Equations of the Viscous Quantum Hydrodynamics

semicon-ductor, and wish to derive evolution equations for particle density n, current density J and energy densityne.

The Poisson Equation The Maxwell’s equations

∇ ×E = 0, divD=ρ inR³, (1.1.1) hold in vanishing magnetic fields. Here E denotes the electric field, D is the displacement vector and ρ is the total space charge density. The first equation implies that there exists a potential V such that E = −∇V. V indicates the electrostatic potential. Dis defined by the following relation

D=εsE, where ε_s is the semiconductor permittivity.

Let n stand for the electron density, C(x) denote the so-called doping pro-file which is a function of the position varible and given by the difference of the number densities of positively charged donor ions and negatively charged acceptor ions. Then the total space charge densityρ is given by

ρ=−qn+qC(x), where q is the elementary charge.

Finally from the second equation of (1.1.1) we infer

εs4V = div(−ε_s∇V) =−divD=q(n− C(x)). (1.1.2) This equation is called thePoisson equation.

1In a bipolar model most of the electrons are valence electrons, i.e., they are responsible for the chemical compound of the semiconductor crystal. When the crytal is electrically neutral, then to each conduction electron there corresponds a ”hole” in the valence band. In this case the total particle density is the difference of the electron density and the hole density.

Madelung Equations

From the quantum view a single electron (or hole) is considered as a wave described by a complex-valued wave function ϕ which is a solution of the Schr¨odinger equation

i~∂tϕ=−~²

2m4ϕ−qV(x, t)ϕ, x∈R^d, t >0, ϕ(x,0) =ϕI(x), x∈R^d.

Hereiis the complex unit withi² =−1,~= h

2π is the reduced Planck constant (leth denote the Planck constant),m is the electron mass,q is the elementary charge, V(x) is the electrostatic potential.

We introduce some reference values depending only upon the considered de-vice. Lis a characteristic length, for instance the device length. The character-istic voltageU is defined byU :=k_BT₀/q,wherek_Bis the Boltzmann constant, T₀ is the lattice temperature. The characteristic time t^∗ is the time needed by a particle to cross the device. We assume that the thermal energy of a particle is equal to the kinetic energy, then

qU =m L

t^∗ 2

. Now we use the scaling

t=t^∗ts, x=Lxs, V =U Vs,

then after some calculations it is easy to verify that the scaled wave function ψ(xs, ts) :=ϕ(x, t) =ϕ(t^∗ts, Lxs)

satisfies the scaled Schr¨odinger equation for single particle (we omit the index

”s”)

i∂_tψ=−²

24ψ−V ψ, x∈R^d, t >0, (1.1.3)

ψ(x,0) =ψ_I(x), x∈R^d, (1.1.4)

whereis the scaled Planck constant which is given by = ~t^∗

mL².

Define the measurable quantitiesn(x, t) andJ(x, t) in the following way n(x, t) :=|ψ(x, t)|², J(x, t) :=−Im(ψ∇ψ).

We make use of the WKB (Wentzel,Kramers,Brillouin) state on initial data:

ψI =√

nIe^iSI/,

where n_I(x) ≥ 0, S_I(x) ∈ R are some functions. Then the electron density n(x, t) and current densityJ(x, t) are satisfying (formally) the scaled Madelung equations [60]

∂_tn−divJ = 0,

∂tJ−div

J ⊗J n

+n∇V +² 2n∇

4√

√ n n

= 0, x∈R^d, t >0 n(x,0) =n_I(x), J(x,0) =J_I(x) : =−n_I∇S_I,

(1.1.5)

which are formally equivalent to the Schr¨odinger equation (1.1.3).

The Wigner-Fokker-Planck Equation

We consider a particle ensemble consisting ofM electrons, then the motion of the particle ensemble is described by the many-particle Schr¨odinger equation

i~∂tψ=−~² 2m

M

X

j=1

4_xjψ−qV(x1,· · · , xM, t)ψ, xj ∈R^d, t >0, ψ(x,0) =ψ_I(x),

whereψis called the wave function of the particle ensemble, it is a function with variables (x₁,· · · , x_M, t), x_j ∈ R^d, j = 1,· · ·, M. Corresponding to the wave functionψ we define the density matrix%:=ψ(r, t)ψ(s, t),r, s∈R^dM, t >0.

Then in order to derive the Wigner equation, we shall use the following assumptions.

• We assume first that the potential V is decomposed into the sum of an external potential and a two-particle interaction potentials:

V(x₁,· · · , x_M, t) =

M

X

l=1

V_ext(x_l, t) +1 2

M

X

l=1 M

X

j=1

V_int(x_l, x_j), and V_int(x_l, x_j) =V_int(x_j, x_l),i.e., V_int is symmetric.

• Next since the particles considered are Fermions, the wave function ψ is antisymmetric, i.e.,

ψ(x₁,· · ·, x_M, t) = sgn(π)ψ(x_π(1),· · ·, x_π(M), t),

for any permutation π of {1,· · ·, M}. This property implies that the density matrix remains invariant underπ. Namely, if the ensemble density matrix% is defined by

%(r₁,· · · , r_M, s₁,· · ·, s_M, t) :=ψ(r₁,· · ·, r_M, t)ψ(s₁,· · · , s_M, t), r_i, s_i∈R^d then

%(r₁,· · · , r_M, s₁,· · · , s_M, t) =%(r_π(1),· · · , r_π(M), s_π(1),· · · , s_π(M), t).

• Finally the density matrices of subensemble consisting of l particles is defined by

ρ^l(r1,· · · , r_l, s1,· · · , s_l, t) :=

Z

R^d(M−l)

ρ(r1,· · · , r_l, u_l+1,· · · , u_M, s1,· · ·, s_l, u_l+1,· · ·, u_M,)du_l+1· · ·du_M. We assume that the particles in the subensemble move independently from each other. The corresponding mathematical description is the so called Hartree ansatz, i.e.,ρ^l(r1,· · · , r_l, s1,· · · , s_l, t) can be factorized

ρ^l(r₁,· · ·, r_l, s₁,· · ·, s_l, t) =

l

Y

i=1

R(r_i, s_i, t).

DefineR:=ρ¹,the effective potentialV(x, t) reads V(x, t) =V_ext(x, t) +

Z

R^d

M R(y, y, t)V_int(x, y)dy.

Then after some reformulations and calculations (see [63], Sec. 1.5) it is easy to verify that the function

W(x, v, t) :=Fη→v⁻¹

M R

x+ ~

2mη, x− ~ 2mη, t

solves the following so-calledVlasov equation [16, 59]

∂_tW +v· ∇_xW + q

mθ_~[V]W = 0, x, v∈R³, (1.1.6) whereθ_~[V] is a pseudo-differential operator defined by

θ_~[V]W(x, v, t) := i (2π)^d

Z

R^d

Z

R^d

m

~

V

x+ ~ 2mη, t

−V

x− ~ 2mη, t

×

×W(x, v⁰, t)e^{i(v−v0)·η}dv⁰dη,

(1.1.7)

v∈R^dindicates the velocity. Furthermore we refer to [63], pp 62, the quantum electron number density (denoted by n(x, t)) can be expressed by

n(x, t) = Z

R^d

W(x, v, t)dv, (1.1.8)

then the macroscopic quantum current density (denoted by J(x, t)) is given by J(x, t) =−q

Z

R^d

vW(x, v, t)dv; (1.1.9)

and the energy density (denoted by ne) is infered as ne(x, t) = m

2 Z

R^d

v²W(x, v, t)dv. (1.1.10) The Vlasov equation presented above doesn’t involve the impact of the semi-conductor crystal lattice on the motion of the particles and collisions of the charged particles with the background oscillators. In order to take into account these aspacts the energy-band and the collision operator need to be considered.

More precisely, we assume a parabolic energy-band E(k) =~²|k|²/2m (k is the wave vector), then it implies

v= 1

~

∇_xE(k) = ~k

m. (1.1.11)

Define the Wigner distribution functionw(x, k, t) := _m¹W(x, v, t). Recall (1.1.6) w(x, k, t) satisfies the equation

∂tw+~k

m · ∇_xw+ q

mθ[V]w= 0, x, k∈R³, (1.1.12) where

θ[V]w(x, k, t) =θ_~[V]W(x, v, t).

We use the scaling ˜η= ~

mη, then substitute ˜η into (1.1.7), we infer θ[V]w(x, k, t) = i

(2π)^d Z

R^d

Z

R^d

m

~

V

x+η˜ 2, t

−V

x−η˜ 2, t

×

×w(x, k⁰, t)e^{i(k−k0)·˜η}dk⁰d˜η.

In order to take into account collisions we introduce the quantumFokker-Planck collision operator [15] which is given by

L(w) := 1

τ₀div_k(kw) +Dpp

~²

4_kw+Dpq

~ divx(∇_kw) +Dqq4_xw, (1.1.13)

and models the interaction of the electrons with the phonons of the crystal lattice (oscillators) with constants:

D_pp= mk_BT₀ τ0

, D_pq= Ω˜~² 6πkBT₀τ0

, D_qq = ~² 12mkBT₀τ0

.

Here τ₀ is the momentum relaxation time, ˜Ω is the cut-off frequency of the reservoir oscillators. Combining (1.1.12) and (1.1.13) we obtain the complete Wigner-Fokker-Planck equation

∂tw(x, k, t) +~k

m · ∇_xw(x, k, t) + q

mθ[V]w(x, k, t) =L(w(x, k, t)), x, k∈R³, t >0.

(1.1.14) The Viscous Quantum Hydrodynamic Equations

We are interested in the macroscopic equations with respect to the particle density n(x, t), the current density J(x, t) and the energy density ne, where n(x, t), J(x, t) and ne(x, t) are related to the Wigner function (according to (1.1.8)-(1.1.10) and (1.1.11)) by

n(x, t) = Z

R^d

w(x, k, t)d(~k), J(x, t) =−q m

Z

R^d

w(x, k, t)(~k)d(~k), ne(x, t) = 1

2m Z

R^d

w(x, k, t)|~k|²d(~k).

In order to derive such equations the moment method as in [40] will be applied. More precisely, the equation (1.1.14) is multiplied by 1,~k, and 1

2|~k|² respectively, and then integrated over R^d with respect to ~k. The resulting system has to be closed by assuming that the Wigner function w is close to a wave vector displaced equilibrium density which was formulated by Wigner [81]. Next we follow [53] then obtain the following approximate equations of (1.1.14), up to orderO(⁴):

∂_tn−1

qdivJ =D_qq4n,

∂_tJ−1 qdiv

J⊗J n

−qk_B

m ∇(nT) +q² mn∇V + q~²

12m²div(n(∇ ⊗ ∇) lnn) =−J

τ₀ +qDpq

m ∇n+Dqq4J,

∂_t(ne)− 1 qdiv

((ne)E_d+P)J n

+J· ∇V =−2 τ₀

ne− d

2nk_BT₀

+2Dpq

q divJ +Dqq4(ne).

HereJ⊗J denotes the matrix with componentJjJk,Edis thed×dunit matrix, and the stress tensorP and energy densityne are given by

P =nkBT E_d− ~²

Notice that the stress tensor consists of the classical pressure and a quantum

”pressure” term. The energy density is the sum of kinetic energy, thermal energy, and quantum energy.

Next we use the following scalings x_s=Lx, t_s=t^∗t, C_s = sup

then after recalling (1.1.2) we obtain the scaled equations (omitting the index

”s”)

where the scaled parameters are the viscosity constant ν₀, the Planck constant , the Debye length λ, the relaxation time τ and the interaction constant µ which are given by

ν0 = ~²

the scaled stress tensor P and energy density have the representations P =nT Ed− ²

If the temperature is invariant, i.e., T is a positive constant denoted by T0, the evolution equation with respect to ne is unnecessary because it is a direct resulting equation from the evolution equations with respect to n, J and the Possion equation. In this case we have the isothermal viscous QHD model which is the simplified version of (1.1.15) and reads















∂tn−divJ =ν04n,

∂_tJ −div

J⊗J n

−T₀∇n+n∇V + ² 2n∇

4√

√ n n

=ν₀4J −J τ, λ²4V =n− C(x), The complete system (1.1.15) is calledthe non-isothermal viscous QHD model.

Im Dokument Solutions to the Multi-Dimensional Viscous Quantum Hydrodynamic Equations for Semiconductors (Seite 12-19)