Download approximation of thermal equilibrium for quantum gases with

SIAM J. APPL. MATH.
Vol. 58, No. 3, pp. 780–805, June 1998
c 1998 Society for Industrial and Applied Mathematics
°
005
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM
GASES WITH DISCONTINUOUS POTENTIALS AND
APPLICATION TO SEMICONDUCTOR DEVICES∗
CARL L. GARDNER† AND CHRISTIAN RINGHOFER†
Abstract. We derive an approximate solution valid to all orders of h̄ to the Bloch equation
for quantum mechanical thermal equilibrium distribution functions via asymptotic analysis for high
temperatures and small external potentials. This approximation can be used as initial data for
transient solutions of the quantum Liouville equation, to derive quantum mechanical correction terms
to the classical hydrodynamic model, or to construct an effective partition function in statistical
mechanics. The validity of the asymptotic solution is investigated analytically and numerically and
compared with Wigner’s O(h̄2 ) solution. Since the asymptotic analysis results in replacing second
derivatives of the potential in the correction to the stress tensor in the original O(h̄2 ) quantum
hydrodynamic model by second derivatives of a smoothed potential, this approach represents a
definite improvement for the technologically important case of piecewise continuous potentials in
quantum 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 through
potential barriers are playing an increasingly important role in advanced microelectronic applications, including multiple-state logic and memory devices and high frequency oscillators. The propagation of electrons and holes in the semiconductor device
can be modeled as the flow of a continuous charged quantum gas in a potential that
has discontinuous jumps at heterojunction barriers. The fluid dynamical equations
are derived by assuming the gas is near thermal equilibrium, but are expected to be
more generally valid.
This investigation is therefore concerned with the approximate description of thermal equilibrium quantum mechanical systems of particles in a potential—especially
in the presence of discontinuous potential barriers. Under the assumption of Boltzmann statistics, thermal equilibrium of a quantum mechanical system is described
by the factor exp{−β ∗ (H(V ) + φ)}, where H(V ) is the Hamiltonian operator, V is
the potential energy, β ∗ is the reciprocal value of the ambient temperature, and φ is
the Fermi level. Depending on the choice of representation of the system, the term
exp{−β ∗ (H(V ) + φ)} can be expressed in various ways. Choosing a representation
via Schrödinger wavefunctions amounts to solving the eigenvalue problem
h̄2
∆x + V (x), x ∈ Rd ,
2m
where d = 1, 2, or 3, m is the particle mass, and the ψλ (λ = 0, 1, 2, . . .) are the
particle wavefunctions. Observable quantities like the thermal equilibrium (denoted
(1)
Eλ ψλ (x) = H(V )ψλ (x),
H(V ) = −
∗ 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. Army
Research Office grant DAAH04-95-1-0122, and the research of the second author was supported in
part 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 momentum
flux density tensor Πe (x) are computed as
X
(2)
exp{−β ∗ (Eλ + φ)}|ψλ (x)|2 ,
ne (x) =
λ
(3)
πe (x) =
h̄ X
exp{−β ∗ (Eλ + φ)}ψλ∗ (x)∇x ψλ (x),
i
λ
(4)
Πe (x) =
h̄2 X
exp{−β ∗ (Eλ + φ)} {∇x ψλ∗ (x)} ∇x ψλ (x).
m
λ
The summation over the index λ is replaced by an integral where the spectrum of the
Hamiltonian is continuous.
Fluid dynamical equations are usually formulated in terms of the stress tensor
Pjk , which is related to the momentum flux density tensor by
(5)
Πjk =
1
πj πk − Pjk ,
mn
where j, k = 1, . . . , d. The energy density W is calculated from the momentum flux
density as
W =
(6)
1
Πjj .
2
We will use the summation convention where repeated Latin indices j, k are summed
over.
For many applications it is advantageous to express the term exp{−β ∗ (H(V )+φ)}
via the density matrix
X
ρe (x, y, β) =
(7)
exp{−β(Eλ + φ)}ψλ (x)ψλ∗ (y).
λ
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
(8)
(9)
∂β ρe =
h̄2
1
(∆x + ∆y )ρe − [V (x) + V (y)] ρe − φρe ,
4m
2
ρe (x, y, β = 0) = δ(x − y).
Thus, solving the eigenvalue problem for the Schrödinger equation is replaced by
solving a parabolic initial value problem in twice as many dimensions. The Bloch
equation is the interpretation of the term exp{−β ∗ (H + φ)} as a semigroup.
The initial condition (9) means that we prescribe randomized behavior at infinite
temperature (β = 0) and solve the Bloch equation up to the specified ambient temperature T0 = 1/β ∗ . (We set Boltzmann’s constant kB = 1.) In the density matrix
representation, the particle, momentum, and momentum flux densities are
(10)
ne (x) = ρe (x, x, β ∗ ),
782
CARL L. GARDNER AND CHRISTIAN RINGHOFER
(11)
πe (x) =
Πe (x) =
(12)
h̄
∇x ρe (x, y, β ∗ )|y=x ,
i
h̄2
∇x ∇y ρe (x, y, β ∗ )|y=x .
m
To relate the quantum mechanical description to the classical picture, we use the
Wigner transform of the Bloch equation. The Wigner (distribution) function [14] is
Z
h̄
h̄
(13)
ρ x − η, x + η, β eiη·p dη,
w(x, p, β) =
2
2
d
Rη
where the new independent variable p is the momentum. Applying the transformation (13) to the Bloch equation (8) yields the Bloch equation for the thermal
equilibrium Wigner function
h̄
h̄
h̄2
|p|2
1
∆x we −
we −
V x + ∇p + V x − ∇p we − φwe ,
(14) ∂β we =
8m
2m
2
2i
2i
we (x, p, β = 0) = h̄−d .
(15)
h̄
The operators V (x ± 2i
∇p ) are defined as
Z Z
h̄
h̄
−d
V x ± η we (x, q, β)eiη·(p−q) dqdη.
V x ± ∇p we (x, p, β) = (2π)
2i
2
d
Rη
Rqd
(16)
In the Wigner representation the thermal equilibrium particle, momentum, and momentum flux densities are
Z
ne (x) =
(17)
we (x, p, β ∗ )dp,
d
Rp
Z
(18)
πe (x) =
(19)
Πe (x) =
1
m
d
Rp
Z
d
Rp
p we (x, p, β ∗ )dp,
ppT we (x, p, β ∗ )dp.
The relationship between the quantum mechanical description and the classical
picture is transparent using the Wigner representation. Setting h̄ to zero formally in
the Bloch equation (14) immediately gives the classical Maxwellian
2
|p|
(20)
+V +φ
.
we (x, p, β) = h̄−d exp −β
2m
The Fermi level φ just plays the role of a scaling parameter and is chosen to determine
the total number Nparticles of particles in the system
Z
X
(21)
ne (x)dx =
exp{−β ∗ (Eλ + φ)} = Nparticles .
d
Rx
λ
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
783
Note that because of the properties of the exponential function in Boltzmann statistics, 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. The
situation becomes more complex for Fermi–Dirac or Bose–Einstein statistics [13].
We will construct approximate solutions to the Bloch equation (8) or its Wigner
transformed version (14). The motivation for this work is threefold. (i) Transient
solutions of quantum kinetic equations usually require a thermal equilibrium Wigner
function as initial data. To obtain the thermal equilibrium solution either via the
eigenvalue problem for the Schrödinger equation [10] or via the Bloch equation [12]
is an expensive computational task. (ii) Approximations to the thermal equilibrium
distribution function can be used to derive quantum mechanical corrections for macroscopic fluid dynamical equations. As in the classical case, equations for the nonequilibrium densities of particles, momentum, and energy can be derived by building the
appropriate moments of the nonequilibrium quantum transport equation, the transient quantum Liouville equation. Closure of the infinite hierarchy of moment equations is obtained by assuming that the nonequilibrium Wigner function is close to
the momentum-displaced thermal equilibrium Wigner function. This procedure leads
to the quantum hydrodynamic (QHD) model, which includes quantum mechanical
correction terms
R to the compressible Euler equations. (iii) The “effective” partition
function Z = Rd ρ(x, x, β)dx derived from the thermal equilibrium density matrix is
x
the starting point for solving problems in quantum statistical mechanics.
Wigner [14] derived an approximate thermal equilibrium solution in 1932:
2
|p|
+V +φ
we (x, p, β ∗ ) = h̄−d exp −β ∗
2m
(22)
× 1 + h̄
2
β ∗2
β ∗3
β ∗3
∆x V +
|∇x V |2 +
−
pj pk ∂x2jk V
8m
24m
24m2
.
The corresponding QHD model has been investigated in [5]. This O(h̄2 ) QHD model
exhibits complex quantum phenomena including quantum tunneling and resonance in
the resonant tunneling diode, and is an inexpensive way to simulate these phenomena
compared with a solution of the full quantum Liouville equation [11, 9]. Alternative
approximations to the thermal equilibrium free energy and density matrix can be
found in [4] and [3], respectively. One of the drawbacks of the O(h̄2 ) QHD equations
is 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 the
validity of the O(h̄2 ) QHD equations near potential discontinuities.
We will take a different approach to the derivation of approximate solutions to
the Bloch equation based on an asymptotic expansion of the solution for “small”
potentials. The main result is that in the corresponding “smooth” QHD model the
potential V is replaced by a smoothed potential S( 1i ∇x )V , where the symbol S(ξ)
of the pseudodifferential operator S behaves like |ξ|−2 for large ξ. This approximation is better suited for dealing with potential discontinuities and incorporates the
nonlocal effects of potential barriers observed in solutions of the full quantum kinetic
equations [11]. In the QHD equations the stress tensor based on the smoothed potential actually cancels the leading singularity in the classical potential at a barrier and
leaves a residual smooth effective potential with a lower potential height in the barrier
784
CARL L. GARDNER AND CHRISTIAN RINGHOFER
region [7]. The smoothing makes the barrier partially transparent to the particle flow
and 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 approximate solution (22) in order to analyze its asymptotic validity. Then we proceed to
derive the improved approximation based on the smoothed potential SV . Section 3
deals with the asymptotic validity of the smoothed potential approximation for small
potentials and high temperatures. We demonstrate the convergence of the approximate Wigner function as well as the particle density. In section 4 we derive the
corresponding QHD model. Section 5 is devoted to the numerical verification of the
asymptotic results. It turns out that, at least for the double barrier structure considered in section 5, the parameter range for which the approximation is valid is actually
much larger than expected.
Our paper [7] presents a derivation and interpretation of the smooth QHD model
from 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 to
the potential V should not change the overall problem and that the Fermi level φ can
be eliminated from the Bloch equation. For the independent variables x, p, and β, we
use the scaling
r
m
(23)
ps , β = β ∗ βs ,
x = Lxs , p =
β∗
where L denotes a characteristic length scale. We scale the potential by
ε
1
V (x) = V0 + ∗ Vs (xs ), V0 =
sup V + inf V .
(24)
β
2 x∈Rd
x∈Rd
An additive constant in the potential V is absorbed into V0 and does not affect Vs . ε
is a dimensionless parameter measuring the size of the potential. The scaled thermal
equilibrium Wigner function and particle density are
we (x, p, β) = h̄−d exp{−β(V0 + φ)}ws (xs , ps , βs ),
(25)
ne (x) =
(26)
m
β ∗ h̄2
d2
exp{−β ∗ (V0 + φ)}ns (xs ).
The normalization condition (1) now reads
Z
(27)
ns (xs )dxs = hd exp{β ∗ (V0 + φ)},
d
Rx
Z
ns (xs ) =
d
Rp
ws (xs , ps , 1)dps ,
where here h denotes the scaled Planck constant
r
β∗
(28)
.
h = h̄
mL2
With this scaling the initial value problem (14) and (15) becomes
h2
|p|2
ε
h
h
∆xs ws −
ws −
Vs xs + ∇ps + Vs xs − ∇ps
ws ,
(29) ∂βs ws =
8
2
2
2i
2i
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
785
ws (xs , ps , 0) = 1
(30)
with 0 ≤ β ≤ 1. The initial value problem (29) and (30) is independent of the Fermi
level φ, and φ can be computed after the fact from (27). After computing φ the
original thermal equilibrium Wigner function can be reconstructed from (25). For
notational convenience we will drop the subscript s from here on.
We will first derive Wigner’s approximate solution. Our derivation differs from
Wigner’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
2
i εh4 h
h2 h
|p|
+ εV w+
∆x w + ε ∂x2j xk V ∂p2j pk w −
rh x, ∇p w,
∂β w = −
2
8
8
2i
(31)
where the symbol of the remainder term rh is
1
h
h
rh x, η = 2 δh2 V x, η − ηj ηk ∂x2j xk V
(32)
2
h
2
with
(33)
δh2 V
h
x, η
2
4
h
h
= 2 V x + η − 2V (x) + V x − η .
h
2
2
If the potential V is sufficiently smooth, the solution w can be expanded in powers of
h2 :
w = w0 + h2 w2 + h4 w4 + · · · .
(34)
Matching powers of h2 in the Bloch equation, we obtain
2
|p|
+ εV w0 , w0 (x, p, 0) = 1,
(35)
∂β w0 = −
2
∂β w2 = −
(36)
|p|2
+ εV
2
w2 +
i
1h
∆x w0 + ε ∂x2j xk V ∂p2j pk w0 ,
8
yielding
(37)
w0 (x, p, β) = exp −β
w2 (x, p, β) =
(38)
|p|2
+ εV
2
,
2
1
|p|
exp −β
+ εV
8
2
ε2 β 3
εβ 3
× −εβ 2 ∆x V +
|∇x V |2 +
pj pk ∂x2j xk V ,
3
3
|p|2
− εV
w(x, p, 1) = exp −
2
w2 (x, p, 0) = 0,
786
CARL L. GARDNER AND CHRISTIAN RINGHOFER
(39)
h2
ε2
ε
−ε∆x V + |∇x V |2 + pj pk ∂x2j xk V
× 1+
+ O(h4 ).
8
3
3
In order to compute the quantum corrections to the fluid dynamical model in the
next section it will be necessary to calculate the thermal equilibrium stress tensor. A
straightforward calculation [5] gives for the O(h2 ) approximation
Pjk = −
(40)
n
h̄2 β ∗ n 2
∂ V + O(h̄4 ).
δ
−
jk
β∗
12m xjk
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 high
temperatures.
• Equation (34) represents only an asymptotic expansion and not a convergent series since the computation of higher order terms involves successive
differentiation.
• The remainder term rh in (31) is bounded independently of h only if the potential is sufficiently smooth. In particular, we can expect this approximation
to break down in the neighborhood of discontinuities in the potential.
We will now take a different tack and approximate the solution of the Bloch
equation not for small values of h but instead for applied potentials which are small
compared to the inverse temperature β ∗ , that is, for small values of the parameter
ε. Expanding the thermal equilibrium Wigner function in powers of ε and matching
powers of ε in the Bloch equation yields
wε =
(41)
∞
X
wλ ελ ,
λ=0
∂β w0 =
(42)
(43)
∂β wλ =
where
(44)
µV
h2
|p|2
∆x w0 −
w0 ,
8
2
h2
|p|2
∆x wλ −
wλ − µV wλ−1 ,
8
2
w0 (x, p, 0) = 1,
wλ (x, p, 0) = 0
(λ = 1, 2, . . .),
h
1
h
h
V x + ∇p + V x − ∇p w(x, p, β).
x, ∇p w(x, p, β) =
2i
2
2i
2i
The solution of (42) is
(45)
β
w0 (x, p, β) = exp − |p|2 .
2
To solve equation (43) for the first-order term w1 , we make use of the equivalency
formula for autonomous pseudodifferential operators:
h
h
(46)
µV x, ∇p w0 (p, β) = µw0 p, ∇x , β V (x),
2i
2i
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
787
where the operator µw0 is
h
h
h
1
µw0 p, ∇x , β =
w0 p + ∇x , β + w0 p − ∇x , β .
(47)
2i
2
2i
2i
It is more convenient to solve the Fourier transformed version of (43). To this end
we denote by f˜ the Fourier transform of the function f with respect to the position
variable x:
Z
−d
˜
2
(48)
f (x)e−iξ·x dx.
f (ξ) = (2π)
d
Rx
After Fourier-transforming in x and using (46), (43) reads, for λ = 1,
(49)
∂β w̃1 = −
h2 2 |p|2
|ξ| +
8
2
h
w̃1 − µw0 p, ξ, β Ṽ (ξ),
2
w̃1 (ξ, p, 0) = 0.
In deriving this expression we have used the fact that the zeroth-order term w0 is actually independent of the position variable x. The first-order term w̃1 in the expansion
is therefore
1
∇x , p, β V (x)
(50)
w̃1 (ξ, p, β) = −βg(ξ, p, β)Ṽ (ξ), w1 (x, p, β) = −βg
i
with the function g defined by
(51)
g(ξ, p, β) =
1
β
Z
0
β
2
h
|p|2
h
|ξ|2 +
µw0 p, ξ, γ dγ.
exp (γ − β)
8
2
2
Thus the O(ε) approximation of the thermal equilibrium Wigner function is
β 2
1
∇x , p, β V (x) + O(ε2 ).
(52)
w(x, p, β) = exp − |p| − εβg
2
i
For the fluid dynamical model in section 4 we will need the corresponding approximations to the particle density and the stress tensor (again π = 0):
d
1
2
∇x V (x) + O(ε2 ),
(53)
n(x) = (2π) 1 − εG0
i
(54)
P (x) = −(2π)
d
2
I − εG2
1
∇x V (x) + O(ε2 ),
i
where the symbols of the pseudodifferential operators G0 and G2 are
Z
d
G0 (ξ) = (2π)− 2
(55)
g(ξ, p, 1)dp,
d
Rp
(56)
d
G2 (ξ) = (2π)− 2
Z
d
Rp
ppT g(ξ, p, 1)dp.
788
CARL L. GARDNER AND CHRISTIAN RINGHOFER
The calculation of G0 and G2 is deferred to Appendix A. The results for the symbols
of the operators G0 and G2 are
h
(57)
|ξ| ,
G0 (ξ) = κ
2
ξξ T
G2 (ξ) = 2 + κ
|ξ|
(58)
ξξ T
h
|ξ| I − 2 ,
2
|ξ|
where
Z
(59)
κ(z) =
1
0
z2
exp (γ − 1)
2
2
dγ.
The thermal equilibrium stress tensor in the O(ε) approximation is
1
∇x V + O(ε2 ).
(60)
P = −nI − εn(G0 I − G2 )
i
In analogy with the Wigner approximation (40), (60) can be written as
(61)
Pjk
εh2 n 2
∂ Sh
= −nδjk −
12 xjk
1
∇x V + O(ε2 ),
i
where the symbol of the smoothing operator Sh is
12
h
(62)
|ξ| .
Sh (ξ) = 2 2 1 − κ
h |ξ|
2
In unscaled form, (61) and (62) read
(63)
(64)
P =−
h̄2 β ∗ n ⊗2
n
∂ Sh̄
I−
∗
β
12m x
1
∇x V + O((β ∗ V )2 ),
i
!#
"
à r
12m
h̄ β ∗
Sh̄ (ξ) = 2 ∗ 2 1 − κ
|ξ|
.
2 m
h̄ β |ξ|
Simple Taylor expansion implies that Sh (ξ) = 1 + O(h2 ) for h → 0. In this sense
the 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 ∂x2 SV
has the same degree of smoothness as the original potential V . In fact, the stress
tensor (63) actually cancels the leading singularity in the classical potential at a
barrier and leaves a residual smooth effective potential with a lower potential height
in the barrier region [7].
One disturbing fact remains: the approximation (52) of the thermal equilibrium
Wigner function does not reduce to the classical Maxwellian
β 2
exp − |p| − εβV
(65)
2
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
789
in the classical limit h → 0. Letting h → 0 in (52) yields
β
lim g(ξ, p, β) = exp − |p|2 ,
(66)
h→0
2
β
lim (w0 + εw1 ) = exp − |p|2 [1 − εβV (x)],
h→0
2
(67)
which is only the O(ε) approximation to the Maxwellian. This discrepancy can be
remedied 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:
β 2
1
w(x, p, β) = exp − |p| − εβg1
∇x , p, β V (x) + O(ε2 ),
(68)
2
i
d
2
(69)
n(x) = (2π) exp −εG0
1
∇x V (x) + O(ε2 ),
i
¡
ª
d
−(2π) 2 exp −εG2jk 1i ∇x V (x) + O(ε2 ), j = k,
¡
(70)
Pjk (x) =
d
(2π) 2 εG2jk 1i ∇x V (x) + O(ε2 ),
j 6= k,
ª
with g1 (ξ, p, β) = exp β|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 derived in section 2 by providing estimates of the remainder term. We define the remainder term r by w = w1 + εw2 + ε2 r and obtain
h
h2
|p|2
∂β r =
∆x r −
r − µV x, ∇p [w1 + εr], r(x, p, 0) = 0.
(71)
8
2
2i
To prove the boundedness of the remainder term r in the L2 (Rxd × Rpd ) norm is a
straightforward exercise in energy inequalities. However, we are more concerned with
the 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 density
tensor. The main result of this section is Theorem 3, which proves the convergence
of the particle density as ε → 0. The convergence of the momentum flux density
tensor remains an unresolved problem at this point, although the numerical evidence
in section 5 indicates that the smoothed potential approximation also produces a good
approximation to Π. We start with
Theorem 1. Let the potential V be bounded. Then
n
o
n
o
(72) max ||r(β)||L2 (Rxd ×Rpd ) ≤ const ||V ||L∞ (Rxd ) max ||w1 (β)||L2 (Rxd ×Rpd ) ,
0≤β≤1
0≤β≤1
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
Z Z 2
1
h
1
2
2
∂β ||r(x, p, β)||L2 (Rxd ×Rpd ) = −
|∇x r| + |pr| + rµV [εr + w1 ] dxdp.
2
8
2
d
d
Rp
Rx
(73)
790
CARL L. GARDNER AND CHRISTIAN RINGHOFER
Using the definition of the pseudodifferential operator µV yields
Z Z
Z Z
h
∗
rµV [εr + w1 ]dxdp =
r̃ (x, η, β)µV x, η [εr̃ + w̃1 ]dxdη
2
d
d
d
d
Rp
Rx
Rη
Rx
(74)
≤ ||V ||L∞ (Rxd ) ||r||L2 (Rxd ×Rpd ) ||εr + w1 ||L2 (Rxd ×Rpd ) .
Combining (73) with (74) implies
h
i
(75) ∂β ||r||L2 (Rxd ×Rpd ) ≤ const ||V ||L∞ (Rxd ) ε||r||L2 (Rxd ×Rpd ) + ||w1 ||L2 (Rxd ×Rpd ) .
The more interesting question, however, is the convergence of the moments of the
remainder term, i.e., the degree of approximation of the particle and momentum flux
densities. The existence of the moments of the Wigner function in the presence of
discontinuous potentials is a challenging problem (see [1, 2] for the treatment of the
transient case). In the thermal equilibrium case described by the Bloch equation, we
are 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 Mλ f (x) be defined by
Z
Mλ f (x) =
(76)
p⊗λ f (x, p)dp, λ = 0, 1, 2, . . . .
d
Rp
Then
α
||Mλ f ||L2 (Rxd ) ≤ const ||(1 + |p|2 ) 2 f (x, p)||L2 (Rxd ×Rpd )
(77)
for α > λ + 12 .
Proof. We have
||Mλ f ||2L2 (Rxd ) =
Z
(78)
≤
d
Rx
"Z
⊗λ 2
d
Rp
|p
d
Rx
2 −α
| (1 + |p| )
"Z
Z
#2
p⊗λ f (x, p)dp
d
Rp
# "Z
dp
dx
#
2 α
d
Rp
2
(1 + |p| ) |f (x, p)| dp dx.
R
For α > λ + 12 the term Rd |p⊗λ |2 (1 + |p|2 )−α dp remains bounded and the result
p
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
α
rα (x, p, β) = (1 + |p|2 ) 2 r(x, p, β).
(79)
rα satisfies the initial value problem
(80)
∂β rα =
h2
|p|2
∆ x rα −
rα − Ωα [w1α + εrα ],
8
2
where
(81)
2
α
2
Ωα = (1 + |p| ) µV
rα (x, p, 0) = 0,
α
h
x, ∇p (1 + |p|2 )− 2
2i
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
791
and
α
w1α (x, p, β) = (1 + |p|2 ) 2 w1 (x, p, β).
(82)
For the estimate of rα we will use a specific norm defined by
Z Z
2
||f (x, p, β)|| =
(83)
max {|f˜(ξ, p, β)|2 }dxdp,
d
Rp
d 0≤β≤1
Rx
where
d
f˜(ξ, p, β) = (2π)− 2
(84)
Z
d
Rx
f (x, p, β)e−iξ·x dx.
In this norm we can estimate the solution of the initial value problem (71) with the
following theorem.
Theorem 2. Let the function r satisfy the initial value problem
(85)
∂β r =
h2
|p|2
∆x r −
r − Ωα [εr + w1α ],
8
2
0 ≤ β ≤ 1,
r(x, p, 0) = 0
with the operator Ωα defined in (81). Let the potential V be smooth enough that its
Fourier transform satisfies |Ṽ (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 potential in Theorem 2 is modest and allows for discontinuous potentials. What remains
to 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 α
provides a bound for the zeroth-order moment (the particle density), a value α > 5/2
would be needed to estimate the second-order moment (the momentum flux density
tensor).
Lemma 2. Let the function w1α be defined by
(86)
α
w1α (x, p, β) = (1 + |p|2 ) 2 w1 (x, p, β)
with w1 given by (50). Then ||w1α || < ∞ for 0 ≤ α < 3/2 with the norm defined
in (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 for
0 ≤ α < 32 . Consequently, ||M0 r||L2 (Rxd ) ≤ const by Lemma 1, and the approximation
of the particle density in section 2 satisfies
(87)
||n − n0 − εn1 ||L2 (Rxd ) = O(ε2 ).
4. QHD models. Next we discuss the QHD models corresponding to the approximations of the thermal equilibrium distribution function derived in section 2.
The nonequilibrium Wigner function is the solution of the quantum Liouville (or
Wigner–Boltzmann) equation
(88)
∂t f +
1
divx (pf ) − θf = Cf,
m
792
CARL L. GARDNER AND CHRISTIAN RINGHOFER
where the operator θ is defined by
i
h̄
h̄
θ=
V x + ∇p − V x − ∇p .
(89)
h̄
2i
2i
The quantum Liouville equation can be derived from the transient Schrödinger equation in the same way the Bloch equation is derived from the eigenvalue problem for
the Schrödinger equation. The derivation makes use of the transformation
X Z
h̄
h̄
f (x, p, t) =
(90)
aλ
ψλ∗ x + η, t ψλ x − η, t eiη·p dη,
2
2
d
Rη
λ
of the transient Schrödinger equation
where the wavefunctions ψλ (x, t) are solutions
P
a
=
Nparticles . The collision operator C
and the aλ ’s are occupation numbers:
λ
λ
in (88) is added after the fact and its form is usually based on some ad hoc assumptions. (Typically either a relaxation time model or a Fokker–Planck collision operator
is 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
Z
hχi (x, t) =
(91)
χ(x, p, t)f (x, p, t)dp.
d
Rp
Integrating the quantum Liouville equation against 1, p, and |p|2 /2m yields conservation laws for particle number, momentum, and energy:
∂t h1i +
(92)
∂t hpi + divx
(93)
ppT
m
+ h1i ∇x V = hpCi ≡ C1 ,
2 DpE
|p|
+ divx
· ∇x V =
C ≡ C2 .
(94)
+
∂t
m
2m
­
­
The system (92)–(94) has to be closed by expressing Π = ppT /m and­Θ ≡ |p|2 p/2m2
in terms of (say) the conserved quantities n = h1i, π = hpi, and W = |p|2 /2m . Note
that the moment equations are identical to those for the classical Boltzmann transport
equation and that quantum effects enter through the closure conditions.
We first change from the moments with respect to p to the centered moments
with respect to the variable p0 defined by
|p|2
2m
1
divx hpi = hCi ≡ C0 ,
m
|p|2 p
2m2
p = p0 + mu,
(95)
hpi = mnu,
where u denotes the macroscopic fluid velocity. This shift gives the transformations
0 0T T
pp
pp
T
= mnuu +
= mnuuT − P,
(96)
Π=
m
m
(97)
W =
|p|2
2m
1
= mn|u|2 +
2
|p0 |2
2m
=
1
1
mn|u|2 − Trace{P },
2
2
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
Θ=
|p|2 p
2m2
=u
|p0 |2
2m
793
+ *
T
1
|p0 |2 p0
p0 p0
2
T
+ mn|u| +u
+
= uW −uT P +q,
2
m
2m2
(98)
where q is the heat flux vector. We obtain
(100)
∂t n + divx (nu) = C0 ,
¡
∂t (mnu) + divx mnuuT − P + n∇x V = C1 ,
(101)
¡
∂t W + divx uW − uT P + q + nu · ∇x V = C2 .
(99)
Closure is obtained by assuming that the distribution function f is close to a
momentum-displaced thermal equilibrium distribution. We assume that
(102)
(103)
f (x, p, t) = A(x, t)we (x, p − mu),
Z
­ 0⊗λ p⊗λ we (x, p)dp
= A(x, t)
p
d
Rp
for a smooth function A(x, t). We will use (103) to express P , W , and q in terms of
the 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 Maxwellian
exp{−|p|2 /2mT }, which yields
(104)
P = −nT I,
1
3
nT + mn|u|2 ,
2
2
and leads to the classical compressible Euler equations. For the Wigner approximation (39) and the smoothed potential approximation (68), the stress tensor P is
calculated in (40) and (63), respectively. The Wigner approximation yields the O(h̄2 )
quantum hydrodynamic model treated in [5]. For the smoothed potential approximation in section 2, we obtain from (63)
(105)
W =
(106)
Pjk = −nT δjk −
h̄2 n 2
∂ (SV ),
12mT xjk
3
1
h̄2 n
nT + mn|u|2 +
∆x (SV ),
2
2
24mT
where the smoothed potential SV is
Z Z
(108)
Sh̄ (ξ)V (y)eiξ·(x−y) dydξ
SV (x) = (2π)−d
(107)
W =
Rξd
with
(109)
12mT0
h̄ |ξ|
√
Sh̄ (ξ) = 2 2 1 − κ
2 mT0
h̄ |ξ|
and
(110)
d
Ry
Z
κ(z) =
1
0
z2
exp (γ − 1)
2
2
dγ.
Note that even though q = 0 for both the Wigner and smoothed potential approximations, we have allowed a nonzero q in the QHD equations since heat conduction plays
a quantitative role in electron and hole propagation in actual semiconductor devices.
794
CARL L. GARDNER AND CHRISTIAN RINGHOFER
1
0.8
V
0.6
0.4
0.2
0
-150
-100
-50
0
x
50
100
150
Fig. 1. Unit double barrier potential. x is in Å for all figures.
5. Numerical results. In this section we present numerical results to verify the
asymptotic expansions of section 2 and to test the limits of their validity. To this end
we consider an electron gas in a GaAs/Alx Ga1−x As double barrier heterostructure
(see Figure 1) at 300 K. (The double barrier heterostructure is the heart of the
resonant tunneling diode.) The barrier width is 25 Å and the well width is 50 Å on
a characteristic length scale of 100 Å. Technologically relevant barrier heights are in
the range 0.1–0.7 eV. Without loss of generality we can assume that the barriers are
perpendicular to the x1 coordinate axis. The salient features of the numerical results
are the following:
• The asymptotic expansion (68) yields a good approximation of both the particle density and the energy density. (Note that the velocity u = 0.)
• The range of validity of the expansion extends far beyond the small ε regime
to ε ≈ 2–10.
• The smoothed potential approximation for the stress tensor (63) is uniformly
better than the one obtained through the Wigner approximation (40).
We will compare the Wigner and smoothed potential approximations with a numerical solution of the Bloch equation. The Bloch equation was solved on a grid
of 200 ∆x using the backward Euler method with homogeneous Neumann boundary
conditions. A detailed description of the basic numerical method plus extensions is
given in [6]. Numerical values for the constants and scaling factors in section 2 are
given in Table 1.
The parameter ε is chosen so that the scaled potential Vs varies between −1 and
1. We have set the effective electron mass in GaAs to m = 0.063 me , where me is the
electron mass.
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
795
Table 1
Numerical values of constants used in the computations. B denotes the barrier height divided
by eV.
Constant
h̄
m
β∗
L
h
ε
Value
6.582 × 10−16 eV sec
3.582 × 10−17 eV sec2 cm−2
38.68 eV−1
10−6 cm
0.6840
19.34 B
Since the potential V is one-dimensional, we can reduce the number of independent variables in the Bloch equation by setting
β
w(x, p, β) = w(x1 , p1 , β) exp − (p22 + p23 ) .
(111)
2
The function w satisfies the initial value problem
h2 2
p2
ε
h
h
∂x w − 1 w −
V x1 + ∂p1 + V x1 − ∂p1
(112)
w,
∂β w =
8
2
2
2i
2i
(113)
w(x1 , p1 , 0) = 1
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 R2 shows the electron density. (The overall scale for the solutions
is 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 the
numerical solution of the full Bloch equation) in the electron momentum flux:
εh2 2
1
Qs =
∂x1 Sh
∂x1 V (x1 ),
(114)
12
i
(115)
Qf = −
P11
− T.
n
The smooth effective potential Q + V for the smoothed potential approximation and
for the numerical solution of the full Bloch equation is shown in Figure 4. In Figures
2–10, the solid lines correspond to the smoothed potential approximations and the
dashed 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 to
ε = 9.67.
The electron density in Figures 2, 5, and 8 is minimized inside the potential
barriers and has a local maximum at the center of the quantum well. The energy
density drops precipitously as the electrons tunnel through the potential barriers and
rises dramatically as the electrons exit from the barriers. The effect of the quantum
correction in Figures 3, 6, and 9 is to partially cancel the effects of the potential
barriers (see Figures 4, 7, and 10).
796
CARL L. GARDNER AND CHRISTIAN RINGHOFER
0.31
Smooth
Bloch
Density
0.3
0.29
0.28
0.27
-150
-100
-50
0
x
50
100
150
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.
0.004
0.002
Q
0
-0.002
-0.004
Smooth
Bloch
-0.006
-150
-100
-50
0
x
50
100
Fig. 3. Quantum correction term in eV for a 0.01 eV double barrier (ε ≈ 0.2).
150
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
797
0.004
Smooth
Bloch
Q + V
0.003
0.002
0.001
0
-150
-100
-50
0
x
50
100
150
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 qualitatively good approximation to the quantum correction term Qf for large values of the
perturbation parameter ε.
In contrast, the quantum correction term Qw obtained from the Wigner approximation (39) is completely dominated by the dipole behavior induced by the potential
discontinuities:
Qw =
(116)
εh2 2
εh2 B 0
∂x1 V (x1 ) =
[δ (x1 + 2a) − δ 0 (x1 + a) + δ 0 (x1 − a) − δ 0 (x1 − 2a)]
12
12
with a = 25 Å. Thus our numerical evidence, convergence results, and the dipole
behavior of Qw demonstrate that the smoothed potential approximation is uniformly
better for potentials with discontinuities than the O(h̄2 ) approximation.
Smooth QHD simulations of the resonant tunneling diode (RTD) and comparison with experimental current-voltage curves are presented in [8]. Future work will
compare the smooth QHD simulations of the RTD with numerical simulations of the
coupled 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
−d
2
Gm (ξ) = (2π)
(117)
Z
0
1
Z
d
Rp
p
⊗m
exp (γ − 1)
h2 2 |p|2
|ξ| +
8
2
h
p, ξ, γ dpdγ
2
µw0
798
CARL L. GARDNER AND CHRISTIAN RINGHOFER
Density
0.4
Smooth
Bloch
0.3
0.2
0.1
0
-150
-100
-50
0
x
50
100
150
Fig. 5. Electron density in 106 cm−3 for a 0.1 eV double barrier in GaAs at 300 K (ε ≈ 2).
0.04
0.02
Q
0
-0.02
-0.04
Smooth
Bloch
-0.06
-150
-100
-50
0
x
50
100
Fig. 6. Quantum correction term in eV for a 0.1 eV double barrier (ε ≈ 2).
150
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
799
0.05
Smooth
Bloch
Q + V
0.04
0.03
0.02
0.01
0
-150
-100
-50
0
x
50
100
150
Fig. 7. Smooth effective potential in eV for a 0.1 eV double barrier (ε ≈ 2).
0.7
Smooth
Bloch
Density
0.6
0.5
0.4
0.3
0.2
0.1
0
-150
-100
-50
0
x
50
100
150
Fig. 8. Electron density in 106 cm−3 for a 0.5 eV double barrier in GaAs at 300 K (ε ≈ 10).
800
CARL L. GARDNER AND CHRISTIAN RINGHOFER
0.2
0.1
Q
0
-0.1
-0.2
Smooth
Bloch
-0.3
-0.4
-150
-100
-50
0
x
50
100
150
Fig. 9. Quantum correction term in eV for a 0.5 eV double barrier (ε ≈ 10).
0.4
Smooth
Bloch
Q + V
0.3
0.2
0.1
0
-150
-100
-50
0
x
50
100
Fig. 10. Smooth effective potential in eV for a 0.5 eV double barrier (ε ≈ 10).
150
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
801
−
with m = 0, 2. Splitting Gm into Gm = 12 (G+
m + Gm ), we define
d
−2
G±
m (ξ) = (2π)
Z
1
h2
h
p⊗m exp (γ − 1) |ξ|2 u(p, γ)w0 p ± ξ, γ dpdγ,
8
2
d
Rp
Z
0
(118)
where
u(p, γ) = exp
(119)
(γ − 1) 2
|p| .
2
For m = 0, integrating with respect to p gives
Z
Z
h
h
u(p, γ)w0 p + ξ, γ dp =
ũ(−η, γ)w̃0 (η, γ) exp i η · ξ dη
2
2
d
d
Rp
Rη
Z
=
−d
2
d
Rη
[γ(1 − γ)]
1
h
2
|η| + i η · ξ dη
exp −
2γ(1 − γ)
2
h2 2
= (2π) exp −γ(1 − γ) |ξ| .
8
d
2
(120)
Thus
G+
0 (ξ)
(121)
Z
=
1
0
h2 2
h
|ξ|
exp (γ − 1) |ξ| dγ = κ
8
2
2
with the function κ defined in (59). G−
0 is obtained by replacing h by −h, and therefore
−
=
G
=
κ(h|ξ|/2).
G0 = G+
0
0
To compute the matrix G+
2 we have to evaluate the integrals
Z
h
pj pk u(p, γ)w0 p + ξ, γ dp
2
d
Rp
Z
=−
d
= [γ(1 − γ)]− 2
h
2
ũ(−η, γ) w̃0 (η, γ) exp i η · ξ dη
∂jk
2
d
Rη
1
1−γ
Z
d
Rη
δjk −
1
ηj ηk
h
exp −
|η|2 + i η · ξ dη
1−γ
2γ(1 − γ)
2
hp
|η|2
+i
(δjk − γηj ηk ) exp −
γ(1 − γ)η · ξ dη
2
2
d
Rη
Z
1
=
1−γ
|z|2
1
2
[δjk + γ∂zj zk ] exp −
= (2π)
1−γ
2
at
z=
hp
γ(1 − γ)ξ
2
γ
|z|2
zj zk exp −
δjk +
1−γ
2
at
z=
hp
γ(1 − γ)ξ.
2
d
2
(122)
= (2π)
d
2
802
CARL L. GARDNER AND CHRISTIAN RINGHOFER
Inserting (122) into (118) gives
G+
2jk (ξ)
(123)
Z
=
1
0
γ 2 h2
h2 2
2
ξj ξk exp (γ − 1) |ξ| dγ,
δjk +
4
8
and integrating by parts yields
h
−2
2
|ξ|
.
(ξ)
=
|ξ|
ξ
+
(|ξ|
δ
−
ξ
ξ
)κ
ξ
G+
j k
jk
j k
2jk
2
(124)
+
−
G−
2 is obtained by replacing h by −h, and therefore G2 = G2 = G2 .
Appendix B. Proof of Theorem 2. To prove Theorem 2 we will need the
following 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
h2
h2
|p|2
|p|2
∆x u −
u + 1 − ∆x +
f,
∂β u =
8
2
8
2
(125)
u(x, p, 0) = 0,
with 0 ≤ β ≤ 1. Then ||u|| ≤ const||f ||, where const denotes a positive constant
independent of h and ε and where the norm || · || is defined in (83).
Proof. The Fourier transform ũ of the solution of (125) is
Z
ũ(ξ, p, β) =
(126)
0
β
exp (γ − β)
h2 2 |p|2
|ξ| +
8
2
h2
|p|2
1 + |ξ|2 +
8
2
f˜(ξ, p, γ)dγ.
Thus
|ũ(ξ, p, β)| ≤ M
≤M
(127)
1+
Z β
2
h2
|p|2
|p|2
h
1 + |ξ|2 +
|ξ|2 +
dγ
exp (γ − β)
8
2
8
2
0
h2 2 |p|2
|ξ| +
8
2
2
2
−1
|p|2
|p|2
h
h
1 − exp −β
|ξ|2 +
|ξ|2 +
,
8
2
8
2
where
(128)
M = max {|f˜(ξ, p, β)|}.
Since the function g(z, β) =
0 ≤ β ≤ 1, we obtain
1+z
z (1
(129)
0≤β≤1
− e−βz ) is uniformly bounded for 0 ≤ z < ∞,
max {|ũ(ξ, p, β)|} ≤ const max {|f˜(ξ, p, β)|}.
0≤β≤1
0≤β≤1
Squaring both sides of (129) and integrating with respect to ξ and p proves the lemma.
Next we estimate the inhomogeneous term in (125).
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
803
Lemma 4. Let the function f (x, p, β) be given by
|p|2
h2
f (x, p, β) = Ωα u
(130)
1 − ∆x +
8
2
for some function u, where the operator Ωα is defined in (81). Then if the Fourier
1
transform of the potential satisfies |Ṽ (ξ)| ≤ const(1 + |ξ|2 )− 2 , ||f || ≤ const||u|| for
0 ≤ α < 3.
Proof. We split the function f and the operator Ωα into f = 12 (f+ + f− ), Ωα =
1
+
−
±
2 (Ωα + Ωα ), where f± and Ωα are defined by
α
h
±
2 α
2
(131)
Ωα = (1 + |p| ) V x ± ∇p (1 + |p|2 )− 2 ,
2i
(132)
|p|2
h2
1 − ∆x +
8
2
f± = Ω±
α u.
Fourier-transforming the function f+ with respect to the position variable x yields
3d
f˜+ (ξ, p, β) = (2π)− 2
Z
Z
×
d
Rx
Z
Rqd
d
Rη
V
x+
d
= (2π)− 2
h2
|p|2
1 + |ξ|2 +
8
2
−1
α
(1 + |p|2 ) 2
α
h
η (1 + |q|2 )− 2 u(x, q, β)eiη·(p−q)−iξ·x dηdqdx
2
1+
h2 2 |p|2
|ξ| +
8
2
−1
α
(1 + |p|2 ) 2
2 !− α2 h
h
×
ũ ξ − ω, p + ω, β dω.
Ṽ (ω) 1 + p + ω 2
2
d
Rω
Ã
Z
(133)
Taking the maximum with respect to β and integrating with respect to p yields
Z
max {|f˜+ (ξ, p, β)|2 }dp
d 0≤β≤1
Rp


|p|2
h2
1 + |ξ|2 +
≤ max
ω,p 
8
2
or
(135)
×

2 !−α 
h
1 + p + ω 
2
Ã
(1 + |p|2 )α (1 + |ω|2 )−1
( 2 )
h
max ũ ξ − ω, p + ω, β dpdω
β
2
d
Rp
Z
Z
(134)
−2
d
Rω
Z
d
Rp
max {|f˜+ (ξ, p, β)|2 }dp ≤ H(ξ)||u||2 ,
0≤β≤1
804
CARL L. GARDNER AND CHRISTIAN RINGHOFER
where
H(ξ) = max
ω,p


1+

2
2
|p|
h
|ξ|2 +
8
2
(136)
Elementary calculus implies that
−2
R
Rξd

2 !−α 
h
1 + p + ω .

2
Ã
(1 + |p|2 )α (1 + |ω|2 )−1
H(ξ)dξ < ∞ for 0 ≤ α < 3. The lemma fol-
lows from integrating (135) with respect to ξ and repeating the same procedure for
f− .
The proof of Theorem 2 consists of the combination of Lemmas 3 and 4 with the
function u in Lemma 4 replaced by εrα + w1α .
Appendix C. Proof of Lemma 2. The x-Fourier transform of the function
w1α is
α
w̃1α (ξ, p, β) = −(1 + |p|2 ) 2 Ṽ (ξ)
Z
(137)
×
β
0
exp (γ − β)
h2 2 |p|2
|ξ| +
8
2
h
p, ξ, γ dγ.
2
µw0
+
−
±
+ w̃1α
), with w1α
defined by
We split the function w̃1α into w̃1α = 12 (w̃1α
(
2 )
Z β
γ h h2 2 |p|2
±
2 α
|ξ| +
− p ± ξ dγ
exp (γ − β)
w̃1α = −(1+|p| ) 2 Ṽ (ξ)
8
2
2
2
0
(138)
+
and obtain for w̃1α
+
w̃1α
"
2 #−1
h2 2
h 2
= −2(1 + |p| ) Ṽ (ξ) p + ξ − |p| + |ξ|
2
4
2
"
(139)
α
2
(
2 )
#
h β h2 2
β
2
|p| + |ξ|
× exp − p + ξ − exp −
.
2
2
2
4
The function g(β) = (e−βb − e−βa )/(a − b) satisfies
1
b ln(b) − a ln(a)
(140)
≤ const
0 ≤ g(β) ≤ exp
a−b
1+a
for 0 ≤ b ≤ 2a. Thus
(141)
−1
α
h2
+
|w̃1α
(ξ, p, β)| ≤ const(1 + |p|2 ) 2 |Ṽ (ξ)| 1 + |p|2 + |ξ|2
4
and
(142)
+ 2
||w1α
||
Z
≤
Rξd
Z
d
Rp
2
α
2
2
(1 + |p| ) |Ṽ (ξ)|
h2
1 + |p| + |ξ|2
4
2
−2
dpdξ.
The integral on the right-hand side of (142) is convergent for 0 ≤ α < 32 . Repeating
−
proves the lemma.
the same argument for w1α
APPROXIMATION OF THERMAL EQUILIBRIUM FOR QUANTUM GASES
805
REFERENCES
[1] A. Arnold, The relaxation-time Wigner equation, in Mathematical Problems in Semiconductor
Physics, Pitman Res. Notes Math. Ser. 340, Rome, 1995, pp. 105–117.
[2] A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas
in the charge neutral case, Math. Methods Appl. Sci., 14 (1991), pp. 595–613.
[3] D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), pp. 7944–7950.
[4] R. P. Feynman and H. Kleinert, Effective classical partition functions, Phys. Rev. A, 34
(1986), pp. 5080–5084.
[5] C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl.
Math., 54 (1994), pp. 409–427.
[6] C. L. Gardner, A. Niemic, and C. Ringhofer, Numerical methods for the Bloch equation,
to appear.
[7] C. L. Gardner and C. Ringhofer, Smooth quantum potential for the hydrodynamic model,
Phys. Rev. E, 53 (1996), pp. 157–167.
[8] C. L. Gardner and C. Ringhofer, Smooth QHD simulation of the resonant tunneling diode,
VLSI Design, (1998), to appear.
[9] H. L. Grubin, T. R. Govindan, J. P. Kreskovsky, and M. A. Stroscio, Transport via the
Liouville equation and moments of quantum distribution functions, Solid-State Electronics,
36 (1993), pp. 1697–1709.
[10] T. Kerkhoven, M. W. Raschke, and U. Ravaioli, Self-consistent simulation of corrugated
layered structures, Superlattices and Microstructures, 12 (1992), pp. 505–508.
[11] N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Self-consistent study of
the resonant-tunneling diode, Phys. Rev. B, 39 (1989), pp. 7720–7735.
[12] N. J. Mauser and C. Ringhofer, Quantum steady states via self-consistent Bloch-type equations, in Proc. International Workshop on Computational Electronics, University of Illinois,
Urbana-Champaign, 1992, pp. 273–276.
[13] N. J. Mauser, C. Ringhofer, and G. Ulich, Numerical methods for Bloch-Poisson type
equations, in Proc. International Workshop on Computational Electronics, University of
Leeds, UK, 1993, pp. 291–295.
[14] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932),
pp. 749–759.