approximation of thermal equilibrium for quantum gases with ...

APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUMGASES WITH DISCONTINUOUS POTENTIALS AND

APPLICATION TO SEMICONDUCTOR DEVICES∗

CARL L. GARDNER† AND CHRISTIAN RINGHOFER†

SIAM J. APPL. MATH. c© 1998 Society for Industrial and Applied MathematicsVol. 58, No. 3, pp. 780–805, June 1998 005

Abstract. We derive an approximate solution valid to all orders of h to the Bloch equationfor quantum mechanical thermal equilibrium distribution functions via asymptotic analysis for hightemperatures and small external potentials. This approximation can be used as initial data fortransient solutions of the quantum Liouville equation, to derive quantum mechanical correction termsto the classical hydrodynamic model, or to construct an effective partition function in statisticalmechanics. The validity of the asymptotic solution is investigated analytically and numerically andcompared with Wigner’s O(h2) solution. Since the asymptotic analysis results in replacing secondderivatives of the potential in the correction to the stress tensor in the original O(h2) quantumhydrodynamic model by second derivatives of a smoothed potential, this approach represents adefinite improvement for the technologically important case of piecewise continuous potentials inquantum semiconductor devices.

Key words. quantum gases, quantum hydrodynamics, nonlinear PDEs, conservation laws,semiconductor device simulation

AMS subject classifications. 76M20, 76W05, 76Y05

PII. S0036139996303907

1. Introduction. Semiconductor devices that rely on quantum tunneling throughpotential barriers are playing an increasingly important role in advanced microelec-tronic applications, including multiple-state logic and memory devices and high fre-quency oscillators. The propagation of electrons and holes in the semiconductor devicecan be modeled as the flow of a continuous charged quantum gas in a potential thathas discontinuous jumps at heterojunction barriers. The fluid dynamical equationsare derived by assuming the gas is near thermal equilibrium, but are expected to bemore generally valid.

This investigation is therefore concerned with the approximate description of ther-mal equilibrium quantum mechanical systems of particles in a potential—especiallyin the presence of discontinuous potential barriers. Under the assumption of Boltz-mann statistics, thermal equilibrium of a quantum mechanical system is describedby the factor exp−β∗(H(V ) + φ), where H(V ) is the Hamiltonian operator, V isthe potential energy, β∗ is the reciprocal value of the ambient temperature, and φ isthe Fermi level. Depending on the choice of representation of the system, the termexp−β∗(H(V ) + φ) can be expressed in various ways. Choosing a representationvia Schrodinger wavefunctions amounts to solving the eigenvalue problem

Eλψλ(x) = H(V )ψλ(x), H(V ) = − h2

2m∆x + V (x), x ∈ Rd,(1)

where d = 1, 2, or 3, m is the particle mass, and the ψλ (λ = 0, 1, 2, . . .) are theparticle wavefunctions. Observable quantities like the thermal equilibrium (denoted

∗Received by the editors May 20, 1996; accepted for publication (in revised form) December 13,1996; published electronically March 24, 1998.

http://www.siam.org/journals/siap/58-3/30390.html†Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804 (gardner@

asu.edu, [email protected]). The research of the first author was supported in part by U.S. ArmyResearch Office grant DAAH04-95-1-0122, and the research of the second author was supported inpart by ARPA grant F49620-93-1-0062.

780

APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES 781

by the subscript e) particle density ne(x), momentum density πe(x), and momentumflux density tensor Πe(x) are computed as

ne(x) =∑λ

exp−β∗(Eλ + φ)|ψλ(x)|2,(2)

πe(x) =h

i

∑λ

exp−β∗(Eλ + φ)ψ∗λ(x)∇xψλ(x),(3)

Πe(x) =h2

m

∑λ

exp−β∗(Eλ + φ) ∇xψ∗λ(x)∇xψλ(x).(4)

The summation over the index λ is replaced by an integral where the spectrum of theHamiltonian is continuous.

Fluid dynamical equations are usually formulated in terms of the stress tensorPjk, which is related to the momentum flux density tensor by

Πjk =1

mnπjπk − Pjk,(5)

where j, k = 1, . . . , d. The energy density W is calculated from the momentum fluxdensity as

W =1

2Πjj .(6)

We will use the summation convention where repeated Latin indices j, k are summedover.

For many applications it is advantageous to express the term exp−β∗(H(V )+φ)via the density matrix

ρe(x, y, β) =∑λ

exp−β(Eλ + φ)ψλ(x)ψ∗λ(y).(7)

It can be easily shown [12] that if the wavefunctions form an orthonormal eigensystem,the density matrix satisfies the initial value problem for the Bloch equation

∂βρe =h2

4m(∆x + ∆y)ρe − 1

2[V (x) + V (y)] ρe − φρe,(8)

ρe(x, y, β = 0) = δ(x− y).(9)

Thus, solving the eigenvalue problem for the Schrodinger equation is replaced bysolving a parabolic initial value problem in twice as many dimensions. The Blochequation is the interpretation of the term exp−β∗(H + φ) as a semigroup.

The initial condition (9) means that we prescribe randomized behavior at infinitetemperature (β = 0) and solve the Bloch equation up to the specified ambient tem-perature T0 = 1/β∗. (We set Boltzmann’s constant kB = 1.) In the density matrixrepresentation, the particle, momentum, and momentum flux densities are

ne(x) = ρe(x, x, β∗),(10)

782 CARL L. GARDNER AND CHRISTIAN RINGHOFER

πe(x) =h

i∇xρe(x, y, β

∗)|y=x ,(11)

Πe(x) =h2

m∇x∇yρe(x, y, β

∗)|y=x .(12)

To relate the quantum mechanical description to the classical picture, we use theWigner transform of the Bloch equation. The Wigner (distribution) function [14] is

w(x, p, β) =

∫Rdη

ρ

(x− h

2η, x+

h

2η, β

)eiη·pdη,(13)

where the new independent variable p is the momentum. Applying the transfor-mation (13) to the Bloch equation (8) yields the Bloch equation for the thermalequilibrium Wigner function

∂βwe =h2

8m∆xwe − |p|2

2mwe − 1

2

[V

(x+

h

2i∇p

)+ V

(x− h

2i∇p

)]we − φwe,(14)

we(x, p, β = 0) = h−d.(15)

The operators V (x± h2i∇p) are defined as

V

(x± h

2i∇p

)we(x, p, β) = (2π)−d

∫Rdη

∫Rdq

V

(x± h

2η

)we(x, q, β)eiη·(p−q)dqdη.

(16)

In the Wigner representation the thermal equilibrium particle, momentum, and mo-mentum flux densities are

ne(x) =

∫Rdp

we(x, p, β∗)dp,(17)

πe(x) =

∫Rdp

pwe(x, p, β∗)dp,(18)

Πe(x) =1

m

∫Rdp

ppTwe(x, p, β∗)dp.(19)

The relationship between the quantum mechanical description and the classicalpicture is transparent using the Wigner representation. Setting h to zero formally inthe Bloch equation (14) immediately gives the classical Maxwellian

we(x, p, β) = h−d exp

−β

( |p|22m

+ V + φ

).(20)

The Fermi level φ just plays the role of a scaling parameter and is chosen to determinethe total number Nparticles of particles in the system∫

Rdx

ne(x)dx =∑λ

exp−β∗(Eλ + φ) = Nparticles.(21)


Note that because of the properties of the exponential function in Boltzmann statis-tics, the effect of the Fermi level is to scale all quantities by the factor exp−β∗φ.This scaling can be performed after solving the Bloch equation with φ = 0. Thesituation becomes more complex for Fermi–Dirac or Bose–Einstein statistics [13].

We will construct approximate solutions to the Bloch equation (8) or its Wignertransformed version (14). The motivation for this work is threefold. (i) Transientsolutions of quantum kinetic equations usually require a thermal equilibrium Wignerfunction as initial data. To obtain the thermal equilibrium solution either via theeigenvalue problem for the Schrodinger equation [10] or via the Bloch equation [12]is an expensive computational task. (ii) Approximations to the thermal equilibriumdistribution function can be used to derive quantum mechanical corrections for macro-scopic fluid dynamical equations. As in the classical case, equations for the nonequi-librium densities of particles, momentum, and energy can be derived by building theappropriate moments of the nonequilibrium quantum transport equation, the tran-sient quantum Liouville equation. Closure of the infinite hierarchy of moment equa-tions is obtained by assuming that the nonequilibrium Wigner function is close tothe momentum-displaced thermal equilibrium Wigner function. This procedure leadsto the quantum hydrodynamic (QHD) model, which includes quantum mechanicalcorrection terms to the compressible Euler equations. (iii) The “effective” partitionfunction Z =

∫Rdxρ(x, x, β)dx derived from the thermal equilibrium density matrix is

the starting point for solving problems in quantum statistical mechanics.Wigner [14] derived an approximate thermal equilibrium solution in 1932:

we(x, p, β∗) = h−d exp

−β∗

( |p|22m

+ V + φ

)

×[1 + h2

−β

∗2

8m∆xV +

β∗3

24m|∇xV |2 +

β∗3

24m2pjpk∂

2xjk

V

].(22)

The corresponding QHD model has been investigated in [5]. This O(h2) QHD modelexhibits complex quantum phenomena including quantum tunneling and resonance inthe resonant tunneling diode, and is an inexpensive way to simulate these phenomenacompared with a solution of the full quantum Liouville equation [11, 9]. Alternativeapproximations to the thermal equilibrium free energy and density matrix can befound in [4] and [3], respectively. One of the drawbacks of the O(h2) QHD equationsis the appearance of second derivatives of the potential V in the approximation (22)and consequently in the quantum correction to the stress tensor and energy density.Since the potential will be discontinuous at heterojunctions, this casts doubt on thevalidity of the O(h2) QHD equations near potential discontinuities.

We will take a different approach to the derivation of approximate solutions tothe Bloch equation based on an asymptotic expansion of the solution for “small”potentials. The main result is that in the corresponding “smooth” QHD model thepotential V is replaced by a smoothed potential S( 1

i∇x)V , where the symbol S(ξ)of the pseudodifferential operator S behaves like |ξ|−2 for large ξ. This approxima-tion is better suited for dealing with potential discontinuities and incorporates thenonlocal effects of potential barriers observed in solutions of the full quantum kineticequations [11]. In the QHD equations the stress tensor based on the smoothed poten-tial actually cancels the leading singularity in the classical potential at a barrier andleaves a residual smooth effective potential with a lower potential height in the barrier


region [7]. The smoothing makes the barrier partially transparent to the particle flowand provides the mechanism for particle tunneling in the smooth QHD model.

The paper is organized as follows. In section 2 we first derive Wigner’s approx-imate solution (22) in order to analyze its asymptotic validity. Then we proceed toderive the improved approximation based on the smoothed potential SV . Section 3deals with the asymptotic validity of the smoothed potential approximation for smallpotentials and high temperatures. We demonstrate the convergence of the approx-imate Wigner function as well as the particle density. In section 4 we derive thecorresponding QHD model. Section 5 is devoted to the numerical verification of theasymptotic results. It turns out that, at least for the double barrier structure consid-ered in section 5, the parameter range for which the approximation is valid is actuallymuch larger than expected.

Our paper [7] presents a derivation and interpretation of the smooth QHD modelfrom a more physical viewpoint.

2. Asymptotics. We start by bringing the initial value problem (14) and (15)into an appropriate scaled and dimensionless form. Note that adding a constant tothe potential V should not change the overall problem and that the Fermi level φ canbe eliminated from the Bloch equation. For the independent variables x, p, and β, weuse the scaling

x = Lxs, p =

√m

β∗ps, β = β∗βs,(23)

where L denotes a characteristic length scale. We scale the potential by

V (x) = V0 +ε

β∗Vs(xs), V0 =

1

2

[supx∈Rd

V + infx∈Rd

V

].(24)

An additive constant in the potential V is absorbed into V0 and does not affect Vs. εis a dimensionless parameter measuring the size of the potential. The scaled thermalequilibrium Wigner function and particle density are

we(x, p, β) = h−d exp−β(V0 + φ)ws(xs, ps, βs),(25)

ne(x) =

(m

β∗h2

) d2

exp−β∗(V0 + φ)ns(xs).(26)

The normalization condition (1) now reads∫Rdx

ns(xs)dxs = hd expβ∗(V0 + φ), ns(xs) =

∫Rdp

ws(xs, ps, 1)dps,(27)

where here h denotes the scaled Planck constant

h = h

√β∗

mL2.(28)

With this scaling the initial value problem (14) and (15) becomes

∂βsws =h2

8∆xsws − |p|2

2ws − ε

2

[Vs

(xs +

h

2i∇ps

)+ Vs

(xs − h

2i∇ps

)]ws,(29)


ws(xs, ps, 0) = 1(30)

with 0 ≤ β ≤ 1. The initial value problem (29) and (30) is independent of the Fermilevel φ, and φ can be computed after the fact from (27). After computing φ theoriginal thermal equilibrium Wigner function can be reconstructed from (25). Fornotational convenience we will drop the subscript s from here on.

We will first derive Wigner’s approximate solution. Our derivation differs fromWigner’s and illustrates the range of validity of this approximation and its limitations.We expand the operator on the right-hand side of (29) in powers of h to obtain

∂βw = −( |p|2

2+ εV

)w+

h2

8

[∆xw + ε

(∂2xjxk

V)∂2pjpk

w]− εh4

8rh

(x,

h

2i∇p

)w,

(31)

where the symbol of the remainder term rh is

rh

(x,h

2η

)=

1

h2

[δ2hV

(x,h

2η

)− ηjηk∂

2xjxk

V

](32)

with

δ2hV

(x,h

2η

)=

4

h2

[V

(x+

h

2η

)− 2V (x) + V

(x− h

2η

)].(33)

If the potential V is sufficiently smooth, the solution w can be expanded in powers ofh2:

w = w0 + h2w2 + h4w4 + · · · .(34)

Matching powers of h2 in the Bloch equation, we obtain

∂βw0 = −( |p|2

2+ εV

)w0, w0(x, p, 0) = 1,(35)

∂βw2 = −( |p|2

2+ εV

)w2 +

1

8

[∆xw0 + ε

(∂2xjxk

V)∂2pjpk

w0

], w2(x, p, 0) = 0,

(36)

yielding

w0(x, p, β) = exp

−β

( |p|22

+ εV

),(37)

w2(x, p, β) =1

8exp

−β

( |p|22

+ εV

)

×(−εβ2∆xV +

ε2β3

3|∇xV |2 +

εβ3

3pjpk∂

2xjxk

V

),(38)

w(x, p, 1) = exp

−|p|

2

2− εV


×[1 +

h2

8

(−ε∆xV +

ε2

3|∇xV |2 +

ε

3pjpk∂

2xjxk

V

)]+O(h4).(39)

In order to compute the quantum corrections to the fluid dynamical model in thenext section it will be necessary to calculate the thermal equilibrium stress tensor. Astraightforward calculation [5] gives for the O(h2) approximation

Pjk = − n

β∗δjk − h2β∗n

12m∂2xjk

V +O(h4).(40)

Note that π = 0 since we are working in a frame comoving with the fluid.There are several facts which should be pointed out here.• The formula (39) represents an approximation for small values of the scaled

Planck constant h, which is equivalent by (28) to large length scales or hightemperatures.

• Equation (34) represents only an asymptotic expansion and not a conver-gent series since the computation of higher order terms involves successivedifferentiation.

• The remainder term rh in (31) is bounded independently of h only if the po-tential is sufficiently smooth. In particular, we can expect this approximationto break down in the neighborhood of discontinuities in the potential.

We will now take a different tack and approximate the solution of the Blochequation not for small values of h but instead for applied potentials which are smallcompared to the inverse temperature β∗, that is, for small values of the parameterε. Expanding the thermal equilibrium Wigner function in powers of ε and matchingpowers of ε in the Bloch equation yields

wε =∞∑λ=0

wλελ,(41)

∂βw0 =h2

8∆xw0 − |p|2

2w0, w0(x, p, 0) = 1,(42)

∂βwλ =h2

8∆xwλ − |p|2

2wλ − µV wλ−1, wλ(x, p, 0) = 0 (λ = 1, 2, . . .),(43)

where

µV

(x,

h

2i∇p

)w(x, p, β) =

1

2

[V

(x+

h

2i∇p

)+ V

(x− h

2i∇p

)]w(x, p, β).(44)

The solution of (42) is

w0(x, p, β) = exp

−β

2|p|2

.(45)

To solve equation (43) for the first-order term w1, we make use of the equivalencyformula for autonomous pseudodifferential operators:

µV

(x,

h

2i∇p

)w0(p, β) = µw0

(p,

h

2i∇x, β

)V (x),(46)


where the operator µw0 is

µw0

(p,

h

2i∇x, β

)=

1

2

[w0

(p+

h

2i∇x, β

)+ w0

(p− h

2i∇x, β

)].(47)

It is more convenient to solve the Fourier transformed version of (43). To this endwe denote by f the Fourier transform of the function f with respect to the positionvariable x:

f(ξ) = (2π)−d2

∫Rdx

f(x)e−iξ·xdx.(48)

After Fourier-transforming in x and using (46), (43) reads, for λ = 1,

∂βw1 = −(h2

8|ξ|2 +

|p|22

)w1 − µw0

(p,h

2ξ, β

)V (ξ), w1(ξ, p, 0) = 0.(49)

In deriving this expression we have used the fact that the zeroth-order term w0 is ac-tually independent of the position variable x. The first-order term w1 in the expansionis therefore

w1(ξ, p, β) = −βg(ξ, p, β)V (ξ), w1(x, p, β) = −βg(

1

i∇x, p, β

)V (x)(50)

with the function g defined by

g(ξ, p, β) =1

β

∫ β

0

exp

(γ − β)

(h2

8|ξ|2 +

|p|22

)µw0

(p,h

2ξ, γ

)dγ.(51)

Thus the O(ε) approximation of the thermal equilibrium Wigner function is

w(x, p, β) = exp

−β

2|p|2

− εβg

(1

i∇x, p, β

)V (x) +O(ε2).(52)

For the fluid dynamical model in section 4 we will need the corresponding ap-proximations to the particle density and the stress tensor (again π = 0):

n(x) = (2π)d2

[1− εG0

(1

i∇x

)V (x)

]+O(ε2),(53)

P (x) = −(2π)d2

[I− εG2

(1

i∇x

)V (x)

]+O(ε2),(54)

where the symbols of the pseudodifferential operators G0 and G2 are

G0(ξ) = (2π)−d2

∫Rdp

g(ξ, p, 1)dp,(55)

G2(ξ) = (2π)−d2

∫Rdp

ppT g(ξ, p, 1)dp.(56)


The calculation of G0 and G2 is deferred to Appendix A. The results for the symbolsof the operators G0 and G2 are

G0(ξ) = κ

(h

2|ξ|),(57)

G2(ξ) =ξξT

|ξ|2 + κ

(h

2|ξ|)[

I− ξξT

|ξ|2],(58)

where

κ(z) =

∫ 1

0

exp

(γ2 − 1)

z2

2

dγ.(59)

The thermal equilibrium stress tensor in the O(ε) approximation is

P = −nI− εn(G0I−G2)

(1

i∇x

)V +O(ε2).(60)

In analogy with the Wigner approximation (40), (60) can be written as

Pjk = −nδjk − εh2n

12∂2xjk

Sh

(1

i∇x

)V +O(ε2),(61)

where the symbol of the smoothing operator Sh is

Sh(ξ) =12

h2|ξ|2[1− κ

(h

2|ξ|)]

.(62)

In unscaled form, (61) and (62) read

P = − n

β∗I− h2β∗n

12m∂⊗2x Sh

(1

i∇x

)V +O((β∗V )2),(63)

Sh(ξ) =12m

h2β∗|ξ|2

[1− κ

(h

2

√β∗

m|ξ|)]

.(64)

Simple Taylor expansion implies that Sh(ξ) = 1 +O(h2) for h→ 0. In this sensethe approximation (63) is consistent with the Wigner approximation (40). However,for large values of ξ, Sh(ξ) behaves like |ξ|−2 and the quantum correction ∂2

xSVhas the same degree of smoothness as the original potential V . In fact, the stresstensor (63) actually cancels the leading singularity in the classical potential at abarrier and leaves a residual smooth effective potential with a lower potential heightin the barrier region [7].

One disturbing fact remains: the approximation (52) of the thermal equilibriumWigner function does not reduce to the classical Maxwellian

exp

−β

2|p|2 − εβV

(65)


in the classical limit h→ 0. Letting h→ 0 in (52) yields

limh→0

g(ξ, p, β) = exp

−β

2|p|2

,(66)

limh→0

(w0 + εw1) = exp

−β

2|p|2

[1− εβV (x)],(67)

which is only the O(ε) approximation to the Maxwellian. This discrepancy can beremedied by replacing terms of the form 1 − εz by e−εz in the formulas (52)–(54),thereby introducing only an additional O(ε2) error term:

w(x, p, β) = exp

−β

2|p|2 − εβg1

(1

i∇x, p, β

)V (x)

+O(ε2),(68)

n(x) = (2π)d2 exp

−εG0

(1

i∇x

)V (x)

+O(ε2),(69)

Pjk(x) =

−(2π)d2 exp

−εG2jk

(1i∇x

)V (x)

+O(ε2), j = k,

(2π)d2 εG2jk

(1i∇x

)V (x) +O(ε2), j 6= k,

(70)

with g1(ξ, p, β) = expβ|p|2/2 g(ξ, p, β). The approximation (68) now reduces to the

classical Maxwellian for ε→ 0 as well as for h→ 0. Since the difference between (68)and (52) is O(ε2), (70) yields the same stress tensor (63).

3. Convergence. We now analyze the asymptotic validity of the expansion de-rived in section 2 by providing estimates of the remainder term. We define the re-mainder term r by w = w1 + εw2 + ε2r and obtain

∂βr =h2

8∆xr − |p|2

2r − µV

(x,

h

2i∇p

)[w1 + εr], r(x, p, 0) = 0.(71)

To prove the boundedness of the remainder term r in the L2(Rdx × Rd

p) norm is astraightforward exercise in energy inequalities. However, we are more concerned withthe approximation quality of the zeroth- and second-order moments of the expansion,i.e., with the convergence of the particle density and the momentum flux densitytensor. The main result of this section is Theorem 3, which proves the convergenceof the particle density as ε → 0. The convergence of the momentum flux densitytensor remains an unresolved problem at this point, although the numerical evidencein section 5 indicates that the smoothed potential approximation also produces a goodapproximation to Π. We start with

Theorem 1. Let the potential V be bounded. Then

max0≤β≤1

||r(β)||L2(Rd

x×Rdp)

≤ const ||V ||L∞(Rd

x) max0≤β≤1

||w1(β)||L2(Rd

x×Rdp)

,(72)

where const denotes a positive constant independent of ε and h.Proof. Multiplying both sides of (71) by r and integrating with respect to x and

p gives

1

2∂β ||r(x, p, β)||L2(Rd

x×Rdp)

= −∫Rdp

∫Rdx

(h2

8|∇xr|2 +

1

2|pr|2 + rµV [εr + w1]

)dxdp.

(73)


Using the definition of the pseudodifferential operator µV yields∫Rdp

∫Rdx

rµV [εr + w1]dxdp =

∫Rdη

∫Rdx

r∗(x, η, β)µV

(x,h

2η

)[εr + w1]dxdη

≤ ||V ||L∞(Rdx)||r||L2(Rd

x×Rdp)||εr + w1||L2(Rd

x×Rdp).(74)

Combining (73) with (74) implies

∂β ||r||L2(Rdx×Rd

p)≤ const ||V ||L∞(Rd

x)

[ε||r||L2(Rd

x×Rdp)

+ ||w1||L2(Rdx×Rd

p)

].(75)

The more interesting question, however, is the convergence of the moments of theremainder term, i.e., the degree of approximation of the particle and momentum fluxdensities. The existence of the moments of the Wigner function in the presence ofdiscontinuous potentials is a challenging problem (see [1, 2] for the treatment of thetransient case). In the thermal equilibrium case described by the Bloch equation, weare aided by the smoothing properties of the operator h2∆x/8− |p|2/2.

Lemma 1. For a function f(x, p) let the λth-order moment Mλf(x) be defined by

Mλf(x) =

∫Rdp

p⊗λf(x, p)dp, λ = 0, 1, 2, . . . .(76)

Then

||Mλf ||L2(Rdx) ≤ const ||(1 + |p|2)α2 f(x, p)||L2(Rd

x×Rdp)

(77)

for α > λ+ 12 .

Proof. We have

||Mλf ||2L2(Rdx) =

∫Rdx

[∫Rdp

p⊗λf(x, p)dp

]2

dx

≤∫Rdx

[∫Rdp

|p⊗λ|2(1 + |p|2)−αdp][∫

Rdp

(1 + |p|2)α|f(x, p)|2dp]dx.(78)

For α > λ + 12 the term

∫Rdp|p⊗λ|2(1 + |p|2)−αdp remains bounded and the result

follows immediately.In order to estimate the moments of the remainder term r, we will estimate the

L2-norm of the function rα defined by

rα(x, p, β) = (1 + |p|2)α2 r(x, p, β).(79)

rα satisfies the initial value problem

∂βrα =h2

8∆xrα − |p|2

2rα − Ωα[w1α + εrα], rα(x, p, 0) = 0,(80)

where

Ωα = (1 + |p|2)α2 µV(x,

h

2i∇p

)(1 + |p|2)−α

2(81)


and

w1α(x, p, β) = (1 + |p|2)α2 w1(x, p, β).(82)

For the estimate of rα we will use a specific norm defined by

||f(x, p, β)||2 =

∫Rdp

∫Rdx

max0≤β≤1

|f(ξ, p, β)|2dxdp,(83)

where

f(ξ, p, β) = (2π)−d2

∫Rdx

f(x, p, β)e−iξ·xdx.(84)

In this norm we can estimate the solution of the initial value problem (71) with thefollowing theorem.

Theorem 2. Let the function r satisfy the initial value problem

∂βr =h2

8∆xr − |p|2

2r − Ωα[εr + w1α], 0 ≤ β ≤ 1, r(x, p, 0) = 0(85)

with the operator Ωα defined in (81). Let the potential V be smooth enough that itsFourier transform satisfies |V (ξ)|2 ≤ const [1 + |ξ|2]−1. Then for ε sufficiently small,||r|| ≤ const ||w1α|| for 0 ≤ α < 3.

The proof is deferred to Appendix B. The condition on the smoothness of the po-tential in Theorem 2 is modest and allows for discontinuous potentials. What remainsto be shown is the bound on the inhomogeneous term w1α in (80). Unfortunately, w1α

is bounded in the norm defined in (83) only for 0 ≤ α < 3/2. While this range of αprovides a bound for the zeroth-order moment (the particle density), a value α > 5/2would be needed to estimate the second-order moment (the momentum flux densitytensor).

Lemma 2. Let the function w1α be defined by

w1α(x, p, β) = (1 + |p|2)α2 w1(x, p, β)(86)

with w1 given by (50). Then ||w1α|| < ∞ for 0 ≤ α < 3/2 with the norm definedin (83).

The proof is deferred to Appendix C.Combining Theorem 2 and Lemma 2 produces the bound on the function rα.Theorem 3. Let the remainder term r and the function rα be the solutions

of the initial value problems (71) and (80), respectively. Then ||rα|| ≤ const for0 ≤ α < 3

2 . Consequently, ||M0r||L2(Rdx) ≤ const by Lemma 1, and the approximation

of the particle density in section 2 satisfies

||n− n0 − εn1||L2(Rdx) = O(ε2).(87)

4. QHD models. Next we discuss the QHD models corresponding to the ap-proximations of the thermal equilibrium distribution function derived in section 2.The nonequilibrium Wigner function is the solution of the quantum Liouville (orWigner–Boltzmann) equation

∂tf +1

mdivx(pf)− θf = Cf,(88)


where the operator θ is defined by

θ =i

h

[V

(x+

h

2i∇p

)− V

(x− h

2i∇p

)].(89)

The quantum Liouville equation can be derived from the transient Schrodinger equa-tion in the same way the Bloch equation is derived from the eigenvalue problem forthe Schrodinger equation. The derivation makes use of the transformation

f(x, p, t) =∑λ

aλ

∫Rdη

ψ∗λ

(x+

h

2η, t

)ψλ

(x− h

2η, t

)eiη·pdη,(90)

where the wavefunctions ψλ(x, t) are solutions of the transient Schrodinger equationand the aλ’s are occupation numbers:

∑λ aλ = Nparticles. The collision operator C

in (88) is added after the fact and its form is usually based on some ad hoc assump-tions. (Typically either a relaxation time model or a Fokker–Planck collision operatoris assumed.) We will leave the form of the collision operator open.

We denote the average value of a quantity χ with the distribution function f by

〈χ〉 (x, t) =

∫Rdp

χ(x, p, t)f(x, p, t)dp.(91)

Integrating the quantum Liouville equation against 1, p, and |p|2/2m yields conserva-tion laws for particle number, momentum, and energy:

∂t 〈1〉+1

mdivx 〈p〉 = 〈C〉 ≡ C0,(92)

∂t 〈p〉+ divx

⟨ppT

m

⟩+ 〈1〉∇xV = 〈pC〉 ≡ C1,(93)

∂t

⟨ |p|22m

⟩+ divx

⟨ |p|2p2m2

⟩+⟨ p

m

⟩· ∇xV =

⟨ |p|22m

C

⟩≡ C2.(94)

The system (92)–(94) has to be closed by expressing Π =⟨ppT /m

⟩and Θ ≡ ⟨|p|2p/2m2

⟩in terms of (say) the conserved quantities n = 〈1〉, π = 〈p〉, and W =

⟨|p|2/2m⟩. Notethat the moment equations are identical to those for the classical Boltzmann transportequation and that quantum effects enter through the closure conditions.

We first change from the moments with respect to p to the centered momentswith respect to the variable p′ defined by

p = p′ +mu, 〈p〉 = mnu,(95)

where u denotes the macroscopic fluid velocity. This shift gives the transformations

Π =

⟨ppT

m

⟩= mnuuT +

⟨p′p′T

m

⟩= mnuuT − P,(96)

W =

⟨ |p|22m

⟩=

1

2mn|u|2 +

⟨ |p′|22m

⟩=

1

2mn|u|2 − 1

2TraceP,(97)


Θ =

⟨ |p|2p2m2

⟩= u

(⟨ |p′|22m

⟩+

1

2mn|u|2

)+uT

⟨p′p′T

m

⟩+

⟨ |p′|2p′2m2

⟩= uW−uTP+q,

(98)

where q is the heat flux vector. We obtain

∂tn+ divx(nu) = C0,(99)

∂t(mnu) + divx(mnuuT − P

)+ n∇xV = C1,(100)

∂tW + divx(uW − uTP + q

)+ nu · ∇xV = C2.(101)

Closure is obtained by assuming that the distribution function f is close to amomentum-displaced thermal equilibrium distribution. We assume that

f(x, p, t) = A(x, t)we(x, p−mu),(102)

⟨p′⊗λ

⟩= A(x, t)

∫Rdp

p⊗λwe(x, p)dp(103)

for a smooth function A(x, t). We will use (103) to express P , W , and q in terms ofthe variables n, u, and T , where β∗ is replaced by 1/T with T the local temperature.In the classical case the thermal equilibrium distribution function is the Maxwellianexp−|p|2/2mT, which yields

P = −nT I,(104)

W =3

2nT +

1

2mn|u|2,(105)

and leads to the classical compressible Euler equations. For the Wigner approxi-mation (39) and the smoothed potential approximation (68), the stress tensor P iscalculated in (40) and (63), respectively. The Wigner approximation yields the O(h2)quantum hydrodynamic model treated in [5]. For the smoothed potential approxima-tion in section 2, we obtain from (63)

Pjk = −nTδjk − h2n

12mT∂2xjk

(SV ),(106)

W =3

2nT +

1

2mn|u|2 +

h2n

24mT∆x(SV ),(107)

where the smoothed potential SV is

SV (x) = (2π)−d∫Rdξ

∫Rdy

Sh(ξ)V (y)eiξ·(x−y)dydξ(108)

with

Sh(ξ) =12mT0

h2|ξ|2[1− κ

(h

2

|ξ|√mT0

)](109)

and

κ(z) =

∫ 1

0

exp

(γ2 − 1)

z2

2

dγ.(110)

Note that even though q = 0 for both the Wigner and smoothed potential approxima-tions, we have allowed a nonzero q in the QHD equations since heat conduction playsa quantitative role in electron and hole propagation in actual semiconductor devices.


-150 -100 -50 0 50 100 150x

0

0.2

0.4

0.6

0.8

1

V

Fig. 1. Unit double barrier potential. x is in A for all figures.

5. Numerical results. In this section we present numerical results to verify theasymptotic expansions of section 2 and to test the limits of their validity. To this endwe consider an electron gas in a GaAs/AlxGa1−xAs double barrier heterostructure(see Figure 1) at 300 K. (The double barrier heterostructure is the heart of theresonant tunneling diode.) The barrier width is 25 A and the well width is 50 A ona characteristic length scale of 100 A. Technologically relevant barrier heights are inthe range 0.1–0.7 eV. Without loss of generality we can assume that the barriers areperpendicular to the x1 coordinate axis. The salient features of the numerical resultsare the following:

• The asymptotic expansion (68) yields a good approximation of both the par-ticle density and the energy density. (Note that the velocity u = 0.)

• The range of validity of the expansion extends far beyond the small ε regimeto ε ≈ 2–10.

• The smoothed potential approximation for the stress tensor (63) is uniformlybetter than the one obtained through the Wigner approximation (40).

We will compare the Wigner and smoothed potential approximations with a nu-merical solution of the Bloch equation. The Bloch equation was solved on a gridof 200 ∆x using the backward Euler method with homogeneous Neumann boundaryconditions. A detailed description of the basic numerical method plus extensions isgiven in [6]. Numerical values for the constants and scaling factors in section 2 aregiven in Table 1.

The parameter ε is chosen so that the scaled potential Vs varies between −1 and1. We have set the effective electron mass in GaAs to m = 0.063 me, where me is theelectron mass.


Table 1Numerical values of constants used in the computations. B denotes the barrier height divided

by eV.

Constant Value

h 6.582× 10−16 eV sec

m 3.582× 10−17 eV sec2 cm−2

β∗ 38.68 eV−1

L 10−6 cmh 0.6840ε 19.34 B

Since the potential V is one-dimensional, we can reduce the number of indepen-dent variables in the Bloch equation by setting

w(x, p, β) = w(x1, p1, β) exp

−β

2(p2

2 + p23)

.(111)

The function w satisfies the initial value problem

∂βw =h2

8∂2xw −

p21

2w − ε

2

[V

(x1 +

h

2i∂p1

)+ V

(x1 − h

2i∂p1

)]w,(112)

w(x1, p1, 0) = 1(113)

with 0 ≤ β ≤ 1.Figures 2–4 show the case of a barrier height of 0.01 eV, which corresponds to

ε = 0.1934. Figure 2 shows the electron density. (The overall scale for the solutionsis set by requiring

∫n(x1)dx1 = 1 cm−2.) Figure 3 shows the quantum mechanical

correction terms Qs (for the smoothed potential approximation) and Qf (for thenumerical solution of the full Bloch equation) in the electron momentum flux:

Qs =εh2

12∂2x1Sh

(1

i∂x1

)V (x1),(114)

Qf = −P11

n− T.(115)

The smooth effective potential Q+ V for the smoothed potential approximation andfor the numerical solution of the full Bloch equation is shown in Figure 4. In Figures2–10, the solid lines correspond to the smoothed potential approximations and thedashed lines correspond to numerical solutions of the full Bloch equation.

Figures 5–7 and 8–10 are a repetition of Figures 2–4 for higher potential barriers.Figures 5–7 show the results for a 0.1 eV double barrier, corresponding to ε = 1.934,and Figures 8–10 show the results for a barrier height of 0.5 eV, corresponding toε = 9.67.

The electron density in Figures 2, 5, and 8 is minimized inside the potentialbarriers and has a local maximum at the center of the quantum well. The energydensity drops precipitously as the electrons tunnel through the potential barriers andrises dramatically as the electrons exit from the barriers. The effect of the quantumcorrection in Figures 3, 6, and 9 is to partially cancel the effects of the potentialbarriers (see Figures 4, 7, and 10).


-150 -100 -50 0 50 100 150x

0.27

0.28

0.29

0.3

0.31

Density

SmoothBloch

Fig. 2. Electron density in 106 cm−3 for a 0.01 eV double barrier in GaAs at 300 K (ε ≈ 0.2).Note that the vertical axis begins at 0.266.

-150 -100 -50 0 50 100 150x

-0.006

-0.004

-0.002

0

0.002

0.004

Q

SmoothBloch

Fig. 3. Quantum correction term in eV for a 0.01 eV double barrier (ε ≈ 0.2).


-150 -100 -50 0 50 100 150x

0

0.001

0.002

0.003

0.004

Q + V

SmoothBloch

Fig. 4. Smooth effective potential in eV for a 0.01 eV double barrier (ε ≈ 0.2).

Although the difference between the smoothed potential approximation and the“exact” solution in Figures 6 and 9 becomes significant, we still obtain a qualita-tively good approximation to the quantum correction term Qf for large values of theperturbation parameter ε.

In contrast, the quantum correction term Qw obtained from the Wigner approxi-mation (39) is completely dominated by the dipole behavior induced by the potentialdiscontinuities:

Qw =εh2

12∂2x1V (x1) =

εh2B

12[δ′(x1 + 2a)− δ′(x1 + a) + δ′(x1 − a)− δ′(x1 − 2a)]

(116)

with a = 25 A. Thus our numerical evidence, convergence results, and the dipolebehavior of Qw demonstrate that the smoothed potential approximation is uniformlybetter for potentials with discontinuities than the O(h2) approximation.

Smooth QHD simulations of the resonant tunneling diode (RTD) and compari-son with experimental current-voltage curves are presented in [8]. Future work willcompare the smooth QHD simulations of the RTD with numerical simulations of thecoupled quantum Liouville/Poisson equations.

Appendix A. Calculation of G0 and G2. According to (51), (55), and (56),the functions G0(ξ) and G2(ξ) are defined by

Gm(ξ) = (2π)−d2

∫ 1

0

∫Rdp

p⊗m exp

(γ − 1)

(h2

8|ξ|2 +

|p|22

)µw0

(p,h

2ξ, γ

)dpdγ

(117)


-150 -100 -50 0 50 100 150x

0

0.1

0.2

0.3

0.4

Density

SmoothBloch

Fig. 5. Electron density in 106 cm−3 for a 0.1 eV double barrier in GaAs at 300 K (ε ≈ 2).

-150 -100 -50 0 50 100 150x

-0.06

-0.04

-0.02

0

0.02

0.04

Q

SmoothBloch

Fig. 6. Quantum correction term in eV for a 0.1 eV double barrier (ε ≈ 2).


-150 -100 -50 0 50 100 150x

0

0.01

0.02

0.03

0.04

0.05

Q + V

SmoothBloch

Fig. 7. Smooth effective potential in eV for a 0.1 eV double barrier (ε ≈ 2).

-150 -100 -50 0 50 100 150x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Density

SmoothBloch

Fig. 8. Electron density in 106 cm−3 for a 0.5 eV double barrier in GaAs at 300 K (ε ≈ 10).


-150 -100 -50 0 50 100 150x

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Q

SmoothBloch

Fig. 9. Quantum correction term in eV for a 0.5 eV double barrier (ε ≈ 10).

-150 -100 -50 0 50 100 150x

0

0.1

0.2

0.3

0.4

Q + V

SmoothBloch

Fig. 10. Smooth effective potential in eV for a 0.5 eV double barrier (ε ≈ 10).


with m = 0, 2. Splitting Gm into Gm = 12 (G+

m +G−m), we define

G±m(ξ) = (2π)−d2

∫ 1

0

∫Rdp

p⊗m exp

(γ − 1)

h2

8|ξ|2

u(p, γ)w0

(p± h

2ξ, γ

)dpdγ,

(118)

where

u(p, γ) = exp

(γ − 1)

2|p|2

.(119)

For m = 0, integrating with respect to p gives∫Rdp

u(p, γ)w0

(p+

h

2ξ, γ

)dp =

∫Rdη

u(−η, γ)w0(η, γ) exp

ih

2η · ξ

dη

=

∫Rdη

[γ(1− γ)]−d2 exp

− 1

2γ(1− γ)|η|2 + i

h

2η · ξ

dη

= (2π)d2 exp

−γ(1− γ)

h2

8|ξ|2

.(120)

Thus

G+0 (ξ) =

∫ 1

0

exp

(γ2 − 1)

h2

8|ξ|2

dγ = κ

(h

2|ξ|)

(121)

with the function κ defined in (59). G−0 is obtained by replacing h by−h, and thereforeG0 = G+

0 = G−0 = κ(h|ξ|/2).To compute the matrix G+

2 we have to evaluate the integrals∫Rdp

pjpku(p, γ)w0

(p+

h

2ξ, γ

)dp

= −∫Rdη

[∂2jku(−η, γ)

]w0(η, γ) exp

ih

2η · ξ

dη

= [γ(1− γ)]−d2

1

1− γ

∫Rdη

[δjk − ηjηk

1− γ

]exp

− 1

2γ(1− γ)|η|2 + i

h

2η · ξ

dη

=1

1− γ

∫Rdη

(δjk − γηjηk) exp

−|η|

2

2+ i

h

2

√γ(1− γ)η · ξ

dη

= (2π)d2

1

1− γ[δjk + γ∂2

zjzk] exp

−|z|

2

2

at z =

h

2

√γ(1− γ)ξ

= (2π)d2

[δjk +

γ

1− γzjzk

]exp

−|z|

2

2

at z =

h

2

√γ(1− γ)ξ.(122)


Inserting (122) into (118) gives

G+2jk

(ξ) =

∫ 1

0

(δjk +

γ2h2

4ξjξk

)exp

(γ2 − 1)

h2

8|ξ|2

dγ,(123)

and integrating by parts yields

G+2jk

(ξ) = |ξ|−2

[ξjξk + (|ξ|2δjk − ξjξk)κ

(h

2|ξ|)]

.(124)

G−2 is obtained by replacing h by −h, and therefore G2 = G+2 = G−2 .

Appendix B. Proof of Theorem 2. To prove Theorem 2 we will need thefollowing two lemmas. First we estimate the principal part of the operator in (71).

Lemma 3. Let the function u(x, p, β) satisfy the initial value problem

∂βu =h2

8∆xu− |p|2

2u+

(1− h2

8∆x +

|p|22

)f, u(x, p, 0) = 0,(125)

with 0 ≤ β ≤ 1. Then ||u|| ≤ const||f ||, where const denotes a positive constantindependent of h and ε and where the norm || · || is defined in (83).

Proof. The Fourier transform u of the solution of (125) is

u(ξ, p, β) =

∫ β

0

exp

(γ − β)

(h2

8|ξ|2 +

|p|22

)(1 +

h2

8|ξ|2 +

|p|22

)f(ξ, p, γ)dγ.

(126)

Thus

|u(ξ, p, β)| ≤M

(1 +

h2

8|ξ|2 +

|p|22

)∫ β

0

exp

(γ − β)

(h2

8|ξ|2 +

|p|22

)dγ

≤M

(1 +

h2

8|ξ|2 +

|p|22

)[1− exp

−β

(h2

8|ξ|2 +

|p|22

)][h2

8|ξ|2 +

|p|22

]−1

,

(127)

where

M = max0≤β≤1

|f(ξ, p, β)|.(128)

Since the function g(z, β) = 1+zz (1 − e−βz) is uniformly bounded for 0 ≤ z < ∞,

0 ≤ β ≤ 1, we obtain

max0≤β≤1

|u(ξ, p, β)| ≤ const max0≤β≤1

|f(ξ, p, β)|.(129)

Squaring both sides of (129) and integrating with respect to ξ and p proves the lemma.

Next we estimate the inhomogeneous term in (125).


Lemma 4. Let the function f(x, p, β) be given by(1− h2

8∆x +

|p|22

)f(x, p, β) = Ωαu(130)

for some function u, where the operator Ωα is defined in (81). Then if the Fourier

transform of the potential satisfies |V (ξ)| ≤ const(1 + |ξ|2)− 12 , ||f || ≤ const||u|| for

0 ≤ α < 3.Proof. We split the function f and the operator Ωα into f = 1

2 (f+ + f−), Ωα =12 (Ω+

α + Ω−α ), where f± and Ω±α are defined by

Ω±α = (1 + |p|2)α2 V(x± h

2i∇p

)(1 + |p|2)−α

2 ,(131)

(1− h2

8∆x +

|p|22

)f± = Ω±αu.(132)

Fourier-transforming the function f+ with respect to the position variable x yields

f+(ξ, p, β) = (2π)−3d2

(1 +

h2

8|ξ|2 +

|p|22

)−1

(1 + |p|2)α2

×∫Rdx

∫Rdq

∫Rdη

V

(x+

h

2η

)(1 + |q|2)−α

2 u(x, q, β)eiη·(p−q)−iξ·xdηdqdx

= (2π)−d2

(1 +

h2

8|ξ|2 +

|p|22

)−1

(1 + |p|2)α2

×∫Rdω

V (ω)

(1 +

∣∣∣∣p+h

2ω

∣∣∣∣2)−α

2

u

(ξ − ω, p+

h

2ω, β

)dω.(133)

Taking the maximum with respect to β and integrating with respect to p yields∫Rdp

max0≤β≤1

|f+(ξ, p, β)|2dp

≤ maxω,p

(

1 +h2

8|ξ|2 +

|p|22

)−2

(1 + |p|2)α(1 + |ω|2)−1

(1 +

∣∣∣∣p+h

2ω

∣∣∣∣2)−α

×∫Rdω

∫Rdp

maxβ

∣∣∣∣u(ξ − ω, p+

h

2ω, β

)∣∣∣∣2dpdω(134)

or ∫Rdp

max0≤β≤1

|f+(ξ, p, β)|2dp ≤ H(ξ)||u||2,(135)


where

H(ξ) = maxω,p

(

1 +h2

8|ξ|2 +

|p|22

)−2

(1 + |p|2)α(1 + |ω|2)−1

(1 +

∣∣∣∣p+h

2ω

∣∣∣∣2)−α

.

(136)

Elementary calculus implies that∫RdξH(ξ)dξ < ∞ for 0 ≤ α < 3. The lemma fol-

lows from integrating (135) with respect to ξ and repeating the same procedure forf−.

The proof of Theorem 2 consists of the combination of Lemmas 3 and 4 with thefunction u in Lemma 4 replaced by εrα + w1α.

Appendix C. Proof of Lemma 2. The x-Fourier transform of the functionw1α is

w1α(ξ, p, β) = −(1 + |p|2)α2 V (ξ)

×∫ β

0

exp

(γ − β)

(h2

8|ξ|2 +

|p|22

)µw0

(p,h

2ξ, γ

)dγ.(137)

We split the function w1α into w1α = 12 (w+

1α + w−1α), with w±1α defined by

w±1α = −(1+|p|2)α2 V (ξ)

∫ β

0

exp

(γ − β)

(h2

8|ξ|2 +

|p|22

)− γ

2

∣∣∣∣p± h

2ξ

∣∣∣∣2dγ

(138)

and obtain for w+1α

w+1α = −2(1 + |p|2)α2 V (ξ)

[∣∣∣∣p+h

2ξ

∣∣∣∣2

−(|p|2 +

h2

4|ξ|2

)]−1

×[exp

−β

2

∣∣∣∣p+h

2ξ

∣∣∣∣2− exp

−β

2

(|p|2 +

h2

4|ξ|2

)].(139)

The function g(β) = (e−βb − e−βa)/(a− b) satisfies

0 ≤ g(β) ≤ exp

b ln(b)− a ln(a)

a− b

≤ const

1

1 + a(140)

for 0 ≤ b ≤ 2a. Thus

|w+1α(ξ, p, β)| ≤ const(1 + |p|2)α2 |V (ξ)|

[1 + |p|2 +

h2

4|ξ|2

]−1

(141)

and

||w+1α||2 ≤

∫Rdξ

∫Rdp

(1 + |p|2)α2 |V (ξ)|2[1 + |p|2 +

h2

4|ξ|2

]−2

dpdξ.(142)

The integral on the right-hand side of (142) is convergent for 0 ≤ α < 32 . Repeating

the same argument for w−1α proves the lemma.


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[3] D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equa-tions for semiconductor device modeling, Phys. Rev. B, 48 (1993), pp. 7944–7950.

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