1.
Introduction

Modern heterojunctions fabricated technology made it possible to fabricate heterostructures with precious dimensions and is used to study spin-dependent properties. The spin separation in heterostructures is due to Dresselhaus spin-orbit interaction (DSOI) and Rashba spin-orbit interaction (RSOI) with zero field. Moreover, the Zeeman effect, giant Zeeman effect, and Dresselhaus spin-dependent mass variation are also reasonable for the separation of spins in the heterojunctions when the field is applied^[1–4]. The spin-injection can be effectively done in semiconductor/semiconductor heterojunctions than ferromagnetic/semiconductor heterojunctions. The spin-dependent hole transport or electron transport is reported in semiconductor/semiconductor heterojunctions^[5–15]. The spin-based devices such as spin-filters and some important observations in spintronics have encouraged this research^[16–19].



From our best knowledge, the spin-transport of LH and HH with externally applied electric and magnetic fields are not found in the literature. In this work, we studied the spin-dependent tunneling of light hole (LH) and heavy hole (HH) with externally applied electric and magnetic fields. In CdTe/Cd_1–xMn_xTe heterostructure, the variation of barrier height as the magnetic field is applied, Zeeman splitting, giant Zeeman splitting, and potential due to the external electric field are accounted. The Dresselhaus mass variation is also included in the theory. The theory and the formalism are given in Section 2. The polarization efficiency of LH and HH as a function of longitudinal energy of incident energy of holes and the magnitude of magnetic field are discussed. The effect of the type of bias on the polarization efficiency is presented. The spin-dependent current density for LH and HH are calculated and the results are given in Section 3 followed by the conclusion.

2.
Theory and formalism

The Hamiltonian of hole motion along the z-axis in the framework of the effective mass approximation can be written as

$$H = frac{1}{{2m}}{left( {{{P}} - e{{A}}} ight)^2} + {V_ { m{eff}}}left( {z,B,{V_ { m{a}}}} ight).$$

(1)

The effective potential of the heterostructure is given by

$${V_ { m{eff}}}left( {z,B,{V_{ m{a}}}} ight) = {V_0}left( {B,z} ight) + {V_{ m{z}}}left( B ight) + {V_{sigma { m{z}}}}left( {B,z} ight) - {V_ { m{E}}}left( {{V_{ m{a}}}} ight).$$

(2)

Here, V₀ (B, z) is the heterostructure potential with an external magnetic field, V_z (B) is the potential due to Zeeman splitting, V_σz (B, z) is the giant Zeeman potential, and V_E (V_a) is the potential due to the applied bias voltage V_a. Fig. 1 shows the heterostructure potential with the applied voltage. For regions I–V, the Eigen function is given by, ${u_{1sigma }}left( z
ight) = $ $ {a_{1sigma }}{{
m e}^{ + i{k_1}z}} + {b_{1sigma }}{{
m e}^{ - i{k_1}z}}$, ${u_{2sigma }}left( z
ight) = {a_{2sigma }}Bileft( {{{
m Z}_{2sigma }}}
ight) + {b_{2sigma }}Aileft( {{{
m Z}_{2sigma }}}
ight)$, ${u_{3sigma }}left( z
ight) = {a_{3sigma }}Bileft( {{{
m Z}_{3sigma }}}
ight) + {b_{3sigma }}Aileft( {{{
m Z}_{3sigma }}}
ight)$, ${u_{4sigma }}left( z
ight) = {a_{4sigma }}Bileft( {{{
m Z}_{4sigma }}}
ight) + $ $ {b_{4sigma }}Aileft( {{{
m Z}_{4sigma }}}
ight) $, and ${u_{5sigma }}left( z
ight) = {a_{5sigma }}{{
m e}^{ + i{k_5}z}} + {b_{5sigma }}{{
m e}^{ - i{k_5}z}}$.






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Figure1.
Potential profile of double barrier heterostructure (a) forward bias and (b) reverse bias.




For the forward bias^[20],

$${k_1}^2 = frac{{2m_ { m{j}}^*{E_ { m{z}}}}}{{{hbar ^2}}},$$

$${k_5}^2 = frac{{2m_ { m{j}}^*left( {{E_ { m{z}}} + e{V_ { m{a}}}} ight)}}{{{hbar ^2}}},$$

$${Z_{{ m{j}}sigma }}left( {{E_ { m{z}}},z} ight) = {left[ {frac{{2m_ { m{j}}^*left( {{E_ { m{z}}},{k_parallel }} ight)e{V_ { m{a}}}}}{{L{hbar ^2}}}} ight]^{1/3}}left[ {frac{{{V_ { m{eff}}}left( {z,B,{V_ { m{a}}}} ight)L}}{{e{V_ { m{a}}}}} - {z_ { m{j}}}} ight].$$

For the reverse bias^[17],

$${k_1}^2 = frac{{2m_{ m{j}}^*left( {{E_{ m{z}}} + e{V_{ m{a}}}} ight)}}{{{hbar ^2}}},$$

$${k_5}^2 = frac{{2m_{ m{j}}^*left( {{E_{ m{z}}}} ight)}}{{{hbar ^2}}},$$

$${{ m Z}_{{ m{j}}sigma }}left( {{E_{ m{z}}},z} ight) = {left[ {frac{{2m_{ m{j}}^*left( {{E_{ m{z}}},{k_parallel }} ight)e{V_{ m{a}}}}}{{L{hbar ^2}}}} ight]^{1/3}}left[ {frac{{{V_{ m{eff}}}left( {z,B,{V_{ m{a}}}} ight)L}}{{e{V_{ m{a}}}}} + {z_{ m{j}}}} ight],$$

where L is the length of the active region in the heterostructure. The Dresselhaus-dependent effective mass of the hole is given by^[6]

$$m_{ m{j}}^*left( {{E_{ m{z}}},{k_parallel }} ight) = {m_{ m{j}}}{left[ {1 + frac{{2gamma sigma {m_{ m{j}}}{k_parallel }}}{{{hbar ^2}}}} ight]^{ - 1}},$$

(3)

where γ is the Dresselhaus spin-orbit constant, k_║ is the in-plane wave vector, and m_j is the effective mass of hole in the heterostructure. According to the spin-dependent boundary conditions,

$$begin{split}&{u_{{text{j}}sigma }}left( z ight) = {u_{{text{j}} + 1sigma }}left( z ight), &frac{1}{{m_{text{j}}^*left( {{E_{text{z}}},{k_parallel }} ight)}}{left{ {frac{d}{{dz}}ln left[ {{u_{{text{j}}sigma }}left( z ight)} ight]} ight}_{z = {z_{text{j}}}}} & - ,frac{1}{{m_{{text{j}} + 1}^*left( {{E_{text{z}}},{k_parallel }} ight)}}{left{ {frac{d}{{dz}}ln left[ {{u_{{text{j}} + 1sigma }}left( z ight)} ight]} ight}_{z = {z_{text{j}}}}} & quad = frac{{2sigma {k_parallel }left( {{beta _{{text{j}} + 1}} - {beta _{text{j}}}} ight)}}{{{hbar ^2}}}, end{split}$$

the coefficient sets {a_jσ, b_jσ} and {a_(j+1)σ, b_(j+1)σ} are related as $left( {begin{array}{*{20}{c}}{{a_{{
m{j}}sigma }}}{{b_{{
m{j}}sigma }}}end{array}}
ight) = M_sigma ^jleft( {begin{array}{*{20}{c}}{{a_{left( {j + 1}
ight)sigma }}}{{b_{left( {j + 1}
ight)sigma }}}end{array}}
ight)$. The interface matrix in the j^th interface M_σ^j is given as^[6]

$$M_sigma ^{ m j} = frac{1}{{varOmega _{ m{j}}^ + - varOmega _{ m{j}}^ - }}left( {begin{array}{*{20}{c}}{varLambda _{{ m{j}} + }^ - }&{varLambda _{{ m{j}} + }^ + }{varLambda _{{ m{j}} - }^ - }&{varLambda _{{ m{j}} - }^ + }end{array}} ight).$$

(4)

Here,

$$begin{split} varOmega _{text{j}}^ + = & {left{ {frac{{ m d}}{{{ m d}z}}ln left[ {phi _{{text{j}}sigma }^ + left( z ight)} ight]} ight}_{z = {z_{text{j}}}}} hfill & - frac{{2sigma m_{text{j}}^*left( {{E_{text{z}}},{k_parallel }} ight){k_parallel }left[ {{beta _{{text{j}} + 1}}left( {{E_{text{z}}},z} ight) - {beta _{text{j}}}left( {{E_{text{z}}},z} ight)} ight]}}{{{hbar ^2}}}, hfill end{split} $$

$$begin{split} varOmega _{text{j}}^ - = & {left{ {frac{{ m d}}{{{ m d}z}}ln left[ {phi _{{text{j}}sigma }^ - left( z ight)} ight]} ight}_{z = {z_{text{j}}}}} hfill &- frac{{2sigma m_{text{j}}^*left( {{E_{text{z}}},{k_parallel }} ight){k_parallel }left[ {{beta _{{text{j}} + 1}}left( {{E_{text{z}}},z} ight) - {beta _{text{j}}}left( {{E_{text{z}}},z} ight)} ight]}}{{{hbar ^2}}}, hfill end{split} $$

$$varLambda _{{ m{j}} + }^ pm = left( {frac{{m_{ m{j}}^*left( {{E_{ m{z}}},{k_parallel }} ight)}}{{m_{{ m{j}} + 1}^*left( {{E_{ m{z}}},{k_parallel }} ight)}}{{left{ {frac{{ m d}}{{{ m d}z}}lnleft[ {phi _{{ m{j}} + 1sigma }^ mp left( z ight)} ight]} ight}}_{z = {z_{ m{j}}}}} - varOmega _{ m{j}}^ - } ight)frac{{phi _{{ m{j}} + 1sigma }^ mp left( z ight)}}{{phi _{{ m{j}}sigma }^ + left( z ight)}},$$

$$varLambda _{{ m{j}} - }^ pm = left( {frac{{m_{ m{j}}^*left( {{E_{ m{z}}},{k_parallel }} ight)}}{{m_{{ m{j}} + 1}^*left( {{E_{ m{z}}},{k_parallel }} ight)}}{{left{ {frac{{ m d}}{{{ m d}z}}lnleft[ {phi _{{ m{j}} + 1sigma }^ mp left( z ight)} ight]} ight}}_{z = {z_{ m{j}}}}} - varOmega _{ m{j}}^ + } ight)frac{{phi _{{ m{j}} + 1sigma }^ mp left( z ight)}}{{phi _{{ m{j}}sigma }^ - left( z ight)}},$$

$$begin{split} beta left( {{E_{text{z}}},z} ight) = & frac{{{P^2}}}{2}left[ {frac{1}{{{E_{text{z}}} - {E_{text{c}}}left( z ight) + {E_{text{g}}}left( z ight) + Vleft( z ight)}}} ight. hfill & left. { -, frac{1}{{{E_{text{z}}} + Delta left( z ight) - {E_{text{c}}}left( z ight) + {E_{text{g}}}left( z ight) + Vleft( z ight)}}} ight] .end{split} $$

The function ?_jσ (z) is chosen as ?_jσ⁺ (z) = Bi(Ζ_jσ) and ?_jσ^? (z) = Ai(Ζ_jσ). From the interface matrices, the transfer matrix is formed and hence the transmission coefficients as well as the polarization efficiency are also calculated.

3.
Results and discussions

The effective heterostructure potential due to the application of electric and magnetic field in CdTe/Cd_1–xMn_xTe double barrier heterostructure is calculated from Eq. (2). The heterostructure potential V₀ (B, z) depends on the externally applied magnetic field and it decreases as the field increases. At the critical magnetic field, V₀ (B, z) is zero. The value of V₀ (B, z) is calculated as given in Ref. [21]. The Zeeman splitting and giant Zeeman splitting are, respectively, m_jgμ_BσB and $frac{{ - 1}}{3}{N_0}beta {x_{
m{eff}}}{S_{
m{z}}}sigma {m_{
m{j}}}$. Here m_j is the quantum number (m_j = 1/2 for light hole and m_j = 3/2 for heavy hole), g is the Mn effective Lande factor (g = 2), μ_B is the Bohr magneton, N₀β is the product between the number of crystalline elementary cells per unit volume and the value of sp-d exchange integral (N₀β = ?0.88 eV), x_eff is the effective ‘Mn’ concentration in Cd_1–xMn_xTe, and < S_z> is the thermodynamic average value of ‘Mn’ spins along thez-axis^{[21, 22]}. The expressions used here to evaluate x_eff and < S_z> are taken from Ref. [19], and they are given as

$$begin{split}{x_{ m{eff}}}! &= frac{2}{5}left( {frac{x}{{1100}}} ight)left[ {718.2{{ m e}^{{ aise0.7exhbox{${ - x}$} !/!lower0.7exhbox{${0.023;07}$}}}} !+! 1988left( {frac{{ - x}}{{0.016;15}}} ight) + 19.66} ight],{S_{ m{z}}} & = - {S_0}{B_{ m{S}}}left( {frac{{gS{mu _{ m{B}}}B}}{{{k_{ m{B}}}{T_{ m{eff}}}}}} ight),end{split}$$

where S₀ accounts for antiferromagnetic super exchange coupling between Mn ions, B_s represents the Brillouin function, and T_eff is the effective temperature. The potential due to the applied external bias voltage is eV_az/L where L is the length of the active layer in the heterostructure^[23]. The effective mass of LH and HH are, respectively m_lh^* = 0.18 + 0.14x and m_hh^* = 0.60 + 0.21x + 0.15x² in the CdTe/Cd_1–xMn_x-Te heterostructure.



Fig. 2 shows the polarization efficiency of holes with magnetic field less than the critical magnetic field (B = 1.90 T) for various applied voltages. The polarization efficiency of light holes approaches zero linearly as the value of E_z increases. But the polarization efficiency of heavy holes is ?1 except at resonances. The resonances of the polarization peaks shift to the lower energy of the incident heavy holes as the applied bias voltage increases. The height of the polarization efficiency peak depends on the applied voltage. Numerically, with these considered values, the polarization of light hole is nearly zero above ~2 meV, whereas the polarization efficiency of heavy holes is ?1 except at resonances.






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Figure2.
(Color?online) Polarization efficiency of (a) LH and (b) HH for various bias voltages.




The polarization efficiency of light hole and heavy hole as a function of applied magnetic field is shown in Fig. 3. As the magnetic field increases, the polarization efficiency of both holes decreases non-linearly. The polarization efficiency of HH becomes ?1 as the magnetic field is ~ 0.2 T or more; however, it reaches only up to ?0.06 for LH, even at critical magnetic field when E_z is 3.5 meV. At low magnetic field, the polarization efficiency depends on the applied voltages for heavy holes, whereas it is almost independent on the applied voltages for light holes. The above mentioned distinct properties of LH and HH are due to (i) different effective mass, and (ii) different potential seen by the LH and HH. A comparative study of polarization efficiency for forward bias and reversed bias is given in Fig. 4. As the bias becomes reversed, the polarization efficiency shifts to the higher energy region. This is due to the fact that the positive bias voltage decreases the effective potential of the barrier, whereas the negative bias voltage increases the effective potential. The energy of resonance polarization shifts to the higher value when the barrier height increases due to the negative bias voltage. This result agrees with the reported results in ZnSe/ZnMnSe heterostructures by Papp et al.^[20].






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Figure3.
(Color?online) Polarization efficiency of (a) LH and (b) HH for various energies of incident electrons.






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Figure4.
(Color?online) Polarization efficiency of (a) LH and (b) HH for forward and reverse bias.




The current density of LH and HH is calculated from

$$begin{split} {J_{{sigma _{text{z}}}}}left( {B,{V_{text{a}}}} ight) = ,& {J_0}Bsumlimits_{n = 0}^infty {intlimits_0^infty {{T_{{sigma _{text{z}}}}}left( {{E_{text{z}}},B,{V_{text{a}}}} ight)} } & times left[ {fleft( {{E_{n,{k_{text{z}}}}} + {V_{text{z}}}} ight) - fleft( {{E_{n,{k_{text{z}}}}} + {V_{text{z}}} + e{V_{text{a}}}} ight)} ight]{ m d}{E_{text{z}}} ,end{split} $$

where J₀ = e²h², and f(E) is the Fermi-Dirac distribution function. The current densities of LH and HH due to spin-down holes are given in Fig. 5. As the bias voltage increases, the current density also increases. The magnetic field influences the current density at high bias voltage, not at low bias. One can obtain the maximum current density at the critical magnetic field (B = 2 T). This is true for both LH and HH, and HH has a much higher current density than LH. Fig. 6 shows the current density of LH due to spin-up holes. The current density of LH due to both spin-up and spin-down components is finite, whereas the current density of HH due to spin-up is almost zero. This is direct evidence that this CdTe/Cd_1–xMn_xTe double barrier heterostructure can be used as a perfect spin filter for HH, which completely allows the spin-down components and blocks the spin-up components with these given heterostructure values. These reports will encourage more research regarding the hole transport with externally applied fields.






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Figure5.
(Color?online) Current density of spin-down (a) LH and (b) HH for various magnetic fields.






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Figure6.
(Color?online) Current density of spin-up LH for various magnetic fields.

4.
Conclusion

A perfect spin filter is proposed using CdTe/Cd_1–xMn_x -Te double barrier heterostructure. The spin-dependent properties are evaluated using the matrix method. The polarization efficiency of LH approaches zero as E_z increases, whereas it is almost ?1 for HH except at resonances. The increase in the magnetic field decreases the polarization efficiency and the magnetic field highly influences the HH more than LH. The negative bias shifts the polarization to the higher energy of resonance. The current densities of spin-up and spin-down LHs are reported; however, the current density of spin-down HH is finite and spin-up HH is almost zero.