Particle continuity equation

In quantum mechanics, calculating current is crucial to applications. A widely accepted approach is using the correspondence regulation from classical to quantum mechanics.

From the Schr¨odinger equation, we have Hψ = i_∂t^∂ψ and (Hψ)^† = −i_∂t^∂ψ . Notice here the transposition in the symbol † only acts on the spin index . We can use the above two equations to get the particle continuity equation.

∂

∂tψ^†ψ = 1 i h

ψ^†(Hψ) − (Hψ)^†ψ i

(3.45)

or

∂

∂tn = −∇ ·^{* *}j (3.46)

where n = ψ^†ψ is the particle density and ^*j is the particle current density.

It describes the conservation of the particle number. In the derivation below, we use the following Hamiltonian:

H = −∇²− κ²β1(kxσx− kyσy) + β3(kxk_y²σx− kyk_x²σy) + V (^*r) (3.47)

In Eq. (3.45) the first term is the kinetic energy. The second and third terms are the Dresselhaus spin-orbit coupling, which has been extensively studied recently. Next we

substitute the Hamiltonian Eq. (3.47) into Eq. (3.45), and Eq. (3.45) becomes

∂

∂tψ^†ψ = 1 i













−[ψ^†∇²ψ − (∇²ψ^†)ψ] − κ²β₁[ψ^†σ_x(k_xψ) + (k_xψ^†)σ_xψ]+

κ²β₁[ψ^†σ_y(k_yψ) + (k_yψ^†)σ_yψ]+β₃[ψ^†σ_x(k_xk²_yψ) + (k_xk²_yψ^†)σ_xψ]

−β₃[ψ^†σ_y(k_yk_x²ψ) + (k_yk_x²ψ^†)σ_yψ]













(3.48)

And we have the

*k = ¹_i∇ =^{* 1}_i(ˆx∇_x+ ˆy∇_y) . After long careful calculations from Eq. (3.45) and Eq. (3.48), we obtain

∂

∂tn = −∇ · (^{* *}j^conv+^*j^extra) (3.49)

where

*j^conv = −i[ψ^†(∇ψ) + (^* ∇ψ^{* †})ψ]

*j^extra =^*j^L+^*j^C

*j^L= −κ²β1[ψ^†(ˆxσx− ˆyσy)ψ]

*j^C = −[β₃ψ^†(ˆyσ_x− ˆxσ_y)(∇_x∇_yψ) + β₃(∇_x∇_yψ^†)(ˆyσ_x− ˆxσ_y)ψ] + [β₃(∇_iψ^†)(ˆxσ_x− ˆyσ_y)(∇_iψ)]

(i = x, y, z)...repeated index connection adopted

(3.50)

Appendix B gives the detailed calculation of the particle continuity equation in Dressel-haus SOI system.

Here ^*j^conv is a ” conventional term ”. This term is general for any potential V (^*r) in Hamiltonian if the potential is position dependent. And the main source to induce an extra term of the particle current^*j^extra comes from Dresselhaus spin orbit coupling which include linear term ^*j^L and cubic term ^*j^C.

In Rashba SOI system in Eq. (3.51), we get the same result in Eq. (3.52) except for

the extra term in Eq. (3.53).

H_R= −∇²+ α(k_xσ_y− k_yσ_x) + V (^*r) (3.51)

*j =^*j^conv+^*j^extra (3.52)

where

*j^extra =^*j^R = αψ^†(^*σ × ˆz)ψ (3.53)

However, we discuss the particle current in some complex system with SOI such as Dresselhaus SOI in Eq. (3.47) and Rashba SOI in Eq. (3.51) in which the momentum appears in the ”potential ” of the Hamiltonian of semiconductors that the extra term must exit. In our case, the number of particle must conserve during the scattering process without any bias or EM fields.

Numerical results and discussions

4.1 Results for linear-k Dresselhaus SOI

We has studied the results of the scattering in linear-k Dresselhaus system. In chapter 3, the spin density of the total wave function would be solved in numerical method as we do in this chapter just like solving the Rashba scattering problem [28]. But we find the recursion relation of the coefficients of the wave function which can be used to simplify the spin density toughly so that we have the analytical form. However, we can compare the numerical results with the analytical results. The plane wave incident a hard wall disk in linear-k Dresselhaus system with the helicity η = +, the incident energy ε=7.7 (in unit of ε^∗), and the incident angle ϕ_k=0 ( the energy unit ε^∗ = 8.977 meV ).The spin density of the total function has the analytical form

Sz(r, ϕ) = ψ_total⁺ σzψtotal = 1 4

X

m,m⁰

sin ((m⁰− m)ϕ)¡

A^∗_+mA+m⁰

¢i^(m0−m+1) (4.1)

where

A_+m= a_mH_m⁽¹⁾(γr) + c_mH_m⁽¹⁾(γ⁰r) + H_m⁽²⁾(γr) (4.2)

whose coefficients are

a_m = −Hm⁽¹⁾(˜γ⁰)H_m−1⁽²⁾ (˜γ) + Hm⁽²⁾(˜γ)H_m−1⁽¹⁾ (˜γ⁰)

Hm⁽¹⁾(˜γ)H_m−1⁽¹⁾ (˜γ⁰) + Hm⁽¹⁾(˜γ⁰)H_m−1⁽¹⁾ (˜γ) (4.3) and

c_m = Hm⁽¹⁾(˜γ)H_m−1⁽²⁾ (˜γ) − Hm⁽²⁾(˜γ)H_m−1⁽¹⁾ (˜γ)

Hm⁽¹⁾(˜γ)H_m−1⁽¹⁾ (˜γ⁰) + Hm⁽¹⁾(˜γ⁰)H_m−1⁽¹⁾ (˜γ) . (4.4) The numerical results of the spin density from 3.2.1 we can solve the boundary con-dition by using numerical method. And then the distribution of the spin density and the probability density of the total wave function are obtained in Fig. 4.1 and Fig. 4.2 respectively.

Figure 4.1: Distribution of the out-of-plane spin density S_z. The plane wave incident a hard wall disk in linear-k Dresselhaus system with the helicity η = +, the incident energy ε=7.7, and the incident angle ϕ_k=0. Lighter regions means the region whose spin density Sz >0 (spin up). And the darker regions represent the region of spin down .

Figure 4.2: The spatial dependence of the magnitude of the total wave function probability density. Out of the white circle is the region with finite Dresselhaus SOI. Apparently, when the plane wave propagates along the x axis, the probability density is almost zero behind the disk due to the hard wall disk .

The spin density function is the odd function of ϕ from Eq. (4.1), i.e. S_z(r, −ϕ) =

−S_z(r, ϕ) . The property from the analytical result is consistent with the distribution of the spin density pattern plotted in Fig. 4.1. Also, Fig. 4.2 show that the total wave function density is almost zero behind the disk due to the hard wall disk.

For a larger incident energy ε=13.23 (in unit of ε^∗) , we obtain the spin density plotted in Fig. 4.3. The total wave function probability density is plotted in Fig. 4.4, where other parameters of the incident plane wave and the linear-k DSOI constant are fixed. From Fig. 4.3 and Fig. 4.2, we can conclude that the concentration of spin density (fringes) become higher by larger incident energy. The energy dispersion of the incident wave with helicity η which we derived in chapter 1 shows that the momentum is independent of the incident angle ϕ_k and then the effective magnetic filed is perpendicular to momentum in linear-k Dresselhaus system. Hence, the distribution of the probability density and spin density in linear-k Dresselhaus system are just rotated by the incident wave angle but the patterns are the same.

Figure 4.3: Distribution of the out-of-plane spin density S_z. The plane wave incident a hard wall disk in linear-k Dresselhaus system with the helicity η = +, the incident energy ε=13.23, and the incident angle ϕ_k=0. Lighter regions means the region whose spin density S_z >0 (spin up) . And the darker regions represent the region of spin down .

Figure 4.4: The spatial dependence of the magnitude of the total wave function probability density. Out of the white circle is the region with finite Dresselhaus SOI. Apparently, when the plane wave propagates along the x axis, the probability density is almost zero behind the disk due to the hard wall disk.

Figure 4.5: The quality the spin density S_z determining the number m as function of ϕ for a central hard wall disk where m represents the number of the partial waves we must sum. Apparently, the line of the spin density corresponding to the distribution of the spin density Fig. 4.1 at r=2 is saturated for a larger m. And the probability density also has the same situation.

4.2 Results for cubic-k Dresselhaus SOI and more