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Angular dependent study on spin transport in magnetic semiconductor heterostructures with Dresselhaus spin–orbit interaction
Author’s Accepted Manuscript Angular dependent study on spin transport in magnetic semiconductor heterostructures with Dresselhaus spin-orbit interaction S.M. Mirzanian, A.A. Shokri, K. Mikaili Agah www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(15)30070-9 http://dx.doi.org/10.1016/j.physe.2015.05.034 PHYSE11986
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 9 February 2015 Revised date: 20 May 2015 Accepted date: 29 May 2015 Cite this article as: S.M. Mirzanian, A.A. Shokri and K. Mikaili Agah, Angular dependent study on spin transport in magnetic semiconductor heterostructures with Dresselhaus spin-orbit interaction, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2015.05.034 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Angular dependent study on spin transport in magnetic semiconductor heterostructures with Dresselhaus spin-orbit interaction S.M. Mirzanian a∗, A. A. Shokri a
b,c
, K. Mikaili Agah
a
Department of Mathematics and Physics,
College of Basic Science, East Tehran Branch, Islamic Azad University, Tehran, Iran b
Department of Physics, Payame Noor University, 19395-3697 Tehran, Iran c
Computational Physical Sciences Research Laboratory,
School of Nano-Science, Institute for studies in theoretical Physics and Mathematics (IPM) P. O. Box 19395-5531, Tehran, Iran
May 30, 2015
Abstract We investigate theoretically the effects of Dresselhaus spin-orbit coupling (DSOC) on the spin-dependent current and shot noise through II-VI diluted magnetic semiconductor/nonmagnetic semiconductor (DMS/NMS) barrier structures. The calculation of transmission probability is based on an effective mass quantum-mechanical approach in the presence of an external magnetic field applied along the growth direction of the junction and also applied voltage. We also study the dependence of spin-dependent properties on external magnetic field and relative angle between the magnetizations of two DMS layers in CdTe/CdMnTe heterostructures with including the DSOC effect. The results show that the DSOC has great different influence on transport properties of electrons with spin up and spin down in the considered system and this aspect may be utilized in designing new spintronics devices.
∗
Corresponding author. E-mail: [email protected]
1
1
Introduction
Spin-dependent transport in the magnetic multilayers and magnetic tunnel junctions (MTJs) has created the new field of spintronics, which involves the study of active control and manipulation of electron spin and charge degrees of freedom. The MTJs are promising candidates for inclusion in magnetic random access memory, magnetic field sensors and quantum computing devices[1, 2]. The performance of such magnetoelectronic devices depends critically on the structure of magnetic junctions and the degree of spin filtering in the multilayers. There is a wide class of semiconducting materials which is characterised by the random substitution of a fraction of the original atoms by magnetic atoms, such as Mn in ordinary III-V and II-VI compounds like GaAs and CdTe, respectively [3, 4, 5, 6]. The materials are commonly known as semimagnetic semiconductors or diluted magnetic semiconductors (DMSs). Spin-polarized transport through II-VI diluted magnetic semiconductor/nonmagnetic semiconductor (DMS/NMS) barrier structures widely use in the spintronics due to their spin-dependent sp-d exchange interaction between the carriers and the local magnetic ions [7, 8, 9, 10, 11, 12, 13]. In the band-gap-matched heterostructures, spin-up electrons see a barrier and spin-down electrons see a quantum well in the presence of the magnetic field, such that, at a given magnetic field the height of the spinup barrier is equal to the depth of the spin-down well. Egues [7] and Guo et al.[13] have investigated the spin filtering in a ZnSe/Zn1−x M nx heterostructure with a single and double paramagnetic layers and found a strong suppression of the spin-up component of current density at higher magnetic fields. As the dimensions of semiconductor devices go on scaling down, the device performance becomes more sensitive to the noise. In this aspect, there has been a resurgence of interest in the study of noise in mesoscopic devices, both experimentally and theoretically [14]. The zero-frequency noise out of equilibrium (shot noise) is a type of electronic noise which originates from the discrete nature of electric charge. It is known that shot noise can yield information on charge fluctuations additional to the average current in mesoscopic systems [15] and also about system parameters that govern the electronic transport [16, 17]. With the development of spintronics, electrical noise studies should be reconsidered with respect to spin-polarization of charge carriers. Guo and coworkers [18] have investigated theoretically the shot noise properties in the ZnSe/ZnMnSe multilayers. They found that the noise shows strongly spin-dependent behavior and is affected considerably by all these internal and external conditions. Spin-orbit coupling (SOC) effect is another spin-dependent potential which has been the focus of much research in the past decade both experimentally and theoretically[19, 20, 21, 22, 23]. In the previous paper, [24] we investigated spin-dependent current and shot noise transport in double magnetic barrier resonant tunneling diodes with including the Rashba spin-orbit coupling (RSOC) effect. It has been found that the RSOC has great influence on the spin transport properties in these nanostructures. In that work, we have considered only the Rashba term of the spin-orbit interaction. The other contribution is called the Dresselhaus spin-orbit coupling(DSOC) term, which is present in semiconductors lacking bulk inversion symmetry. Similar to the RSOC, the DSOC gives rise to an enhancement of the spin-splitting of the Landau levels. Thus, we expect that not only RSOC but also DSOC one can affect spin dependent transport in these barrier structures. Zhu and coworker [25] have investigated the zero-frequency shot noise of 2
current through resonant double-barrier structures in the presence of DSOC effect, they found that the DSOC can greatly affect the average current, shot noise, and Fano factor of resonant diode structures. In this work, spin-dependent current and shot noise transport in double magnetic barrier resonant tunneling diodes is theoretically investigated with including the DSOC effect in the presence of a magnetic field and also applied voltage. Our calculations are based on parabolic energy band approximation and transfer matrix method in the presence of external magnetic field and also applied voltage, within the phase coherent transport regime. We also examine the dependence of spin transport on the relative angle between the magnetizations of two DMS layers in CdTe/CdMnTe heterostructures. This effect may be considered because in some experiments, spin conservation no longer holds, and the spin-flip scattering may affect the transport properties [26, 27]. The paper is organized as follows. In the next section, we introduce our model and formalism, such as the effective-mass Hamiltonian for the double DMS barrier resonant tunneling diodes and transfer matrix method. In Section 3, our numerical results for the DSOC effect on the spin-dependent transport. Finally, we conclude in Section 4 with a brief summary.
2
Theoretical framework
We consider spin-polarized electron transport through a NMS/DMS double barrier heterostructure in the presence of a magnetic field B and also applied voltage Va . Here, the DMS and NMS layers are assumed from the type (II, Mn) VI (e.g., Cd1−x M nx T e) and II-VI (e.g., CdT e) semiconductors, respectively. For simplicity, we assume that the two electrodes and the spacer layer are made of the same material and all the interfaces are flat shown in Figure 1 of Ref. [10]. The Hamiltonian of an electron in the framework of the parabolic-band effective-mass approximation can be written as H=
eVa z 1 2 + V (z) + (P + eA) + V (z) + V + VDresselhaus (z), off sp−d Z σ z 2m∗ (z) L
(1)
where m∗ is the electron effective mass, e is the absolute electron charge and P is the momentum operator. The vector potential is taken as A = (−By/2, Bx/2, 0). In Mn-based layers the conduction electrons interact with the d-electrons of the localized magnetic moments of the Mn ions via the sp − d exchange interaction that causes the giant Zeeman splitting effect. It can be written as Vsp−d (z) = −N0 ασz xef f < Sz > Θ(z)Θ(L−z) in mean-field approximation. Where N0 α is the electronic sp-d exchange constant, σz = ±1/2 are zth component of the spin of the itinerant electron; xef f = x(1−x)12 shows the reduced effective M n2+ concentration. The thermal average of magnetic ion spins < Sz > can be written in terms of Brillouin function SBS [5µB /KB (T + T0 )], where T0 is a parameter representing Mn-Mn interaction phenomenaligically; Θ(z) is the Heaviside function, and L is the width of the paramagnetic layer. In the absence of any kind of electron scattering, the motion along the z-axis is decoupled from that of the x - y plane, which is quantized in the Landau levels with energies En = (n+ 21 )¯ hωc , where n = 0, 1, 2,... 1 ∗ and h ¯ ωc = eB/m . Here VZσz = 2 gs µB σ.B describes the Zeeman spliting of the conduction electrons, where σ is the conventional Pauli spin operator. Voff (z) is the conduction band offset under zero magnetic field, which depends on the Mn concentration x and is 3
the difference between NMS and DMS energy gaps. The Hamiltonian of the Dresselhaus spin-orbit interaction can be written as[28] VDresselhaus = −2σz γ(K 2 kF − kF3 /4),qwhere γ is the Dresselhaus parameter denoting the strong of the coupling and kF (B) ≈ 2eBn/¯h is the Fermi wave vector. Also, K 2 ∼ [π/ω]2 is a measure of the kinetic energy in the z direction due to surface confinement, and ω is the effective width of the potential well confining the electrons. In the model, the spins of electrons during tunneling are not supposed to be conserved, that is, the tunneling of the spin-up and spin-down electrons are not an independent process. The Schr¨odinger equation for a biased barrier layer can be simplified by a coordinate transformation whose solution is the linear combination of the Airy function Ai[ρ(z)] and its complement Bi[ρ(z)] [29]. Considering all five regions of the NMS/DMS/NMS/DMS/NMS junction, the eigenfunctions of the Hamiltonian (1) with eigenvalue Ez by including the angular dependence of magnetization in the DMS layers have the following forms: A1↑ eik1↑ z + B1↑ e−ik1↑ z + A1↓ eik1↓ z + B1↓ e−ik1↓ z , A2↑ Ai[ρ(z)] + B2↑ Bi[ρ(z)] + A2↓ Ai[ρ(z)] + B2↓ Bi[ρ(z)],
ψj (z) =
z<0 0 < z < L1 A3↑ Ai[ρ(z)] + B3↑ Bi[ρ(z)] + A3↓ Ai[ρ(z)] + B3↓ Bi[ρ(z)], L1 < z < L1 + L2 A4↑ Ai[ρ(z)] + B4↑ Bi[ρ(z)] + A4↓ Ai[ρ(z)] + B4↓ Bi[ρ(z)], L1 + L2 < z < L A5↑ eik5↑ x + B5↑ e−ik5↑ z + A5↓ eik5↓ z + B5↓ e−ik5↓ z , z>L (2)
where,
k5σ =
q
2m∗ (Ez − VZσz )/¯ h,
(3)
2m∗ (Ez − VZσz + eVa )/¯ h,
(4)
k1σ = q
are the hole wave vectors along the z axis. The coefficients Ajσ and Bjσ are constants to be determined from the boundary conditions, while ρ(z) = −
L eVa z (VZσz + Voff (z) + Vsp−d (z) + VDresselhaus − ), eVa λ L
with −¯ h2 L λ= 2m∗ eVa "
(5)
#1/3
.
(6)
The wave functions and their first derivatives in the five regions are matched at the interfaces between the regions, and also, the change in quantization axis at z = L1 + L2 requires the spinor transformation [30] ψ3↑ (z) = ψ4↑ (z)cos( 2θ ) + ψ4↓ (z)sin( 2θ ), ψ3↓ (z) = −ψ4↑ (z)sin( 2θ ) + ψ4↓ (z)cos( 2θ ),
(7)
and, similarly for the derivatives. The matrix formula can be written as
A1↑ B1↑ A1↓ B1↓
= M
4
A5↑ B5↑ A5↓ B5↓
,
(8)
where, M is the total transfer matrix [30, 31]. The spin-dependent transmission coefficient of conduction electrons, Tσσ0 (Ez , Va , B, θ), is obtained from solving a one dimensional Schr¨odinger equation, by transfer matrix method [30, 31]. The spin dependent current density is connected with the transmission coefficients via Jσσ0 (Va , B, θ) =
∞ Z +∞ eB X dEz Tσσ0 (Ez ) [fL − fR ] , 4π 2 h ¯ 2 n=0 0
(9)
where fL = 1/1 + exp{[(Ez + (n + 1/2)¯ hωc + VZσz ) − EF ]/kB T } and fR = 1/1 + exp{[(Ez + (n+1/2)¯ hωc +VZσz −eVa )−EF ]/kB T } are the Fermi-Dirac distribution function of the left and right electrodes, respectively. Here, EF denotes the Fermi energy of the left electrode and ωc = eB/m∗ is the cyclotron frequency of electrons in magnetic field. The general result for the noise power of the current fluctuations in a two terminal conductor is given by [14]
∞ Z +∞ e2 B X dEz {Tσσ0 (Ez )[fL (1 − fL ) + fR (1 − fR )] + Tσσ0 (Ez )[1 − Tσσ0 (Ez )][fL − fR ]2 } 4π 2 h ¯ 2 n=0 0 (10) where, the first two terms represent the contribution of the thermal noise, and the third one term, proportional to Tσσ0 [1 − Tσσ0 ], is exactly the shot noise spectral density. Also, the Fano factor can be defined as:
Sσσ0 (Va , B, θ) =
Fσσ0 (Va , B, θ) =
3
Sσσ0 2eJσσ0
(11)
Numerical results and discussions
Numerical calculations are performed for a typical CdTe/Cd1−x Mnx Te/CdTe/Cd1−x Mnx Te/CdTe double DMS barrier resonant tunneling diode (x = 0.02), taking into account the DSOC effect. The effective mass of all carriers for the structure are taken is m∗ =0.096me , where me is the free-electron mass. The calculation have been performed at a fixed temperature T = 4.2 K. The zero-field conduction band offset is set to be Voff = 10meV , which can be found in the relevant literature [23, 32, 33]. Experimentally, the observed values of DSOC parameter lie in the range γ = 0 − 187 eV ˚ A3 for large variety of systems [34, 35, 36, 37, 38]. For comparison, we choose the Dresselhaus constant γ = 0, 50 and 150 eV ˚ A3 . The other parameters used in our calculation are as follows: gs = 1.64, gMn = 2, L1 = L2 = L3 = 5nm ,T0 = 3.1 K, N0 α = 0.22 eV [23, 32, 33]. In the structure, the magnetization direction of the left DMS electrode stays fixed, but the spin orientation of the right DMS electrode is free and may be switched from 0 to 2π with respect to the left DMS magnetization direction by an external magnetic field. Fig. 1 shows the spin-dependent current density, shot noise and Fano Factor as a function of magnetic field with different Dresselhaus constants γ. In Figs. 1a, 1c and 1e, the curves have been plotted when the spins of electrons during tunneling process are conserved, namely, the cases for (↓↓) and (↑↑). Fig. 1(a) displays the dependence of the current density as a function of magnetic field. In the presence of a positive zero-field conduction band offset as high as Voff = 10meV , the paramagnetic layer behaves as a potential barrier for both spin-down and spin-up electrons in the absence of the magnetic field. 5
In this case, as the magnetic field increases , because of the sp-d exchange and Zeeman interaction, the spin-down electrons see a smaller barrier while the spin-up electrons see a larger barrier. Therefore, J↓↓ increases and J↑↑ decreases rapidly when the magnetic field changes. The DSOC gives rise to further decrease of the barrier height seen by spin-down electrons, this leads to the enhancement of J↓↓ . However, DSOC causes further increase of the barrier height seen by the spin-up electrons which leads to the reduction of J↑↑ . Thus, as the strength of DSOC increases, J↑↑ is further suppressed and J↓↓ is significantly increased. As it can be seen in Fig. 1(c), the shot noise has a similar behavior to the current density, namely, as the magnetic field increases, the potential in the DMS layers goes up for spin-up electrons and goes down for spin down ones. So, it was found that the spin-dependent shot noise varies by the potential change. Also, we can see the effect of DRSC on Fano factor in Fig. 1e when the magnetic moments of two DMS layers are parallel. These three panels show the importance of Dresselhaus effect on spin filtering in DMS/NMS structures. Figs. 1b, 1d and 1f, display the dependence current density, shot noise and Fano Factor as a function of magnetic field when the magnetic moments of two DMS layers are antiparallel in the presence of DSOC effect. In this case (θ = π), J↓↓ (S↓↓ ) and J↑↑ (S↑↑ ) are zero, and the spin-flip effect is very important. The origin of this effect can be understood by the effect of angular dependence of the magnetizations in the DMSs, which changes the spin of the transmitted electron spin through the system. These results indicate the DSOC and the magnitude of external magnetic field are the main factors that determine the Fano factors of the spin-polarized electrons in these structures. Fig. 2 shows the Dresselhaus strength dependence of the spin-dependent current density in a typical double barrier DMS heterostructure. It can be seen that the spin transport strongly depends on the spin orientation of tunneling electrons. In Fig. 2(a) the curve was plotted when the spins of electrons were conserved during tunneling process (θ = 0). When the Dresselhaus strength increases the current density significantly increases for spin down electrons that tunnel from left electrode into right electrode, while it obviously decreases for spin up ones. The difference of the current density between two different spin electrons can reach up to several orders of magnitude. An increasing Dresselhaus strength makes the quantum barrier higher and higher (smaller and smaller) For spin-up (spin-down) electrons. This figure shows the importance of Dresselhaus effect on spin filtering in DMS/NMS structure. Fig. 2(b) displays the dependence of the current density as a function of Dresselhaus strength in the heterostructure when the magnetic moments configuration of two DMS layers are antiparallel (θ = π). In this case (θ = π), J↓↓ and J↑↑ are zero, and the spin-flip effect is very important. The DSOC has remarkable influence on spin-flip effect, such that by changing the barrier height it can enhance J↓↑ and reduce J↑↓ . In Fig. 3, we also calculated the dependence of current density, shot noise and Fano Factor on the relative angle between the magnetization of two electrodes for sevral different Dresselhaus constants. In Figs. 3a, 3c and 3e, the features were plotted when the spins of tunneling electrons during tunneling process are conserved, namely, the cases for (↓↓) and (↑↑). These curves show that the lowest values of J↓↓ (J↑↑ ), S↓↓ (S↑↑ ) and also the highest values of F↓↓ (F↑↑ ) are obtained in θ = π. When the magnetic moments of two DMS electrodes are antiparallel (θ = π), the current densities and shot noises are zero, therefore the spin-flip effect is very important. The results show that the highest values of J↓↓ (J↑↑ ), S↓↓ (S↑↑ ) and also the lowest values of F↓↓ (F↑↑ ) are obtained in (θ = 0). It 6
means that the spin-filtering effect is very important. Further, the difference of the spindependent quantities between two different spin electrons is enhanced for increasing the Dresselhaus constant. The curves in Figs. 3b, 3d, 3f were plotted when the spin orientation of carrier is switched from 0 to π, namely, (↓↑) and (↑↓) cases. Here the highest values of J↓↑ (J↑↓ ), S↓↑ (S↑↓ ) and the lowest values of F↓↑ (F↑↓ ) are obtained in θ = π. When the magnetic moments of two DMS electrodes are parallel (θ = 0), the current densities and shot noises are zero and therefore there is no spin-flip effect. For other angles, all components of current densities and shot noises change and become nonzero. Continuing to increase the angle from θ = π to 2π, the variation of the physical quantities show similar behavior as that from θ = 0 to π. This figure indicates that the DSOC has significant effects on the spin-polarized transport for both spin-up electrons and spin-down ones in every relative angle between the magnetizations of two DMS layers.
4
Conclusions
The effects of DSOC on the current density and shot noise in DMS/NMS heterostructures have been investigated. Our calculations are based on the transfer-matrix method and the effective mass approach in the presence of an external magnetic field and also an applied voltage. In this model, the spins of electrons are not supposed to be conserved during tunneling, that is, the tunneling of the spin-up and spin-down electrons are not an independent process. The results show that the DSOC effect has great different influence on the current density, the shot noise and the Fano factor for all components. Also, with considering the results of this paper and the previous work [24], we find that not only RSOC but also DSOC has remarkable influence on spin filtering and spin flipping effect in these structures. It is found that the spin polarization can be controlled very efficiently by taking into account the SOC effect and including the angle relative magnetization of the DMS barriers. The results obtained in this work may be a base for the development of future spintronics devices.
5
Acknowledgements
We would like to thank the Islamic Azad University (East Tehran Branch) for supporting us with a grant.
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9
Figure 1
0
a
b
1.0
0.6
0.8 0.5
-21
J/10 J0
0.6
0.4
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.4
0.2
0.0
c
0.3
0
1
2
3
4
5
0.2
0
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
1
2
3
4
5
3
4
5
d 0.35
0.30
0.30
0.25
S/2e
0.20
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.25
0.20
0.15 0.15 0.10 0.05
0
1
2
3
4
5
e
0
1
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
2
f 0.7
0.65
0.6
Fano Factor
0.10
0.5
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.60
0.55
0.50
0.4
0.3
0
1
2
3
4
5
0.45
0
1
2
3
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
4
5
B(T)
B(T)
Figure 1: Spin-dependent current densities, shot noises and Fano factors as a function of magnetic field with different Dresselhaus constants γ.
10
Figure 2 a 1.2
0 1.0
-21
J/10 J0
0.8
0.6 0.4 0.2 0.0
b 0.6
0.5
-21
J/10 J0
0.4 0.3 0.2 0.1 0.0
0
50
100
150
200
250
300
( eV 3 )
Figure 2: The Dresselhaus strength dependence of the spin-dependent current density in a typical double barrier DMS heterostructure. 11
Figure 3 a
b 1.0
0.5
0.8
-21
J/10 J0
0.6
c
0.4
0.3
0.4
0.2
0.2
0.1
0.0
0.0
50 eV 3 150 eV 3 50 eV 3 150 eV 3 50 eV 3 150eV 3
d
0.30
0.25 0.20
0.25
3
0 eV 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.15
0.20 0.15 S/2e
S/2e
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.10
0.10
0.05
0.05
50 eV 3 150 eV 3 50 eV 3 150 eV 3 50 eV 3 150eV 3
50 eV 3 150 eV 3 50 eV 3 150 eV 3 50 eV 3 150eV 3
0.00
0.00
e
f
1.0
0.75
0.8 0.70 Fano Factor
0.6
0.65
0.4
0.2
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.60 0.55 0.50
0.0 0.0
0.5
1.0
1.5
2.0
/
0.0
0.5
1.0
1.5
2.0
/
Figure 3: Variation of the tunneling current densities, shot noise and Fano factor on the relative angle between the magnetization of two DMS layers. 12
PII: DOI: Reference:
S1386-9477(15)30070-9 http://dx.doi.org/10.1016/j.physe.2015.05.034 PHYSE11986
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 9 February 2015 Revised date: 20 May 2015 Accepted date: 29 May 2015 Cite this article as: S.M. Mirzanian, A.A. Shokri and K. Mikaili Agah, Angular dependent study on spin transport in magnetic semiconductor heterostructures with Dresselhaus spin-orbit interaction, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2015.05.034 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Angular dependent study on spin transport in magnetic semiconductor heterostructures with Dresselhaus spin-orbit interaction S.M. Mirzanian a∗, A. A. Shokri a
b,c
, K. Mikaili Agah
a
Department of Mathematics and Physics,
College of Basic Science, East Tehran Branch, Islamic Azad University, Tehran, Iran b
Department of Physics, Payame Noor University, 19395-3697 Tehran, Iran c
Computational Physical Sciences Research Laboratory,
School of Nano-Science, Institute for studies in theoretical Physics and Mathematics (IPM) P. O. Box 19395-5531, Tehran, Iran
May 30, 2015
Abstract We investigate theoretically the effects of Dresselhaus spin-orbit coupling (DSOC) on the spin-dependent current and shot noise through II-VI diluted magnetic semiconductor/nonmagnetic semiconductor (DMS/NMS) barrier structures. The calculation of transmission probability is based on an effective mass quantum-mechanical approach in the presence of an external magnetic field applied along the growth direction of the junction and also applied voltage. We also study the dependence of spin-dependent properties on external magnetic field and relative angle between the magnetizations of two DMS layers in CdTe/CdMnTe heterostructures with including the DSOC effect. The results show that the DSOC has great different influence on transport properties of electrons with spin up and spin down in the considered system and this aspect may be utilized in designing new spintronics devices.
∗
Corresponding author. E-mail: [email protected]
1
1
Introduction
Spin-dependent transport in the magnetic multilayers and magnetic tunnel junctions (MTJs) has created the new field of spintronics, which involves the study of active control and manipulation of electron spin and charge degrees of freedom. The MTJs are promising candidates for inclusion in magnetic random access memory, magnetic field sensors and quantum computing devices[1, 2]. The performance of such magnetoelectronic devices depends critically on the structure of magnetic junctions and the degree of spin filtering in the multilayers. There is a wide class of semiconducting materials which is characterised by the random substitution of a fraction of the original atoms by magnetic atoms, such as Mn in ordinary III-V and II-VI compounds like GaAs and CdTe, respectively [3, 4, 5, 6]. The materials are commonly known as semimagnetic semiconductors or diluted magnetic semiconductors (DMSs). Spin-polarized transport through II-VI diluted magnetic semiconductor/nonmagnetic semiconductor (DMS/NMS) barrier structures widely use in the spintronics due to their spin-dependent sp-d exchange interaction between the carriers and the local magnetic ions [7, 8, 9, 10, 11, 12, 13]. In the band-gap-matched heterostructures, spin-up electrons see a barrier and spin-down electrons see a quantum well in the presence of the magnetic field, such that, at a given magnetic field the height of the spinup barrier is equal to the depth of the spin-down well. Egues [7] and Guo et al.[13] have investigated the spin filtering in a ZnSe/Zn1−x M nx heterostructure with a single and double paramagnetic layers and found a strong suppression of the spin-up component of current density at higher magnetic fields. As the dimensions of semiconductor devices go on scaling down, the device performance becomes more sensitive to the noise. In this aspect, there has been a resurgence of interest in the study of noise in mesoscopic devices, both experimentally and theoretically [14]. The zero-frequency noise out of equilibrium (shot noise) is a type of electronic noise which originates from the discrete nature of electric charge. It is known that shot noise can yield information on charge fluctuations additional to the average current in mesoscopic systems [15] and also about system parameters that govern the electronic transport [16, 17]. With the development of spintronics, electrical noise studies should be reconsidered with respect to spin-polarization of charge carriers. Guo and coworkers [18] have investigated theoretically the shot noise properties in the ZnSe/ZnMnSe multilayers. They found that the noise shows strongly spin-dependent behavior and is affected considerably by all these internal and external conditions. Spin-orbit coupling (SOC) effect is another spin-dependent potential which has been the focus of much research in the past decade both experimentally and theoretically[19, 20, 21, 22, 23]. In the previous paper, [24] we investigated spin-dependent current and shot noise transport in double magnetic barrier resonant tunneling diodes with including the Rashba spin-orbit coupling (RSOC) effect. It has been found that the RSOC has great influence on the spin transport properties in these nanostructures. In that work, we have considered only the Rashba term of the spin-orbit interaction. The other contribution is called the Dresselhaus spin-orbit coupling(DSOC) term, which is present in semiconductors lacking bulk inversion symmetry. Similar to the RSOC, the DSOC gives rise to an enhancement of the spin-splitting of the Landau levels. Thus, we expect that not only RSOC but also DSOC one can affect spin dependent transport in these barrier structures. Zhu and coworker [25] have investigated the zero-frequency shot noise of 2
current through resonant double-barrier structures in the presence of DSOC effect, they found that the DSOC can greatly affect the average current, shot noise, and Fano factor of resonant diode structures. In this work, spin-dependent current and shot noise transport in double magnetic barrier resonant tunneling diodes is theoretically investigated with including the DSOC effect in the presence of a magnetic field and also applied voltage. Our calculations are based on parabolic energy band approximation and transfer matrix method in the presence of external magnetic field and also applied voltage, within the phase coherent transport regime. We also examine the dependence of spin transport on the relative angle between the magnetizations of two DMS layers in CdTe/CdMnTe heterostructures. This effect may be considered because in some experiments, spin conservation no longer holds, and the spin-flip scattering may affect the transport properties [26, 27]. The paper is organized as follows. In the next section, we introduce our model and formalism, such as the effective-mass Hamiltonian for the double DMS barrier resonant tunneling diodes and transfer matrix method. In Section 3, our numerical results for the DSOC effect on the spin-dependent transport. Finally, we conclude in Section 4 with a brief summary.
2
Theoretical framework
We consider spin-polarized electron transport through a NMS/DMS double barrier heterostructure in the presence of a magnetic field B and also applied voltage Va . Here, the DMS and NMS layers are assumed from the type (II, Mn) VI (e.g., Cd1−x M nx T e) and II-VI (e.g., CdT e) semiconductors, respectively. For simplicity, we assume that the two electrodes and the spacer layer are made of the same material and all the interfaces are flat shown in Figure 1 of Ref. [10]. The Hamiltonian of an electron in the framework of the parabolic-band effective-mass approximation can be written as H=
eVa z 1 2 + V (z) + (P + eA) + V (z) + V + VDresselhaus (z), off sp−d Z σ z 2m∗ (z) L
(1)
where m∗ is the electron effective mass, e is the absolute electron charge and P is the momentum operator. The vector potential is taken as A = (−By/2, Bx/2, 0). In Mn-based layers the conduction electrons interact with the d-electrons of the localized magnetic moments of the Mn ions via the sp − d exchange interaction that causes the giant Zeeman splitting effect. It can be written as Vsp−d (z) = −N0 ασz xef f < Sz > Θ(z)Θ(L−z) in mean-field approximation. Where N0 α is the electronic sp-d exchange constant, σz = ±1/2 are zth component of the spin of the itinerant electron; xef f = x(1−x)12 shows the reduced effective M n2+ concentration. The thermal average of magnetic ion spins < Sz > can be written in terms of Brillouin function SBS [5µB /KB (T + T0 )], where T0 is a parameter representing Mn-Mn interaction phenomenaligically; Θ(z) is the Heaviside function, and L is the width of the paramagnetic layer. In the absence of any kind of electron scattering, the motion along the z-axis is decoupled from that of the x - y plane, which is quantized in the Landau levels with energies En = (n+ 21 )¯ hωc , where n = 0, 1, 2,... 1 ∗ and h ¯ ωc = eB/m . Here VZσz = 2 gs µB σ.B describes the Zeeman spliting of the conduction electrons, where σ is the conventional Pauli spin operator. Voff (z) is the conduction band offset under zero magnetic field, which depends on the Mn concentration x and is 3
the difference between NMS and DMS energy gaps. The Hamiltonian of the Dresselhaus spin-orbit interaction can be written as[28] VDresselhaus = −2σz γ(K 2 kF − kF3 /4),qwhere γ is the Dresselhaus parameter denoting the strong of the coupling and kF (B) ≈ 2eBn/¯h is the Fermi wave vector. Also, K 2 ∼ [π/ω]2 is a measure of the kinetic energy in the z direction due to surface confinement, and ω is the effective width of the potential well confining the electrons. In the model, the spins of electrons during tunneling are not supposed to be conserved, that is, the tunneling of the spin-up and spin-down electrons are not an independent process. The Schr¨odinger equation for a biased barrier layer can be simplified by a coordinate transformation whose solution is the linear combination of the Airy function Ai[ρ(z)] and its complement Bi[ρ(z)] [29]. Considering all five regions of the NMS/DMS/NMS/DMS/NMS junction, the eigenfunctions of the Hamiltonian (1) with eigenvalue Ez by including the angular dependence of magnetization in the DMS layers have the following forms: A1↑ eik1↑ z + B1↑ e−ik1↑ z + A1↓ eik1↓ z + B1↓ e−ik1↓ z , A2↑ Ai[ρ(z)] + B2↑ Bi[ρ(z)] + A2↓ Ai[ρ(z)] + B2↓ Bi[ρ(z)],
ψj (z) =
z<0 0 < z < L1 A3↑ Ai[ρ(z)] + B3↑ Bi[ρ(z)] + A3↓ Ai[ρ(z)] + B3↓ Bi[ρ(z)], L1 < z < L1 + L2 A4↑ Ai[ρ(z)] + B4↑ Bi[ρ(z)] + A4↓ Ai[ρ(z)] + B4↓ Bi[ρ(z)], L1 + L2 < z < L A5↑ eik5↑ x + B5↑ e−ik5↑ z + A5↓ eik5↓ z + B5↓ e−ik5↓ z , z>L (2)
where,
k5σ =
q
2m∗ (Ez − VZσz )/¯ h,
(3)
2m∗ (Ez − VZσz + eVa )/¯ h,
(4)
k1σ = q
are the hole wave vectors along the z axis. The coefficients Ajσ and Bjσ are constants to be determined from the boundary conditions, while ρ(z) = −
L eVa z (VZσz + Voff (z) + Vsp−d (z) + VDresselhaus − ), eVa λ L
with −¯ h2 L λ= 2m∗ eVa "
(5)
#1/3
.
(6)
The wave functions and their first derivatives in the five regions are matched at the interfaces between the regions, and also, the change in quantization axis at z = L1 + L2 requires the spinor transformation [30] ψ3↑ (z) = ψ4↑ (z)cos( 2θ ) + ψ4↓ (z)sin( 2θ ), ψ3↓ (z) = −ψ4↑ (z)sin( 2θ ) + ψ4↓ (z)cos( 2θ ),
(7)
and, similarly for the derivatives. The matrix formula can be written as
A1↑ B1↑ A1↓ B1↓
= M
4
A5↑ B5↑ A5↓ B5↓
,
(8)
where, M is the total transfer matrix [30, 31]. The spin-dependent transmission coefficient of conduction electrons, Tσσ0 (Ez , Va , B, θ), is obtained from solving a one dimensional Schr¨odinger equation, by transfer matrix method [30, 31]. The spin dependent current density is connected with the transmission coefficients via Jσσ0 (Va , B, θ) =
∞ Z +∞ eB X dEz Tσσ0 (Ez ) [fL − fR ] , 4π 2 h ¯ 2 n=0 0
(9)
where fL = 1/1 + exp{[(Ez + (n + 1/2)¯ hωc + VZσz ) − EF ]/kB T } and fR = 1/1 + exp{[(Ez + (n+1/2)¯ hωc +VZσz −eVa )−EF ]/kB T } are the Fermi-Dirac distribution function of the left and right electrodes, respectively. Here, EF denotes the Fermi energy of the left electrode and ωc = eB/m∗ is the cyclotron frequency of electrons in magnetic field. The general result for the noise power of the current fluctuations in a two terminal conductor is given by [14]
∞ Z +∞ e2 B X dEz {Tσσ0 (Ez )[fL (1 − fL ) + fR (1 − fR )] + Tσσ0 (Ez )[1 − Tσσ0 (Ez )][fL − fR ]2 } 4π 2 h ¯ 2 n=0 0 (10) where, the first two terms represent the contribution of the thermal noise, and the third one term, proportional to Tσσ0 [1 − Tσσ0 ], is exactly the shot noise spectral density. Also, the Fano factor can be defined as:
Sσσ0 (Va , B, θ) =
Fσσ0 (Va , B, θ) =
3
Sσσ0 2eJσσ0
(11)
Numerical results and discussions
Numerical calculations are performed for a typical CdTe/Cd1−x Mnx Te/CdTe/Cd1−x Mnx Te/CdTe double DMS barrier resonant tunneling diode (x = 0.02), taking into account the DSOC effect. The effective mass of all carriers for the structure are taken is m∗ =0.096me , where me is the free-electron mass. The calculation have been performed at a fixed temperature T = 4.2 K. The zero-field conduction band offset is set to be Voff = 10meV , which can be found in the relevant literature [23, 32, 33]. Experimentally, the observed values of DSOC parameter lie in the range γ = 0 − 187 eV ˚ A3 for large variety of systems [34, 35, 36, 37, 38]. For comparison, we choose the Dresselhaus constant γ = 0, 50 and 150 eV ˚ A3 . The other parameters used in our calculation are as follows: gs = 1.64, gMn = 2, L1 = L2 = L3 = 5nm ,T0 = 3.1 K, N0 α = 0.22 eV [23, 32, 33]. In the structure, the magnetization direction of the left DMS electrode stays fixed, but the spin orientation of the right DMS electrode is free and may be switched from 0 to 2π with respect to the left DMS magnetization direction by an external magnetic field. Fig. 1 shows the spin-dependent current density, shot noise and Fano Factor as a function of magnetic field with different Dresselhaus constants γ. In Figs. 1a, 1c and 1e, the curves have been plotted when the spins of electrons during tunneling process are conserved, namely, the cases for (↓↓) and (↑↑). Fig. 1(a) displays the dependence of the current density as a function of magnetic field. In the presence of a positive zero-field conduction band offset as high as Voff = 10meV , the paramagnetic layer behaves as a potential barrier for both spin-down and spin-up electrons in the absence of the magnetic field. 5
In this case, as the magnetic field increases , because of the sp-d exchange and Zeeman interaction, the spin-down electrons see a smaller barrier while the spin-up electrons see a larger barrier. Therefore, J↓↓ increases and J↑↑ decreases rapidly when the magnetic field changes. The DSOC gives rise to further decrease of the barrier height seen by spin-down electrons, this leads to the enhancement of J↓↓ . However, DSOC causes further increase of the barrier height seen by the spin-up electrons which leads to the reduction of J↑↑ . Thus, as the strength of DSOC increases, J↑↑ is further suppressed and J↓↓ is significantly increased. As it can be seen in Fig. 1(c), the shot noise has a similar behavior to the current density, namely, as the magnetic field increases, the potential in the DMS layers goes up for spin-up electrons and goes down for spin down ones. So, it was found that the spin-dependent shot noise varies by the potential change. Also, we can see the effect of DRSC on Fano factor in Fig. 1e when the magnetic moments of two DMS layers are parallel. These three panels show the importance of Dresselhaus effect on spin filtering in DMS/NMS structures. Figs. 1b, 1d and 1f, display the dependence current density, shot noise and Fano Factor as a function of magnetic field when the magnetic moments of two DMS layers are antiparallel in the presence of DSOC effect. In this case (θ = π), J↓↓ (S↓↓ ) and J↑↑ (S↑↑ ) are zero, and the spin-flip effect is very important. The origin of this effect can be understood by the effect of angular dependence of the magnetizations in the DMSs, which changes the spin of the transmitted electron spin through the system. These results indicate the DSOC and the magnitude of external magnetic field are the main factors that determine the Fano factors of the spin-polarized electrons in these structures. Fig. 2 shows the Dresselhaus strength dependence of the spin-dependent current density in a typical double barrier DMS heterostructure. It can be seen that the spin transport strongly depends on the spin orientation of tunneling electrons. In Fig. 2(a) the curve was plotted when the spins of electrons were conserved during tunneling process (θ = 0). When the Dresselhaus strength increases the current density significantly increases for spin down electrons that tunnel from left electrode into right electrode, while it obviously decreases for spin up ones. The difference of the current density between two different spin electrons can reach up to several orders of magnitude. An increasing Dresselhaus strength makes the quantum barrier higher and higher (smaller and smaller) For spin-up (spin-down) electrons. This figure shows the importance of Dresselhaus effect on spin filtering in DMS/NMS structure. Fig. 2(b) displays the dependence of the current density as a function of Dresselhaus strength in the heterostructure when the magnetic moments configuration of two DMS layers are antiparallel (θ = π). In this case (θ = π), J↓↓ and J↑↑ are zero, and the spin-flip effect is very important. The DSOC has remarkable influence on spin-flip effect, such that by changing the barrier height it can enhance J↓↑ and reduce J↑↓ . In Fig. 3, we also calculated the dependence of current density, shot noise and Fano Factor on the relative angle between the magnetization of two electrodes for sevral different Dresselhaus constants. In Figs. 3a, 3c and 3e, the features were plotted when the spins of tunneling electrons during tunneling process are conserved, namely, the cases for (↓↓) and (↑↑). These curves show that the lowest values of J↓↓ (J↑↑ ), S↓↓ (S↑↑ ) and also the highest values of F↓↓ (F↑↑ ) are obtained in θ = π. When the magnetic moments of two DMS electrodes are antiparallel (θ = π), the current densities and shot noises are zero, therefore the spin-flip effect is very important. The results show that the highest values of J↓↓ (J↑↑ ), S↓↓ (S↑↑ ) and also the lowest values of F↓↓ (F↑↑ ) are obtained in (θ = 0). It 6
means that the spin-filtering effect is very important. Further, the difference of the spindependent quantities between two different spin electrons is enhanced for increasing the Dresselhaus constant. The curves in Figs. 3b, 3d, 3f were plotted when the spin orientation of carrier is switched from 0 to π, namely, (↓↑) and (↑↓) cases. Here the highest values of J↓↑ (J↑↓ ), S↓↑ (S↑↓ ) and the lowest values of F↓↑ (F↑↓ ) are obtained in θ = π. When the magnetic moments of two DMS electrodes are parallel (θ = 0), the current densities and shot noises are zero and therefore there is no spin-flip effect. For other angles, all components of current densities and shot noises change and become nonzero. Continuing to increase the angle from θ = π to 2π, the variation of the physical quantities show similar behavior as that from θ = 0 to π. This figure indicates that the DSOC has significant effects on the spin-polarized transport for both spin-up electrons and spin-down ones in every relative angle between the magnetizations of two DMS layers.
4
Conclusions
The effects of DSOC on the current density and shot noise in DMS/NMS heterostructures have been investigated. Our calculations are based on the transfer-matrix method and the effective mass approach in the presence of an external magnetic field and also an applied voltage. In this model, the spins of electrons are not supposed to be conserved during tunneling, that is, the tunneling of the spin-up and spin-down electrons are not an independent process. The results show that the DSOC effect has great different influence on the current density, the shot noise and the Fano factor for all components. Also, with considering the results of this paper and the previous work [24], we find that not only RSOC but also DSOC has remarkable influence on spin filtering and spin flipping effect in these structures. It is found that the spin polarization can be controlled very efficiently by taking into account the SOC effect and including the angle relative magnetization of the DMS barriers. The results obtained in this work may be a base for the development of future spintronics devices.
5
Acknowledgements
We would like to thank the Islamic Azad University (East Tehran Branch) for supporting us with a grant.
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9
Figure 1
0
a
b
1.0
0.6
0.8 0.5
-21
J/10 J0
0.6
0.4
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.4
0.2
0.0
c
0.3
0
1
2
3
4
5
0.2
0
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
1
2
3
4
5
3
4
5
d 0.35
0.30
0.30
0.25
S/2e
0.20
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.25
0.20
0.15 0.15 0.10 0.05
0
1
2
3
4
5
e
0
1
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
2
f 0.7
0.65
0.6
Fano Factor
0.10
0.5
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.60
0.55
0.50
0.4
0.3
0
1
2
3
4
5
0.45
0
1
2
3
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
4
5
B(T)
B(T)
Figure 1: Spin-dependent current densities, shot noises and Fano factors as a function of magnetic field with different Dresselhaus constants γ.
10
Figure 2 a 1.2
0 1.0
-21
J/10 J0
0.8
0.6 0.4 0.2 0.0
b 0.6
0.5
-21
J/10 J0
0.4 0.3 0.2 0.1 0.0
0
50
100
150
200
250
300
( eV 3 )
Figure 2: The Dresselhaus strength dependence of the spin-dependent current density in a typical double barrier DMS heterostructure. 11
Figure 3 a
b 1.0
0.5
0.8
-21
J/10 J0
0.6
c
0.4
0.3
0.4
0.2
0.2
0.1
0.0
0.0
50 eV 3 150 eV 3 50 eV 3 150 eV 3 50 eV 3 150eV 3
d
0.30
0.25 0.20
0.25
3
0 eV 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.15
0.20 0.15 S/2e
S/2e
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.10
0.10
0.05
0.05
50 eV 3 150 eV 3 50 eV 3 150 eV 3 50 eV 3 150eV 3
50 eV 3 150 eV 3 50 eV 3 150 eV 3 50 eV 3 150eV 3
0.00
0.00
e
f
1.0
0.75
0.8 0.70 Fano Factor
0.6
0.65
0.4
0.2
0 eV 3 50 eV 3 150 eV 3 0 eV 3 50 eV 3 150 eV 3
0.60 0.55 0.50
0.0 0.0
0.5
1.0
1.5
2.0
/
0.0
0.5
1.0
1.5
2.0
/
Figure 3: Variation of the tunneling current densities, shot noise and Fano factor on the relative angle between the magnetization of two DMS layers. 12