

Tup

0.6

Tdown

0.4 0.2

0.0

0.0 0.8

with Rashba spin−orbit term

0.8

Polarization

Transmission

0.4

with both Dresselhaus and Rashba spin−orbit terms

0.8 Transmission

0.8

Tup

0.6

Tdown

0.4

0.4

with both Dresselhaus and Rashba spin−orbit terms

0.0 with Dresselhaus spin−orbit term

−0.4

0.2 0.0

with Rashba spin−orbit term

−0.8 0.0

0.1 0.2 0.3 0.4 The barrier height (Vb) in eV

0.5

0.0

0.1 0.2 0.3 0.4 The barrier height (Vb) in eV

0.5

Fig. 4. The transmission probability and the spin polarization P as a function of the barrier height, where we take the electron energy, E ¼160 meV and the well width, ˚ b¼ 30 A.

J.-D. Lu / Physica E 43 (2010) 142–145

strikingly higher than the right peak, this can be ascribed to the strong suppression of the barrier. It also can be clearly seen that the considerable spin splitting occurs in the transmitted curves for electrons with opposite spin orientations mainly due to Rashba spin–orbit coupling for about the barrier height, Vb o 0:25 eV, and Dresselhaus spin–orbit interaction for about the barrier height, Vb 40:25 eV. This character can be more clearly seen from the spin polarization curve as plotted in Fig. 4(d). Therefore, we can conclude that the large spin polarization can be achieved in such a non-magnetic heterostructure due to both Dresselhaus and Rashba spin–orbit interactions.

4. Conclusion In summary, the spin-dependent electron transport has been numerically investigated through a non-magnetic heterostructure including both Dresselhaus and Rashba spin–orbit coupling interactions. It is found that the large spin polarization can be achieved in such a heterostructure, and the barrier obviously suppresses the electron transmission. It is also found that the transmission probability and the spin polarization both show a periodic profile with the increase of the well width. These features could be useful for the fabrication of spin-dependent electric devices based on non-magnetic semiconductors.

145

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