- Email: [email protected]
Properties of the resonant tunneling diode in external magnetic field with inclusion of the Rashba effect
Solid State Communications 189 (2014) 52–57
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Properties of the resonant tunneling diode in external magnetic field with inclusion of the Rashba effect Nemanja Niketić, Vitomir Milanović, Jelena Radovanović n School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia
art ic l e i nf o
a b s t r a c t
Article history: Received 27 January 2014 Received in revised form 27 March 2014 Accepted 28 March 2014 by E.Y. Andrei Available online 2 April 2014
Influence of the Rashba effect on electronic properties of resonant tunneling diode in an external magnetic field is analyzed in this paper. Wave functions and energies, as well as expressions for currents densities, are determined for electrons of both spins. Appearances of many modes due to the external magnetic field induce irregularities in the current–voltage characteristics, which are observable in case when the thermal energy is lower than, or comparable to, the energy difference of two consecutive Landau levels. Current density through the heterostructure is investigated with emphasis on the degree of spin polarization; further, spin transfer is shown to depend on the direction of external magnetic field. & 2014 Elsevier Ltd. All rights reserved.
Keywords: A. Resonant tunneling diode D. Perpendicular magnetic field D. Rashba effect D. Transmission
1. Introduction Spintronics presents one of the fastest-growing and most attractive fields of modern science [1]. The reason to believe that spintronics will become an important branch of the future technology lies in specific advantages of spin over charge of electron. Some of these are increased processing speed, lower power consumption and increased device density on a chip. Further, experiments prove that single electron can exist much longer in coherent superposition of two orthogonal spin states than it is the case with electron charge. This fact is of importance in quantum computing, where spin states could be used to store qubits. Therefore it is of high interest to examine and investigate spin related processes in nanoelectronic structures and exploit those results in practical usage (as in the case of giant magnetoresistance that is now in everyday use in today's computers). The Rashba effect has been studied in multiple quantum well structures, quantum wires [2] as well as in nonmagnetic heterostructures [3–11]. Interesting calculations have been done in order to investigate tunneling times in quantum wells in the presence of the Rashba effect [12,13]. The authors of Ref. [9] consider Rashba spin–orbit constant by using weak antilocalization analysis, and find that these values have a strong correlation with the degree of inversion asymmetry of the quantum wells. In particular, theoretical predictions are in good agreement with experimental findings. Spin-
n
Corresponding author. Tel.: þ 381 113370088. E-mail address: [email protected] (J. Radovanović).
http://dx.doi.org/10.1016/j.ssc.2014.03.021 0038-1098/& 2014 Elsevier Ltd. All rights reserved.
filter device based on the Rashba effect using a nonmagnetic resonant tunneling diode has been analyzed in [10]. It entails a symmetric structure, comprising, apart from the emitter and collector, two quantum wells and a middle barrier. The paper provides a detailed analysis of the described structure in the presence of the Rashba effect, as well as the procedure for evaluating the current between the emitter and the collector and the polarization. The paper [11] deals with InGaAs/GaAsSb based resonant tunneling diode in the presence of magnetic field parallel to the planes of the layers. In-depth theoretical study of the output characteristics (emitter– collector current, and spin polarization) together with a temperature dependence of these quantities is presented. In this paper we focus on theoretical investigation of the resonant tunneling diode, which comprises a quantum well and two barriers with different heights, in addition to the emitter and collector. In contrast to symmetric structures considered in [10,11], taking into account the asymmetry of the structural profile here leads to the dependence of output properties on the direction of the magnetic field (which is considered to be reversible), applied perpendicularly to the planes of the layers. We present in detail a general procedure for evaluating the spin-dependent current and spin-polarization for the case of perpendicular magnetic field, and describe the possibility of modifying these properties by switching the orientation of the field. Similar procedure has been given in Refs. [10,11], but it is expanded here to include application of the normal magnetic field and the Zeeman effect – although it is not dominant in our case. Barriers of resonant tunneling diode (RTD) are made of GaAs and A1As, while emitter and collector regions, as well as quantum well regions, are made of InAs, Fig. 1.
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
53
oscillator, j is the index of the Landau level and ηj7 ðzÞ is the wave functions for spin up and spin down electrons as a conse^ SIA in quence of the Zeeman splitting. It is convenient to present H þ ^ SIA ¼ the form of creation and annihilation operators a^ and a^ : H pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 0 ia^ 0 , AðzÞ ¼ ð 2=lb ÞαðzÞ where lb ¼ ℏ=eB. AðzÞ þ ia^
The assumed solution to Eq. (1) would read [16]: 2 3 C jþ ηjþ ðzÞ eikx x þ 1 5 Ψ ¼ pffiffiffiffiffi ∑ Φj ðyÞ4 C j ηj ðzÞ Lx j ¼ 0
Fig. 1. (Color online) The schematics of the resonant tunneling heterostructure.
2. Theoretical consideration Spin–orbit interaction that arises as a consequence of the structural inversion asymmetry in a semiconductor heterostructure under consideration is called the Rashba effect. In particular, an external electric field applied to the RTD bends the potential profile and the structural inversion asymmetry arises. Apart from external influences, asymmetry could emerge as an intrinsic property of the material itself (the Dresselhaus effect) which will not be considered here. Since the bulk properties remain the same, the Rashba effect is felt only inside the heterostructure. Our analysis is based on the Schrö dinger equation for the envelope wave functions: ^ 0 UI þH ^ g þH ^ SIA ÞΨ ¼ εΨ ðH
ð1Þ
^ 0 ¼ ð1=2mn ðzÞÞðp^ þ eByÞ2 þð1=2mn ðzÞp^ 2 Þ þ p^ ð1=2mn ðzÞÞp^ þ where H x z z y ! ! UðzÞ, Landau gauge A ¼ By e x is used to describe the influence 0 ^ g ¼ gðzÞ μ B=2 1 of magnetic field [14]; H provides splitB 0 1 ting of spin states in an external magnetic field [15]; " # 0 ik ^ SIA ¼ αðzÞ is the Rashba Hamiltonian. Furthermore H ik þ 0 αðzÞ is the Rashba parameter, and in the presence of magnetic field we have k 7 ¼ ðkx ðeBy=ℏÞÞ 7 iky . The Rashba parameter and the Landé g factor are given by the following expressions: ℏ2 Ep ðzÞ d 1 1 αðzÞ ¼ 4m0 dz E Ec ðzÞ VðzÞ þ Eg ðzÞ E Ec ðzÞ VðzÞ þ Eg ðzÞ þ ΔðzÞ ð2Þ Ep ðzÞ 1 1 gðzÞ ¼ 2 1 þ 3 E Ec ðzÞ VðzÞ þ Eg ðzÞ þ ΔðzÞ E Ec ðzÞ VðzÞ þ Eg ðzÞ
ð3Þ where Ep(z) is the momentum matrix element, Ec(z) is the bottom of the conduction band, V(z) is the potential energy due to applied voltage, Eg(z) is the band gap between the conduction and the valence band, and Δ(z) is the spin–orbit gap. The solution of Eq. (1) determines the electronic structure of the resonant tunneling diode (energies and wave functions). In order to find the wave function Ψ, we will assume it in the form of a superposition of the wave functions determined from the following equation: ^ 0 UI þH ^ g ÞΨ ¼ EΨ ðH
ð4Þ "
e ffiffiffiffi where Ψ ¼ p Φj ðyÞ ikx x
Lx
ηjþ ðzÞ ηj ðzÞ
# , Lx is the structure length in the
x-direction, Φj(y) is the wave functions of the linear harmonic
ð5Þ
The constants C jþ and C j are determined by inserting Eq. (5) into Eq. (1). Here we use the fact that Ψ is constructed in the basis ^ 0g Hamiltonian ðH ^ 0g ¼ H ^ 0 UI þH ^ g Þ, where E þ of eigenfunctions of H j and Ej are the energies of spin up and spin down states respectively without the Rashba effect. Therefore we obtain the next system of equations: þ1
þ1
þ1
j¼0
j¼0
j¼0
^ þ ∑ Φj η þ C þ þ ia^ A ∑ Φj η C ¼ ε ∑ Φj η þ C þ H j j j j 0g j j þ1
þ1
þ1
j¼0
j¼0
j¼0
þ ^ ∑ Φj η C ¼ ε ∑ Φj η C ia^ A ∑ Φj ηjþ C jþ þ H j j 0g j j
ð6Þ
ð7Þ
^ 7 Φj η 7 ¼ E 7 Φj η 7 and for For the basic functions we have: H 0g j j j pffiffiffiffiffiffiffiffiffi þ creation and annihilation operators: a^ Φj ¼ jþ 1Φj þ 1 and pffi a^ Φj ¼ jΦj 1 respectively. Taking these into account, Eqs. (6) and (7) could be transformed into a more practical form: þ1
þ1
j¼0
j¼0
∑ ðEjþ εÞΦj ηjþ C jþ þiA ∑ þ1
iA ∑
j¼0
pffi jΦj 1 ηj C j ¼ 0
þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi j þ 1Φj þ 1 ηjþ C jþ þ ∑ ðEj εÞΦj ηj C j ¼ 0
ð8Þ
ð9Þ
j¼0
Multiplying Eqs. (8) and (9) by appropriate complex conjugate wave functions and taking an integral over the whole domain, we obtain pffiffiffiffiffiffiffiffiffi ð10Þ ðEjþ εÞC jþ þi jþ 1Ajþ C jþ 1 ¼ 0 pffi ð11Þ i jAj C jþ 1 þ ðEj εÞC j ¼ 0 R R where Ajþ ¼ z ηjþ 1 AðzÞðηjþ Þn dz and Aj ¼ z ηjþ 1 AðzÞðηj Þn dz. If we substitute s¼ jþ1 in Eq. (10), we obtain a system of two equations with two unknown constants C sþ 1 and C s : pffiffi ðEsþ 1 εÞC sþ 1 þ i sAsþ 1 C s ¼ 0 ð12Þ pffiffi i sAs C sþ 1 þ ðEs εÞC s ¼ 0
ð13Þ
The only unpaired equation left is given by writing Eq. (11) for j ¼ 0, from which we get ε ¼ E0 . This solution corresponds to the first Landau level, with the absence of spin–orbit interaction. The 0 x eikxffiffiffi ffiΦ0 ðyÞ η ðzÞ. wave function of this state is Ψ 0 ¼ p Lx 1 0 For sZ 1, solution to the system (12) and (13) is " # iDsþ Φs 1 ðyÞηsþ 1 ðzÞ eikx x 1 Ψ sþ ¼ pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Φs ðyÞηs ðzÞ Lx 1 þ D þ 2 s
ð14Þ
" # Φs 1 ðyÞηsþ 1 ðzÞ eikx x 1 ð15Þ Ψ s ¼ pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lx 1 þ D 2 iDs Φs ðyÞηs ðzÞ s pffiffi pffiffi where Dsþ ¼ ð sAsþ 1 Þ=ðEsþ 1 εsþ Þ and Ds ¼ ð sAs Þ=ðεs Es Þ. The appropriate energy values that correspond to Ψ sþ and Ψ s
54
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
are given by the next expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Esþ 1 þ Es 7 ðEsþ 1 Es Þ2 þ 4sAsþ 1 As 7 εs ¼ 2
ð16Þ
According to Eqs. (14) and (15) we see that each energy level is populated with electrons of both spins. The mixing of spin states is a consequence of the Rashba effect, and it is present only inside the structure. In case of the opposite direction of the magnetic field, energy values are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ E þ E 7 ðEs 1 Esþ Þ2 þ 4sAs 1 As s 1 s εs7 ¼ ð17Þ 2 R þ R þ n n where As ¼ z ηs þ 1 AðzÞðηs Þ dz and As ¼ z ηs 1 AðzÞðηsþ Þ dz. The corresponding wave functions are of the same form as in Eqs. (14) pffiffi þ and (15) but with different coefficients: Dsþ ¼ sAs 1 = Es 1 εsþ pffiffi þ and Ds ¼ sAs = εs Es . The unpaired solution without spin– orbit interaction has the energy ε ¼ E0þ , while the corresponding 0 þ x eikxffiffiffi ffiΦ0 ðyÞ wave function of this state is Ψ 0 ¼ p η ðzÞ. Lx 1 0 2.1. Current density In the previous section we have determined the electronic structure and now we are able to derive expressions for current density of different states described by Eqs. (14) and (15). Inside the resonant tunneling diode we have mixing of spin currents, which can be given by þ J up ¼ J up þ J up
ð18Þ
þ þ J down J down ¼ J dowm
ð19Þ
Each current density is determined by the transmission probability, the Fermi–Dirac distributions and the probability for electron to be in the state defined by s, kx and kz. Therefore the current density for spin up electrons in ε þ state is given by the following expression: þ J up ¼e
∑
s;kx ;kz 4 0
þ þ νz ðkz ÞT up ðkz Þff f d ðεsþ ; EF Þ f f d ðεsþ ; EF eUÞgjΨ ups;k j2 x ;kz
ð20Þ The velocity of electron in the z-direction, vz(kz), is given by vz ðkz Þ ¼
1 ∂εsþ 1 ∂εsþ ∂Ez ¼ ℏ ∂kz ℏ ∂Ez ∂kz
ð21Þ
2 ℏ2 kz
We use a parabolic model where Ez ¼ ulus squared to the wave function amounts to 0 12 þ jΨ ups;k j2 ¼ x ;kz
=2mn . The mod-
1 B Dsþ C 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Φs 1 ðyÞ Lx Lz þ2 1 þ Ds
ð22Þ
where Lz is the structure length in the z-direction. By following a standard procedure and changing from summation to integration, we obtain the expression for current density in a more appropriate form: 0 12 þ J up ¼
e ∂ε þ ∂Ez þ Dsþ B C 2 ∑ ∬kx ;kz s T up ðkz ÞFD þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Φs 1 ðyÞ dkx dkz 2 ∂Ez ∂kz 4π ℏ s þ2 1 þ Ds
where J 0 ¼ e2 = 4π 2 ℏ2 and FD þ ¼ f f d ðεsþ ; EF Þ f f d ðεsþ ; EF eUÞ. The three remaining current densities are derived in a similar manner. The appropriate expressions are
¼ J 0 B∑ J up
0
Z
s
þ1 0
þ ¼ J 0 B∑ J down
Z
s
¼ J 0 B∑ J down
0 þ1
0
Z
s
12
∂εs 1 B C T ðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz ∂Ez up 1 þDs 2 12
∂εsþ þ 1 B C T ðEz ÞFD þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz ∂Ez down 1 þ Dsþ 2 0
þ1
0
ð24Þ
ð25Þ
12
∂εs Ds B C T down ðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A dEz ∂Ez 1 þ Ds 2
ð26Þ
As already pointed out, an external electric field applied to the semiconductor heterostructure bends the potential profile and asymmetry inside the structure arises. Outside the analyzed structure energy levels stay unchanged and therefore the Rashba parameter will be zero in the bulk material. Since Dsþ and Ds are þ directly proportional to α parameter we can conclude that J up and J down are zero outside the structure.
2.2. Approximation of constant parameters and the numerical results In this paper we present numerical analysis in case of constant parameters inside the heterostructure. All parameters are taken to be equal to the ones corresponding to InAs, Table 1. According to Eqs. (2) and (3) the Rashba parameter α and the Landé g factor are energy and coordinate dependent. In our approximation the Landé g factor is calculated for zero energy and zero voltage, while the Rashba parameter α is calculated for zero energy and averaged over the coordinate for each applied voltage. For constant values of effective mass and Landé g factor we obtain a much simpler form for energies: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ℏωc gμB B2 2sα2 7 εs ¼ Ez þ ðs 1Þℏωc 7 t ð27Þ þ 2 2 l B
while the expressions for electron wave functions remain formally the same, just with different coefficients pffiffiffi pffiffi pffiffi Dsþ ¼ sA= Esþ 1 εsþ and Ds ¼ sA= εs Es ; A ¼ α 2=lB ; n n ωc ¼ eB=m . Here, m is the effective mass at the bottom of the conduction band in InAs, while the z-part of the wave function becomes independent of the Landau level s. The energies in the absence of the Rashba effect are 1 gμ B Es7 ¼ Ez þ s ℏωc 7 B ð28Þ 2 2 The appropriate expressions for current densities read
J up ¼ J 0 B∑ s
0
Z
þ1 0
12
1 B C TðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz 1 þ Ds 2
ð29Þ
ð23Þ After evaluating the integral over kx we obtain 0 12 Z þ1 þ ∂εs þ Dsþ C þ þB J up ¼ J 0 B∑ T ðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz ∂Ez up þ2 s 0 1 þ Ds
Table 1 Values of parameters used in numerical calculations, which are taken to be constant and the same as in InAs. Parameter Value
mn 0:023m0
g 14.83
Ep 22.2 eV
Ec ðzÞ 0 eV
Eg 0.418 eV
D 0.38 eV
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
þ J down
0
Z ¼ J 0 B∑ s
þ1 0
12
þB
1 C TðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz þ2 1 þDs
ð30Þ
Since we are using the constant parameters approximation, we have η 7 ðz; sÞ ¼ ηðzÞ, and therefore the transmission coefficients for each electron have the same shape (regardless of the spin value) and are only shifted in energy position. In case of the opposite direction of the magnetic field, the values of energies become vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ℏωc þgμB B2 2sα2 7 ð31Þ εs ¼ Ez þ ðs 1Þℏωc 7 t þ 2 2 l B
while the expressions for current densities remain formally pffiffi same as Eqs. (29) and (30), with changes in Dsþ ¼ sA= Es 1 εsþ pffiffi and Ds ¼ sA= εs Esþ . Next, we will present numerical results obtained on the basis of the theoretical model developed in the previous sections, by using the constant parameters approximation. In Fig. 2 we observe current–voltage characteristics for both spins at low temperatures, where we notice step-like features of the current density. At very low temperatures where the electron thermal energy (Ekt) is much smaller than the energy difference between two consecutive Landau levels (ΔEB), quantization caused by the magnetic field is observable in the current–voltage dependence. Current density is primarily determined by the electron transmission coefficient and therefore this behavior of current–voltage characteristics is best explained on that basis. In Fig. 3 we see the
55
overlap of transmission coefficient with the Fermi–Dirac (FD) function for different voltages. If magnetic field is applied, the electron states are characterized by a quantum number s. In this case, the electron wave vector depends on the magnetic field: kz ¼ kz ðε; ωc ; sÞ. By increasing the quantum number s, transmission coefficient is shifted to higher energies. The applied electric field has qualitatively opposite influence and shifts the transmission coefficient towards lower energies. As temperatures increase, the FD function ceases to have a steplike character, and therefore different Landau levels start to overlap with it. That is why at higher temperatures there is no appearance of different modes on current–voltage characteristics, Fig. 4. As a combined consequence of the Zeeman and the Rashba effects, we observe differences in spin up and spin down currents, Fig. 4. If we define a degree of polarization as P ¼ J u J d = J u þ J d , we are able to assess the percentage of each type of current for different applied voltages. The direction of magnetic field dictates the spin of electrons that is “filtered”; if magnetic field is in the direction of electron propagation, the spin up electrons are predominantly transmitted. For the opposite direction of magnetic field, the total current density has more electrons with spin down. In Fig. 5(a and b) we observe the polarization at high temperatures. The oscillatory character is preserved even at high temperatures where Ekt⪢EB. The polarization maximum is 21% for spin down and 22% for electrons with spin up. Optimization at low temperatures yields better results, where best achieved polarization is 58% for spin down and 57% for spin up, Fig. 5(c and d). In Fig. 6 we illustrate the current–voltage dependence for both positive and negative voltages.
Fig. 2. (Color online) Current–voltage characteristics of spin up and spin down electrons for B¼5 T at low temperatures (T ¼ 1 K): (a) taking three Landau levels and (b) taking two Landau levels.
Fig. 3. (Color online) The transmission coefficient and FD function at T ¼ 1 K and B¼ 5 T: (a) s ¼ 1 and (b) s ¼ 2.
56
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
Fig. 4. (Color online) Spin up and spin down currents for the two opposite directions of magnetic field at high temperatures (T ¼ 300 K) and B ¼5 T.
Fig. 5. (Color online) Polarization for B¼ 5 T at different temperatures. (a,b) For T1 ¼1 K, the achieved polarization is up to 22% for spin up and 21% for spin down electrons. (c,d) For T2 ¼300 K, the highest polarization for spin up electrons is 58%, while for spin down it amounts to 57%.
Fig. 6. (Color online) The current density and polarization for positive and negative voltages at low temperatures, for B¼ 5 T.
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
57
3. Conclusions
References
In this paper we have provided a detailed theoretical analysis, together with numerical results, of current density and polarization in semiconductor heterostructure placed in an external magnetic field, with the influence of the Rashba effect included. Although the Rashba effect causes mixing of different spin states, it occurs only inside the inner layers of the structure and therefore we are able to observe the spin polarization of electrons propagating through the entire structure. By using the approximation of constant parameters we obtained results for current densities of spin up and spin down electrons, as well as for the degree of electron spin polarization. Different signs of polarization could be obtained by switching the direction of applied magnetic field. At very low temperatures, when the thermal energy is much lower than the difference between two consecutive Landau levels, we observe step-like features in the current–voltage curve. Even at room temperature, the quantization due to the presence of magnetic field is observable in figures of spin polarization.
[1] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, I. Zutic, Acta Phys. Slovaca 57 (2007) 565–907. [2] R. Khordad, J. Lumin. 134 (2013) 201–207. [3] G. Isic, D. Indjin, V. Milanovic, J. Radovanovic, Z. Ikonic, P. Harrison, J. Nanophotonics 5 (2011) 051819. [4] G. Isic, D. Indjin, V. Milanovic, J. Radovanovic, Z. Ikonic, P. Harrison, J. Appl. Phys. 110 (2011) 064507. [5] G. Isic, D. Indjin, V. Milanovic, J. Radovanovic, Z. Ikonic, P. Harrison, J. Appl. Phys. 108 (2010) 044506. [6] J. Radovanovic, G. Isic, V. Milanovic, Opt. Mater. 30 (2008) 1134–1138. [7] G. Isic, J. Radovanovic, V. Milanovic, J. Appl. Phys. 102 (2007) 123704. [8] D. Grundler, Phys. Rev. Lett. 84 (2000) 6074–6077. [9] T. Koga, J. Nitta, T. Akazaki, H. Takayanagi, Phys. Rev. Lett. 89 (2002) 046801. [10] T. Koga, J. Nitta, H. Takayanagi, Phys. Rev. Lett. 88 (2002) 126601. [11] J.S. de Sousa, J. Smoliner, Phys. Rev. B 85 (2012) 085303. [12] G. Isic, V. Milanovic, J. Radovanovic, D. Indjin, Z. Ikonic, Microelectron. J. 40 (2009) 611–614. [13] M. Eric, J. Radovanovic, V. Milanovic, Z. Ikonic, S. Indjin, J. Appl. Phys. 103 (2008) 083701. [14] A. Gharoati, R Khordad, Superlattices Microstruct. 51 (2012) 194–202. [15] David Z.-Y. Ting, X. Cartoixa, Phys. Rev. B 68 (2003) 235320. [16] XF. Wang, P. Vasilopoulos, Phys. Rev. B 72 (2005) 085344.
Acknowledgments This work was supported by the Ministry of Science (Republic of Serbia), ev. no III 45010.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Properties of the resonant tunneling diode in external magnetic field with inclusion of the Rashba effect Nemanja Niketić, Vitomir Milanović, Jelena Radovanović n School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia
art ic l e i nf o
a b s t r a c t
Article history: Received 27 January 2014 Received in revised form 27 March 2014 Accepted 28 March 2014 by E.Y. Andrei Available online 2 April 2014
Influence of the Rashba effect on electronic properties of resonant tunneling diode in an external magnetic field is analyzed in this paper. Wave functions and energies, as well as expressions for currents densities, are determined for electrons of both spins. Appearances of many modes due to the external magnetic field induce irregularities in the current–voltage characteristics, which are observable in case when the thermal energy is lower than, or comparable to, the energy difference of two consecutive Landau levels. Current density through the heterostructure is investigated with emphasis on the degree of spin polarization; further, spin transfer is shown to depend on the direction of external magnetic field. & 2014 Elsevier Ltd. All rights reserved.
Keywords: A. Resonant tunneling diode D. Perpendicular magnetic field D. Rashba effect D. Transmission
1. Introduction Spintronics presents one of the fastest-growing and most attractive fields of modern science [1]. The reason to believe that spintronics will become an important branch of the future technology lies in specific advantages of spin over charge of electron. Some of these are increased processing speed, lower power consumption and increased device density on a chip. Further, experiments prove that single electron can exist much longer in coherent superposition of two orthogonal spin states than it is the case with electron charge. This fact is of importance in quantum computing, where spin states could be used to store qubits. Therefore it is of high interest to examine and investigate spin related processes in nanoelectronic structures and exploit those results in practical usage (as in the case of giant magnetoresistance that is now in everyday use in today's computers). The Rashba effect has been studied in multiple quantum well structures, quantum wires [2] as well as in nonmagnetic heterostructures [3–11]. Interesting calculations have been done in order to investigate tunneling times in quantum wells in the presence of the Rashba effect [12,13]. The authors of Ref. [9] consider Rashba spin–orbit constant by using weak antilocalization analysis, and find that these values have a strong correlation with the degree of inversion asymmetry of the quantum wells. In particular, theoretical predictions are in good agreement with experimental findings. Spin-
n
Corresponding author. Tel.: þ 381 113370088. E-mail address: [email protected] (J. Radovanović).
http://dx.doi.org/10.1016/j.ssc.2014.03.021 0038-1098/& 2014 Elsevier Ltd. All rights reserved.
filter device based on the Rashba effect using a nonmagnetic resonant tunneling diode has been analyzed in [10]. It entails a symmetric structure, comprising, apart from the emitter and collector, two quantum wells and a middle barrier. The paper provides a detailed analysis of the described structure in the presence of the Rashba effect, as well as the procedure for evaluating the current between the emitter and the collector and the polarization. The paper [11] deals with InGaAs/GaAsSb based resonant tunneling diode in the presence of magnetic field parallel to the planes of the layers. In-depth theoretical study of the output characteristics (emitter– collector current, and spin polarization) together with a temperature dependence of these quantities is presented. In this paper we focus on theoretical investigation of the resonant tunneling diode, which comprises a quantum well and two barriers with different heights, in addition to the emitter and collector. In contrast to symmetric structures considered in [10,11], taking into account the asymmetry of the structural profile here leads to the dependence of output properties on the direction of the magnetic field (which is considered to be reversible), applied perpendicularly to the planes of the layers. We present in detail a general procedure for evaluating the spin-dependent current and spin-polarization for the case of perpendicular magnetic field, and describe the possibility of modifying these properties by switching the orientation of the field. Similar procedure has been given in Refs. [10,11], but it is expanded here to include application of the normal magnetic field and the Zeeman effect – although it is not dominant in our case. Barriers of resonant tunneling diode (RTD) are made of GaAs and A1As, while emitter and collector regions, as well as quantum well regions, are made of InAs, Fig. 1.
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
53
oscillator, j is the index of the Landau level and ηj7 ðzÞ is the wave functions for spin up and spin down electrons as a conse^ SIA in quence of the Zeeman splitting. It is convenient to present H þ ^ SIA ¼ the form of creation and annihilation operators a^ and a^ : H pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 0 ia^ 0 , AðzÞ ¼ ð 2=lb ÞαðzÞ where lb ¼ ℏ=eB. AðzÞ þ ia^
The assumed solution to Eq. (1) would read [16]: 2 3 C jþ ηjþ ðzÞ eikx x þ 1 5 Ψ ¼ pffiffiffiffiffi ∑ Φj ðyÞ4 C j ηj ðzÞ Lx j ¼ 0
Fig. 1. (Color online) The schematics of the resonant tunneling heterostructure.
2. Theoretical consideration Spin–orbit interaction that arises as a consequence of the structural inversion asymmetry in a semiconductor heterostructure under consideration is called the Rashba effect. In particular, an external electric field applied to the RTD bends the potential profile and the structural inversion asymmetry arises. Apart from external influences, asymmetry could emerge as an intrinsic property of the material itself (the Dresselhaus effect) which will not be considered here. Since the bulk properties remain the same, the Rashba effect is felt only inside the heterostructure. Our analysis is based on the Schrö dinger equation for the envelope wave functions: ^ 0 UI þH ^ g þH ^ SIA ÞΨ ¼ εΨ ðH
ð1Þ
^ 0 ¼ ð1=2mn ðzÞÞðp^ þ eByÞ2 þð1=2mn ðzÞp^ 2 Þ þ p^ ð1=2mn ðzÞÞp^ þ where H x z z y ! ! UðzÞ, Landau gauge A ¼ By e x is used to describe the influence 0 ^ g ¼ gðzÞ μ B=2 1 of magnetic field [14]; H provides splitB 0 1 ting of spin states in an external magnetic field [15]; " # 0 ik ^ SIA ¼ αðzÞ is the Rashba Hamiltonian. Furthermore H ik þ 0 αðzÞ is the Rashba parameter, and in the presence of magnetic field we have k 7 ¼ ðkx ðeBy=ℏÞÞ 7 iky . The Rashba parameter and the Landé g factor are given by the following expressions: ℏ2 Ep ðzÞ d 1 1 αðzÞ ¼ 4m0 dz E Ec ðzÞ VðzÞ þ Eg ðzÞ E Ec ðzÞ VðzÞ þ Eg ðzÞ þ ΔðzÞ ð2Þ Ep ðzÞ 1 1 gðzÞ ¼ 2 1 þ 3 E Ec ðzÞ VðzÞ þ Eg ðzÞ þ ΔðzÞ E Ec ðzÞ VðzÞ þ Eg ðzÞ
ð3Þ where Ep(z) is the momentum matrix element, Ec(z) is the bottom of the conduction band, V(z) is the potential energy due to applied voltage, Eg(z) is the band gap between the conduction and the valence band, and Δ(z) is the spin–orbit gap. The solution of Eq. (1) determines the electronic structure of the resonant tunneling diode (energies and wave functions). In order to find the wave function Ψ, we will assume it in the form of a superposition of the wave functions determined from the following equation: ^ 0 UI þH ^ g ÞΨ ¼ EΨ ðH
ð4Þ "
e ffiffiffiffi where Ψ ¼ p Φj ðyÞ ikx x
Lx
ηjþ ðzÞ ηj ðzÞ
# , Lx is the structure length in the
x-direction, Φj(y) is the wave functions of the linear harmonic
ð5Þ
The constants C jþ and C j are determined by inserting Eq. (5) into Eq. (1). Here we use the fact that Ψ is constructed in the basis ^ 0g Hamiltonian ðH ^ 0g ¼ H ^ 0 UI þH ^ g Þ, where E þ of eigenfunctions of H j and Ej are the energies of spin up and spin down states respectively without the Rashba effect. Therefore we obtain the next system of equations: þ1
þ1
þ1
j¼0
j¼0
j¼0
^ þ ∑ Φj η þ C þ þ ia^ A ∑ Φj η C ¼ ε ∑ Φj η þ C þ H j j j j 0g j j þ1
þ1
þ1
j¼0
j¼0
j¼0
þ ^ ∑ Φj η C ¼ ε ∑ Φj η C ia^ A ∑ Φj ηjþ C jþ þ H j j 0g j j
ð6Þ
ð7Þ
^ 7 Φj η 7 ¼ E 7 Φj η 7 and for For the basic functions we have: H 0g j j j pffiffiffiffiffiffiffiffiffi þ creation and annihilation operators: a^ Φj ¼ jþ 1Φj þ 1 and pffi a^ Φj ¼ jΦj 1 respectively. Taking these into account, Eqs. (6) and (7) could be transformed into a more practical form: þ1
þ1
j¼0
j¼0
∑ ðEjþ εÞΦj ηjþ C jþ þiA ∑ þ1
iA ∑
j¼0
pffi jΦj 1 ηj C j ¼ 0
þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi j þ 1Φj þ 1 ηjþ C jþ þ ∑ ðEj εÞΦj ηj C j ¼ 0
ð8Þ
ð9Þ
j¼0
Multiplying Eqs. (8) and (9) by appropriate complex conjugate wave functions and taking an integral over the whole domain, we obtain pffiffiffiffiffiffiffiffiffi ð10Þ ðEjþ εÞC jþ þi jþ 1Ajþ C jþ 1 ¼ 0 pffi ð11Þ i jAj C jþ 1 þ ðEj εÞC j ¼ 0 R R where Ajþ ¼ z ηjþ 1 AðzÞðηjþ Þn dz and Aj ¼ z ηjþ 1 AðzÞðηj Þn dz. If we substitute s¼ jþ1 in Eq. (10), we obtain a system of two equations with two unknown constants C sþ 1 and C s : pffiffi ðEsþ 1 εÞC sþ 1 þ i sAsþ 1 C s ¼ 0 ð12Þ pffiffi i sAs C sþ 1 þ ðEs εÞC s ¼ 0
ð13Þ
The only unpaired equation left is given by writing Eq. (11) for j ¼ 0, from which we get ε ¼ E0 . This solution corresponds to the first Landau level, with the absence of spin–orbit interaction. The 0 x eikxffiffiffi ffiΦ0 ðyÞ η ðzÞ. wave function of this state is Ψ 0 ¼ p Lx 1 0 For sZ 1, solution to the system (12) and (13) is " # iDsþ Φs 1 ðyÞηsþ 1 ðzÞ eikx x 1 Ψ sþ ¼ pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Φs ðyÞηs ðzÞ Lx 1 þ D þ 2 s
ð14Þ
" # Φs 1 ðyÞηsþ 1 ðzÞ eikx x 1 ð15Þ Ψ s ¼ pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lx 1 þ D 2 iDs Φs ðyÞηs ðzÞ s pffiffi pffiffi where Dsþ ¼ ð sAsþ 1 Þ=ðEsþ 1 εsþ Þ and Ds ¼ ð sAs Þ=ðεs Es Þ. The appropriate energy values that correspond to Ψ sþ and Ψ s
54
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
are given by the next expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Esþ 1 þ Es 7 ðEsþ 1 Es Þ2 þ 4sAsþ 1 As 7 εs ¼ 2
ð16Þ
According to Eqs. (14) and (15) we see that each energy level is populated with electrons of both spins. The mixing of spin states is a consequence of the Rashba effect, and it is present only inside the structure. In case of the opposite direction of the magnetic field, energy values are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ E þ E 7 ðEs 1 Esþ Þ2 þ 4sAs 1 As s 1 s εs7 ¼ ð17Þ 2 R þ R þ n n where As ¼ z ηs þ 1 AðzÞðηs Þ dz and As ¼ z ηs 1 AðzÞðηsþ Þ dz. The corresponding wave functions are of the same form as in Eqs. (14) pffiffi þ and (15) but with different coefficients: Dsþ ¼ sAs 1 = Es 1 εsþ pffiffi þ and Ds ¼ sAs = εs Es . The unpaired solution without spin– orbit interaction has the energy ε ¼ E0þ , while the corresponding 0 þ x eikxffiffiffi ffiΦ0 ðyÞ wave function of this state is Ψ 0 ¼ p η ðzÞ. Lx 1 0 2.1. Current density In the previous section we have determined the electronic structure and now we are able to derive expressions for current density of different states described by Eqs. (14) and (15). Inside the resonant tunneling diode we have mixing of spin currents, which can be given by þ J up ¼ J up þ J up
ð18Þ
þ þ J down J down ¼ J dowm
ð19Þ
Each current density is determined by the transmission probability, the Fermi–Dirac distributions and the probability for electron to be in the state defined by s, kx and kz. Therefore the current density for spin up electrons in ε þ state is given by the following expression: þ J up ¼e
∑
s;kx ;kz 4 0
þ þ νz ðkz ÞT up ðkz Þff f d ðεsþ ; EF Þ f f d ðεsþ ; EF eUÞgjΨ ups;k j2 x ;kz
ð20Þ The velocity of electron in the z-direction, vz(kz), is given by vz ðkz Þ ¼
1 ∂εsþ 1 ∂εsþ ∂Ez ¼ ℏ ∂kz ℏ ∂Ez ∂kz
ð21Þ
2 ℏ2 kz
We use a parabolic model where Ez ¼ ulus squared to the wave function amounts to 0 12 þ jΨ ups;k j2 ¼ x ;kz
=2mn . The mod-
1 B Dsþ C 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Φs 1 ðyÞ Lx Lz þ2 1 þ Ds
ð22Þ
where Lz is the structure length in the z-direction. By following a standard procedure and changing from summation to integration, we obtain the expression for current density in a more appropriate form: 0 12 þ J up ¼
e ∂ε þ ∂Ez þ Dsþ B C 2 ∑ ∬kx ;kz s T up ðkz ÞFD þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Φs 1 ðyÞ dkx dkz 2 ∂Ez ∂kz 4π ℏ s þ2 1 þ Ds
where J 0 ¼ e2 = 4π 2 ℏ2 and FD þ ¼ f f d ðεsþ ; EF Þ f f d ðεsþ ; EF eUÞ. The three remaining current densities are derived in a similar manner. The appropriate expressions are
¼ J 0 B∑ J up
0
Z
s
þ1 0
þ ¼ J 0 B∑ J down
Z
s
¼ J 0 B∑ J down
0 þ1
0
Z
s
12
∂εs 1 B C T ðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz ∂Ez up 1 þDs 2 12
∂εsþ þ 1 B C T ðEz ÞFD þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz ∂Ez down 1 þ Dsþ 2 0
þ1
0
ð24Þ
ð25Þ
12
∂εs Ds B C T down ðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A dEz ∂Ez 1 þ Ds 2
ð26Þ
As already pointed out, an external electric field applied to the semiconductor heterostructure bends the potential profile and asymmetry inside the structure arises. Outside the analyzed structure energy levels stay unchanged and therefore the Rashba parameter will be zero in the bulk material. Since Dsþ and Ds are þ directly proportional to α parameter we can conclude that J up and J down are zero outside the structure.
2.2. Approximation of constant parameters and the numerical results In this paper we present numerical analysis in case of constant parameters inside the heterostructure. All parameters are taken to be equal to the ones corresponding to InAs, Table 1. According to Eqs. (2) and (3) the Rashba parameter α and the Landé g factor are energy and coordinate dependent. In our approximation the Landé g factor is calculated for zero energy and zero voltage, while the Rashba parameter α is calculated for zero energy and averaged over the coordinate for each applied voltage. For constant values of effective mass and Landé g factor we obtain a much simpler form for energies: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ℏωc gμB B2 2sα2 7 εs ¼ Ez þ ðs 1Þℏωc 7 t ð27Þ þ 2 2 l B
while the expressions for electron wave functions remain formally the same, just with different coefficients pffiffiffi pffiffi pffiffi Dsþ ¼ sA= Esþ 1 εsþ and Ds ¼ sA= εs Es ; A ¼ α 2=lB ; n n ωc ¼ eB=m . Here, m is the effective mass at the bottom of the conduction band in InAs, while the z-part of the wave function becomes independent of the Landau level s. The energies in the absence of the Rashba effect are 1 gμ B Es7 ¼ Ez þ s ℏωc 7 B ð28Þ 2 2 The appropriate expressions for current densities read
J up ¼ J 0 B∑ s
0
Z
þ1 0
12
1 B C TðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz 1 þ Ds 2
ð29Þ
ð23Þ After evaluating the integral over kx we obtain 0 12 Z þ1 þ ∂εs þ Dsþ C þ þB J up ¼ J 0 B∑ T ðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz ∂Ez up þ2 s 0 1 þ Ds
Table 1 Values of parameters used in numerical calculations, which are taken to be constant and the same as in InAs. Parameter Value
mn 0:023m0
g 14.83
Ep 22.2 eV
Ec ðzÞ 0 eV
Eg 0.418 eV
D 0.38 eV
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
þ J down
0
Z ¼ J 0 B∑ s
þ1 0
12
þB
1 C TðEz ÞFD @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA dEz þ2 1 þDs
ð30Þ
Since we are using the constant parameters approximation, we have η 7 ðz; sÞ ¼ ηðzÞ, and therefore the transmission coefficients for each electron have the same shape (regardless of the spin value) and are only shifted in energy position. In case of the opposite direction of the magnetic field, the values of energies become vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ℏωc þgμB B2 2sα2 7 ð31Þ εs ¼ Ez þ ðs 1Þℏωc 7 t þ 2 2 l B
while the expressions for current densities remain formally pffiffi same as Eqs. (29) and (30), with changes in Dsþ ¼ sA= Es 1 εsþ pffiffi and Ds ¼ sA= εs Esþ . Next, we will present numerical results obtained on the basis of the theoretical model developed in the previous sections, by using the constant parameters approximation. In Fig. 2 we observe current–voltage characteristics for both spins at low temperatures, where we notice step-like features of the current density. At very low temperatures where the electron thermal energy (Ekt) is much smaller than the energy difference between two consecutive Landau levels (ΔEB), quantization caused by the magnetic field is observable in the current–voltage dependence. Current density is primarily determined by the electron transmission coefficient and therefore this behavior of current–voltage characteristics is best explained on that basis. In Fig. 3 we see the
55
overlap of transmission coefficient with the Fermi–Dirac (FD) function for different voltages. If magnetic field is applied, the electron states are characterized by a quantum number s. In this case, the electron wave vector depends on the magnetic field: kz ¼ kz ðε; ωc ; sÞ. By increasing the quantum number s, transmission coefficient is shifted to higher energies. The applied electric field has qualitatively opposite influence and shifts the transmission coefficient towards lower energies. As temperatures increase, the FD function ceases to have a steplike character, and therefore different Landau levels start to overlap with it. That is why at higher temperatures there is no appearance of different modes on current–voltage characteristics, Fig. 4. As a combined consequence of the Zeeman and the Rashba effects, we observe differences in spin up and spin down currents, Fig. 4. If we define a degree of polarization as P ¼ J u J d = J u þ J d , we are able to assess the percentage of each type of current for different applied voltages. The direction of magnetic field dictates the spin of electrons that is “filtered”; if magnetic field is in the direction of electron propagation, the spin up electrons are predominantly transmitted. For the opposite direction of magnetic field, the total current density has more electrons with spin down. In Fig. 5(a and b) we observe the polarization at high temperatures. The oscillatory character is preserved even at high temperatures where Ekt⪢EB. The polarization maximum is 21% for spin down and 22% for electrons with spin up. Optimization at low temperatures yields better results, where best achieved polarization is 58% for spin down and 57% for spin up, Fig. 5(c and d). In Fig. 6 we illustrate the current–voltage dependence for both positive and negative voltages.
Fig. 2. (Color online) Current–voltage characteristics of spin up and spin down electrons for B¼5 T at low temperatures (T ¼ 1 K): (a) taking three Landau levels and (b) taking two Landau levels.
Fig. 3. (Color online) The transmission coefficient and FD function at T ¼ 1 K and B¼ 5 T: (a) s ¼ 1 and (b) s ¼ 2.
56
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
Fig. 4. (Color online) Spin up and spin down currents for the two opposite directions of magnetic field at high temperatures (T ¼ 300 K) and B ¼5 T.
Fig. 5. (Color online) Polarization for B¼ 5 T at different temperatures. (a,b) For T1 ¼1 K, the achieved polarization is up to 22% for spin up and 21% for spin down electrons. (c,d) For T2 ¼300 K, the highest polarization for spin up electrons is 58%, while for spin down it amounts to 57%.
Fig. 6. (Color online) The current density and polarization for positive and negative voltages at low temperatures, for B¼ 5 T.
N. Niketić et al. / Solid State Communications 189 (2014) 52–57
57
3. Conclusions
References
In this paper we have provided a detailed theoretical analysis, together with numerical results, of current density and polarization in semiconductor heterostructure placed in an external magnetic field, with the influence of the Rashba effect included. Although the Rashba effect causes mixing of different spin states, it occurs only inside the inner layers of the structure and therefore we are able to observe the spin polarization of electrons propagating through the entire structure. By using the approximation of constant parameters we obtained results for current densities of spin up and spin down electrons, as well as for the degree of electron spin polarization. Different signs of polarization could be obtained by switching the direction of applied magnetic field. At very low temperatures, when the thermal energy is much lower than the difference between two consecutive Landau levels, we observe step-like features in the current–voltage curve. Even at room temperature, the quantization due to the presence of magnetic field is observable in figures of spin polarization.
[1] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, I. Zutic, Acta Phys. Slovaca 57 (2007) 565–907. [2] R. Khordad, J. Lumin. 134 (2013) 201–207. [3] G. Isic, D. Indjin, V. Milanovic, J. Radovanovic, Z. Ikonic, P. Harrison, J. Nanophotonics 5 (2011) 051819. [4] G. Isic, D. Indjin, V. Milanovic, J. Radovanovic, Z. Ikonic, P. Harrison, J. Appl. Phys. 110 (2011) 064507. [5] G. Isic, D. Indjin, V. Milanovic, J. Radovanovic, Z. Ikonic, P. Harrison, J. Appl. Phys. 108 (2010) 044506. [6] J. Radovanovic, G. Isic, V. Milanovic, Opt. Mater. 30 (2008) 1134–1138. [7] G. Isic, J. Radovanovic, V. Milanovic, J. Appl. Phys. 102 (2007) 123704. [8] D. Grundler, Phys. Rev. Lett. 84 (2000) 6074–6077. [9] T. Koga, J. Nitta, T. Akazaki, H. Takayanagi, Phys. Rev. Lett. 89 (2002) 046801. [10] T. Koga, J. Nitta, H. Takayanagi, Phys. Rev. Lett. 88 (2002) 126601. [11] J.S. de Sousa, J. Smoliner, Phys. Rev. B 85 (2012) 085303. [12] G. Isic, V. Milanovic, J. Radovanovic, D. Indjin, Z. Ikonic, Microelectron. J. 40 (2009) 611–614. [13] M. Eric, J. Radovanovic, V. Milanovic, Z. Ikonic, S. Indjin, J. Appl. Phys. 103 (2008) 083701. [14] A. Gharoati, R Khordad, Superlattices Microstruct. 51 (2012) 194–202. [15] David Z.-Y. Ting, X. Cartoixa, Phys. Rev. B 68 (2003) 235320. [16] XF. Wang, P. Vasilopoulos, Phys. Rev. B 72 (2005) 085344.
Acknowledgments This work was supported by the Ministry of Science (Republic of Serbia), ev. no III 45010.