Download 1 Hot Electron Modeling I: Extended Drift–Diffusion Models

ECE539 - Advanced Theory of Semiconductors and Semiconductor Devices
Numerical Methods and Simulation / Umberto Ravaioli
Review of Conventional Semiconductor Device Models
Based on Partial Differential Equations - II
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Hot Electron Modeling I: Extended Drift–Diffusion Models
Due to the relative simplicity of the drift–diffusion equations, it would be very appealing, to extend
the validity of drift–diffusion–like models well into the hot electron regime. We have seen that the
simplest attempt to include high–field effects is to make the mobility and the diffusion coefficient
field dependent. The current in 1–D is
∂n(x, t)
(1)
∂x
Here µ(E)E = v(E) is the (steady state) drift velocity for the case of homogeneous field E. The
field is also space and time dependent, i.e. E = E(x, t). Mobility and diffusion coefficients are
steady state quantities, but the carrier velocity may differ considerably from the steady state value
v(E), due to abrupt space or time changes of the electric field. v(E) can be considered accurate
only if space or time variations of the field are very smooth (adiabatic).
Velocity overshoot occurs when the average electron velocity exceeds the steady state (bulk) velocity. A modified version of the drift diffusion equation to include velocity overshoot was proposed
by Thornber.
J(x, t) = qn(x, t)µ(E)E + qD(E)
∂E
∂E
+ B(E)
∂x
∂t
∂n(x, t)
∂n(x, t)
+qD(E)
+ qA(E)
∂x
∂t
J(x, t) = qn(x, t) v(E) + W (E)
(2)
where three new terms have been added. The term with W (E) contains the field gradient and
corrects the local drift velocity for spatial velocity overshoot effects. The term with B(E) contains the time derivative of the field and corrects for temporal velocity overshoot. The last term
with A(E) preserves the invariances of the total current (note that this term does not represent
generation, recombination, trapping, etc.), effects which may be incorporated with an addition
term premultiplied by A(E) also. The quantities W (E), B(E), and A(E) must be tabulated from
detailed transport calculations.
In steady state the current equation simply becomes
∂E
∂n
J(x) = qn(x) v(E) + W (E)
+ qD(E)
∂x
∂x
(3)
and the resulting continuity equation is
∂n(x)
∂
∂E
∂n
=
n(x)v(E) + n(x)W (E)
+ D(E)
∂t
∂x
∂x
∂x
(4)
Equation (4) does not describe a real transient, since the time derivatives in the current equation
have been neglected. Therefore (4) is valid in the steady state limit ∂n/∂t = 0, i.e. t → ∞. The
equation may be used, however, to solve the pseudo–time–dependent problem until steady state is
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achieved. Since the Poisson equation is solved at each time step, the fields and the related variable
are continuously updated in space. Alternatively, one may solve the steady–state equation obtained
with ∂n/∂t = 0, using Newton’s method.
Developing the space derivatives, (4) becomes
∂n
∂t
∂n
∂v(E)
∂n
∂
∂E
v(E) +
n+
W (E)Ex + n [W (E)
]
∂x
∂x
∂x
∂x
∂x
∂ 2 n ∂D(E) ∂n
+D(E) 2 +
∂x
∂x ∂x
∂v(E)
∂
∂E
∂E ∂D(E) ∂n
∂2n
= (
+
[W (E)
])n + [v(E) + W (E)
+
]
+ D(E) 2
∂x
∂x
∂x
∂x
∂x ∂x
∂x
=
(5)
Finally, the pseudo–time–dependent equation has form
∂n
∂2n
∂n
= a(x, t) 2 + b(x, t)
+ c(x, t)n
(6)
∂t
∂x
∂x
Since from Gauss law f rac∂E∂x = −ρ/ and ρ depends on the carrier density n(x), the coefficients
b(x, t) and c(x, t) are also functions of n(x) and eq. is nonlinear.
Often the overshoot parameter W (E) is rewritten in terms of mobility, as W (E) = µ(E)L(E),
where L(E) is called length coefficient. Monte Carlo calculations as well as analytical models for
the length coefficient have been presented in the literature. Extension to 2–D is not trivial when
confining fields (barriers) besides accelerating fields are present. An approximate approach allows
the extension to 2–D by using the gradient of the quasi–Fermi levels (very small inside barrier
regions and equal to the field in accelerating regions) as the functional parameter for the length
coefficient.
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Hot Electron Modeling II: Hydrodynamic Model
Transport equations which express conservation of particles, momentum, and energy may be obtained from the moments of the Boltzmann equation
∂n
∂n
+ ∇·(vn) =
∂t
∂t coll
∂P
∂P
+ ∇·(vP) = −qnE − ∇(nkB T) +
∂t
∂t coll
∂W
∂W
+ ∇·(vW ) = −qnv · E − ∇·(nkB Tv) − ∇·Q +
∂t
∂t coll
(7)
(8)
(9)
v is the average velocity, P is the momentum density, W is the kinetic energy density, T is the
carrier temperature tensor, Q is the heat flow vector. No assumption has been made on the nature
of the collision terms, in the above equations. One set of equations can be written for each valley in
the conduction band of the semiconductor considered, with different set of variables n, v, P, Q, W ,
and T.
Equation (7) is a form of the carrier continuity equation, stating that the local increase of
carrier density plus the outflow of carriers is equal to the density increase due to collisions. The
momentum balance equation (8) states that the rate of change plus the outflow of the momentum
density equals the change due to the electric forces and to the electric pressure nkB T plus the
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momentum variation caused by collisions. The energy balance equation (9) similarly states that
the rate of change plus the outflow of kinetic energy density W equals the sum of the energy
supplied by the electric field, plus the work performed by the electric pressure, the divergence of
the heat flow Q and the rate of kinetic energy density change due to collisions. In terms of the
electron distribution function, the various variables are defined as
Z
n =
Z
nv =
f (r, k, t)d3 k
(10)
v(k)f (r, k, t)d3 k
(11)
m∗
v2 (k)f (r, k, t)d3 k
2 Z
Z
W
nkB Tij
=
= m∗
Qi =
[vi (k) − vi ][vj (k) − vj ]f (r, k, t)d3 k
m∗
n
2
Z
[vi (k) − vi ][v(k) − v]2 f (r, k, t)d3 k
(12)
(13)
(14)
The subscripts i and j indicate the conduction band valleys. The number of unknowns may be
reduced to simplfy the equations. Heat conduction has been traditionally assumed to obey the
classical law
Q = −κ∇T
(15)
where κ is the heat conductivity of the electron gas. However, it has been recognized in recent times
that this approach is not correct for semiconductors, particularly in correspondence of junctions,
where the heat flow is much more complicated. The temperature is a measure of the average
thermal energy per electron, and can be related to the average total kinetic energy w as
W
3
m|v|2
= kB T +
n
2
2
The momentum density is also simply related to the drift velocity according to
w=
P = m∗ nv
(16)
(17)
(16) assumes that T is a local property of the valley rather than a tensor, and therefore in (13)
only i = j is considered.
The assessment of the collision terms is a rather difficult task. These terms are responsible
for momentum and energy relaxation and for intervalley transfer. The simplest way is to adopt a
phenomenological approach which introduces relaxation times
ni
nj
∂ni
= =−
+
∂t coll
τij
τji
∂Pi
Pi
= −
∂t coll
(τm )i
∂Wi
∂t
"
coll
#
(18)
(19)
"
#
n i wi
n j wj
wi − 3kB To /2
+
− ni
= −
(τe )ij
(τe )ji
(τe )ij
To is the lattice temperature. If the average total kinetic energy w is used, then
3
(20)
∂wi
∂t
=
coll
1
ni
∂Wi
∂t
−
coll
wi
ni
∂ni
∂t
(21)
coll
and similarly, if the drift velocity replaces the momentum,
∂vi
∂t
coll
1
= ∗
mi ni
∂Pi
∂t
vi
−
ni
coll
∂ni
∂t
(22)
coll
The particle relaxation time for scattering from the ith to the j th valley depends on the average
kinetic energy of carriers on the ith valley. The momentum relaxation time includes both intravalley
and intervalley scattering, therefore one can write
1
(τm )i
=
1
+
(τm )ii
1
(τm )ij
(23)
The first two terms in (20) express the energy relaxation due to intervalley transfer out of and
into the ith valley, while the third term describes the relaxation towards the equilibrium with lattice
temperature To . This last term is the only one to be considered in a single valley semiconductor.
It is interesting now to look at the simplest form of momentum and energy balance equation,
valid for a uniform 1–dimensional, single band semiconductor
d[m∗ (w)v]
dt
dw
dt
m∗ (w)v
τm (w)
w − wo
= qEv −
τe (w)
= qE −
(24)
(25)
Note that the effective mass is a function of energy also, to account for a non–parabolic band, and
wo is the average equilibrium electron energy 3k2B T .
Equation (24) and (25) can be used to study bulk properties, like velocity overshoot, as done
for instance by Nougier et al. This simple set of equations has also been used to study multivalley
semiconductors, using special average effective mass in different valleys, which takes into account
the fraction βi of the total electrons residing in each valley
m∗av =
X
βi m∗i
(26)
i
This approach is not too accurate, and tends to overestimate the average mass (10 to 20% too high
for average fields in GaAs). A better way is to treat separately the valleys and obtain the coupling
through the intervalley particle relaxation in the continuity equations. If one really wants to use
an average mass to account for a multivalley situation, a Monte Carlo experiment should be run
for each value of the field, relating effective mass to field with the elementary formula
m∗i
X
∆vi (n) = qE
n
X
∆ti (n)
(27)
n
where n indicates, ∆vi (n) is the velocity change during the flight, ∆ti (n) is the time of flight. If a
simulation is run for a time
ttot =
XX
∆ti (n)
n
i
recording the evolution of one electron, the average effective mass is found as
4
(28)
qEttot
m∗av = P P
i
n ∆ti (n)
(29)
and the fraction of electrons and the average mass in the ith valley are
βi =
X ∆ti (n)
ttot
n
m∗i
P
∆ti (n)
= qE P n
n ∆vi (n)
(30)
(31)
therefore
m∗av =
P
i βi
m∗i
(32)
To include non–parabolicity, the energy–momentum relation becomes
h̄2 |k|2
= w(a + αw)
(33)
2m∗o
where α is the non–parabolicity factor and m∗o is the effective mass at the energy minimum. The
energy dependent effective mass for motion parallel to to the E field is
m∗ =
1 ∂2w
h̄2 ∂k 2
!−1
= mo (1 + 2αw)3
(34)
To solve a system of balance equations for device simulation, very frequently assumptions
are made on the shape of the distribution function in momentum (velocity) space. A displaced
Maxwellian is often postulated, or simply a symmetric distribution function
f (r, v, t) = f (r, |v − hv(r)i |, t)
(35)
One should note that normally a hot carrier distribution is likely to approach a displaced
Maxwellian if considerable carrier–carrier scattering is present, i.e. the carrier concentration is
higher than a given critical concentration. This is therefore very questionable in the case of pinch–
off regions. Because of the symmetry of the distribution, one can say that the average kinetic
energy is dominated by carrier temperature effects, rather than drift, i.e.
3
1
kB Te m∗ |v|2
(36)
2
2
where Te refers to an electron distribution. In steady state we may write the simplified system of
equations
w≈
∇·(nv) = 0
τm (w)
2w 2w
qE
−
∇
v =
−
∇n
m∗
3
3n
2w
w − wo
v · ∇w + ∇
= qv · E −
3
τw (w)
q(n − ND )
∇·E =
5
(37)
(38)
(39)
(40)
Although considerably simplified, the equations are still fairly difficult to solve, particularly if
one wants to use the method for engineering applications, involving intensive computations. Further
approximations are possible if the transport properties of a particular material (e.g. Si or GaAs)
are well known, or if a special geometry allows one to make assumptions on the electric fields or on
depletion regions. An example for MESFET simulation may be found in the mentioned paper by
Cook and Frey.
A general solution of the original equations (7)–(9) is in principle possible if a real accurate
answer is actually sought. However, it is very difficult to find a solution without assuming a particular form of the distribution function, unless one has a way to calculate the auxilliary equations
(10)–(14) and the collision terms accurately. A possible approach is to use Monte Carlo methods
to generate extensive tables of the various parameters, for different fields and concentrations. Since
the Monte Carlo method is stochastic in nature, no assumptions need to be made on the distribution function, and the scattering rates may be input directly without assuming any relaxation
time. Even if all the needed parameters may be evaluated accurately, a complete solution of the
(7)–(9) plus Poisson’s equation for a general device simulation remains a formidable task. If the
three conduction valleys of GaAs are included, for instance, one is left with 9 nonlinear partial
differential equations, with strong hyperbolic nature, coupled to Poisson’s equation.
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Hot Electron Modeling III: Energy Transport Models
The energy transport models offer an alternative to hydrodynamic approaches. Instead of taking the
moments of Boltzmann equation and then approximating the collision integrals by using average
relaxation times for momentum and energy, in this approach a macroscopic relaxation time is
assumed in the Boltzmann equation. In steady state for electrons in a single energy valley, the
Boltzmann equation becomes
For electrons, the zeroth and second order moments of the steady-state Boltzmann transport
equation (BTE) give the carrier continuity and energy balance equations:
∇ · J = −q(G − R) ;
(41)
∂E ∇·S = F·J−n<
>
∂t coll
(42)
where the currentR J and the energy flow S are defined by J = −qn < v >= −q d3 kvf and
S = n < Ev >= d3 k Evf . Here, G − R stands for the carrier generation-recombination rate, F
is the electric field, q the elementary charge, n the carrier concentration, v the carrier velocity, k
the crystal momentum, f the carrier distribution function and E the carrier
energy.
The angular
R 3
4 R
3
brackets denote averages over the distribution function, i.e., < A > = d kAf / d kf . The last
term in (42) is the energy loss rate due to collisions. By decomposing f into its even and odd parts,
f0 and f1 , and using the microscopic relaxation time approximation (∂f1 /∂t)coll = −f1 /τ [?], the
odd part of the BTE becomes
qτ F
∇k f0 − τ v∇r f0 .
(43)
f1 =
h̄
Substitution of (43) into the expressions for J and S yields
R
J = −q
Z
S =
Z
d3 kvf1 = q nµ̂F + ∇ · (nD̂) ;
(44)
d3 k Evf1 = −nµ̂E F − ∇ · (nD̂E ) .
(45)
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where all the transport coefficients µ̂, D̂, µ̂E , and D̂E are tensors and their components are defined
by
q
∂f0
µi,j = < τ vi
>0 ; Di,j =< τ vi vj >0 ;
(46)
f0 ∂kj
h̄
µE i,j =
q
∂f0
>0 ; DE i,j =< τ Evi vj >0 .
< τ Evi
f0 ∂kj
h̄
(47)
Here, < >0 denotes average over f0 . The divergence of the tensors in (44) and (45) means to
take the divergence of each row of the tensor as a vector element. It is worth noticing that the
constraint f1 f0 , which is the basis of the Legendre polynomial expansion approach in [?] and
the perturbed distribution approach in [?], is not necessary in this model.
For materials like Si, where the mean kinetic energy m < v >2 /2 is much smaller than the
mean thermal energy m < v2 > /2, the correction from the field-induced anisotropy is generally
not important [?]. Therefore, f0 can be approximated by an isotropic distribution f0 (E) and
µ̂, D̂, µ̂E , and D̂E become scalar functions of the mean carrier energy < E >. We assume in
the following τ = τ (E) ∝ E −p , where p is determined by the dominant scattering mechanism and
ranges from 0.5 to 1 according to our preliminary bulk MC simulations. Additional hot–carrier
effects are included by incorporating the following first–order approximations: (a) non–parabolic
bands: h̄2 k2 /2m∗ = E(1 + αE); (b) non-Maxwellian effects: f0 = (1 + γE/kB Te )fm (E). Here,
fm (E) is the Maxwellian distribution at an elevated temperature Te and γ is the non-Maxwellian
parameter. With these assumptions and from Eqs. (46) and (47), the modified Einstein relations
are valid not only for µ and D but also for µE and DE , i.e., D = kB Tm µ/q and DE = kB Tm µE /q,
where the equivalent carrier temperature Tm = (1 + γ)Te is related to the mean carrier energy by
5
3
5
3
< E > = (1 + γ + αkB Te ) kB Te ' (1 + αkB Tm ) kB Tm .
2
2
2
2
(48)
Then, we can recast (44) and (45) into
J = q(−µn∇ψ + kB ∇(nµTm ) ;
S = −Ce (−µnkB Tm ∇ψ +
2
kB
∇(nµTm 2 ) ;
q
(49)
(50)
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where ψ is the electrical potential and Ce = µE /(kB Tm µ) = (5/2 − p)(1 − αkB Tm /2) is the energy
flow factor Ce . Ce is in general a smoothly varying function of the mean carrier energy and for
nearly parabolic bands Ce ' 5/2 − p. It is important to notice that in our approach the WiedmannFranz law for heat flow is never invoked, as it is usually necessary in the conventional HD method
to obtain a closure of the moment equations. Based on the similarity of the functional expressions
in (49) and (50), an extended Scharfetter-Gummel method [?] was developed for self-consistent
discretization of both carrier continuity equation and energy balance equation. By assuming that
˜ S̃, and (1/T )(dψ/dl) are constant, we can solve 1-D forms of both (49) and
along each mesh line J,
(50) on the mesh lines and perform the standard box-integration around each node [?]. With the
following normalized notation: J̃ = J/q, S̃ = −S/qCe , Ñ = nµ, T̃ = kB Tm /q, Eqs. (49) and (50)
are evaluated on the mesh line connecting the nodes i and j by
J̃l =
S̃l =
1
(B(ũij )Nj T˜j − B(−ũij )Ni T̃i )
Lij
(51)
1
2
2
(B(ũij )Nj T˜j − B(−ũij )Ni T̃i ) .
Lij
(52)
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Lij is the distance between nodes i and j, B(x) = x/(exp(x) − 1) is the Bernoulli function, and
4 R xj
T̃ dx/Lij . Usually, the carrier temperature varies much more
ũij = (ψj − ψi )/T̄ , with T̄ = xi
slowly than the potential does. Therefore, the linear approximation T̄ = (T̃i + T˜j )/2, is reasonable
in practice. This also implies a quadratic approximation in space for the potential according to the
above discretization assumptions.
Since F = −∇ψ, by substituting F · J with −∇ · (ψJ) + ψ∇ · J and employing a macroscopic
relaxation time approximation for (∂E/∂t)coll , we can rewrite (42) as
∇ · H = ψ · (G − R) − n
< E > −E0
τ
(53)
where E0 = 3kB T0 /2, with T0 being the lattice temperature, τ is the energy relaxation time, and
H = S + ψJ is the total energy flow, consisting of both the thermal energy flow S and the electrical
energy flow ψJ. From Eqs. (51) and (52), and using ψ̄ = (ψi + ψj )/2, we can express the total
energy flow on the mesh line as
H
=
=
q(ψ J̃l − Ce S̃l )
q
(B(ũij )Nj T̃j (ψ̄ − Ce T̃j ) − B(−ũij )Ni T̃i (ψ̄ − Ce T̃i )).
Lij
(54)
In this improved ET model, the necessary transport parameters µ(Tm ), Ce (Tm ) and τ (Tm ) can
all be determined, for instance, from bulk MC simulations by evaluating the following ensemble
averages
<v>
3 < Ev >
< E > −E0
µ=−
; Ce =
; τ = −
.
(55)
F
2 < E >< v >
qF < v >
The energy transport approach is appealing for practical simulation applications, because the
numerical procedure developed for drift–diffusion problems can be easily adapted. The numerical
difficulties encountered with hydrodynamic approaches are not present here, because instead of
hyperbolic nature, the energy transport equations have parabolic nature.
References
K. Blotekjaer, “Transport equations for electrons in two–valley semiconductors,” IEEE Transactions on Electron Devices, vol. ED-17, pp. 38–47, 1970.
K. Blotekjaer and E. B. Lunde, “Collision integral for displaced Maxwellian distributions,” Physical
Status Solidi, vol. 35, pp. 581–592, 1969.
R. Stratton, “Diffusion of hot and cold electrons in semiconductor barriers,” Physical Review, vol.
126, pp. 2002–2014, 1962.
R. Cook and J. Frey, “Two–dimensional numerical simulation of energy transport effects in Si and
GaAs MESFETs,” IEEE Transactions on Electron Devices, volt ED-29, pp. 970–977, 1982.
J. P. Nougier et al., “Determination of the transient regime in semiconductor devices using relaxation time approximation,” Journal of Applied Physics, vol. 52, pp.825–832, 1981.
K. K. Thornber, “Current equations for velocity overshoot,” IEEE Electron Device Letters, volt.
EDL-3, pp. 69–71, 1982.
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