Drift-Diffusion model
The drift-diffusion model is a fundamental concept in the field of
semiconductor physics, semiconductor devices and simulation in
electronics, and drift-diffusion models are a powerful tool for simulating
the electrical behavior of semiconductor devices.
1. Derivation of the System of Equations:
The derivation includes transport equations from semiconductor
physics including carrier motion, diffusion coefficient, electric field to
model carrier flow under an electric field and carrier diffusion due to
concentration gradients.
1.1. Derivation of the carrier transport equations
The popular drift-diffusion current equations can be easily derived
from the Boltzmann transport equation (BTE).
The Boltzmann transport equation (BTE) is a more fundamental
equation that describes the behavior of electrons in a semiconductor
material. It takes into account the effects of electric fields, diffusion, and
scattering on the electron distribution function, which is the probability of
finding an electron with a given energy. The BTE can be expressed as:
With the use of a relaxation time approximation, the Boltzmann
transport equation may be written:
(1)
The general definition of current density may be written:
(2)
Where the integral on the right-hand side represents the first
moment of the distribution function. This definition of current can be
related to (1) after multiplying both sides by v and integrating over v. From
the left-hand side, we get:
(3)
The equilibrium distribution function is symmetric in v, and its
integral is zero. Therefore, we have from (1)
(4)
Integrating by parts we have
(5)
and we can write
(6)
where is ⟨𝜈 2 ⟩ the average of the square of the velocity defined as
(7)
The drift-diffusion equations are derived introducing the mobility
𝑘 𝑇
µ = ∗ and replacing ⟨𝜈 2 ⟩ with its average equilibrium value 𝐵∗ ,
𝑒𝜏
𝑚
therefore neglecting thermal effects. The diffusion coefficient 𝐷 =
𝑚
µ𝑘𝐵 𝑇𝑜
𝑒
(Einstein’s relation) is also introduced, and the resulting drift-diffusion
current is:
(8)
(9)
where q is used to indicate the absolute value of the electronic
charge. Although no direct assumptions on the non-equilibrium
distribution function f (v, x) was made in the derivation of (8) and (9), in
effect, the choice of equilibrium (thermal) velocity means that the driftdiffusion equations are only valid for very small perturbations of the
equilibrium state (low fields). The validity of the drift-diffusion equations
is empirically extended by introducing field-dependent mobility µ(E) and
diffusion coefficient D(E), obtained from empirical models or detailed
calculations.
1.2. Poisson’s equation
We begin with two of Maxwell's equations relevant to
electrostatics:
Gauss's law for electricity:
∇·E=ρ/ε
(10)
where:
E is the electric field vector
ρ is the charge density
ε is the permittivity of the material
Faraday's law of induction (for static fields):
∇×E=0
(10)
This equation implies that the electric field is conservative,
meaning it can be derived from a scalar potential function, denoted by Φ:
E = -∇Φ
(11)
Combining the Equations:
Substituting the expression for E from Faraday's law into Gauss's
law, we get:
∇·(-∇Φ) = ρ / ε
(12)
Expanding the gradient operator, we obtain:
-∇²Φ = ρ / ε
(13)
This is the Laplacian equation, which relates the potential
function to the charge density.
Poisson's Equation with Material Properties:
The permittivity of a semiconductor material is usually not constant
and depends on various factors like temperature and doping concentration.
To account for this, we can introduce the relative permittivity, ε_r, defined
as:
εr = ε/ε0
where ε0 is the permittivity of free space.
(14)
Substituting this into the Laplacian equation, we get:
-∇²Φ = ρ / (εr * ε0)
(15)
This is the Poisson equation in its most common form used for
semiconductor materials. It explicitly incorporates the relative permittivity,
enabling a more accurate description of the electric field distribution for a
given charge density.
1.3. Continuity equations
(16)
(17)
2. Physical Limitations on Numerical Drift-Diffusion Schemes:
The continuity equations are the conservation laws for the charge
carriers, which may be easily derived taking the zeroth moment of the time
dependent BTE. A numerical scheme which solves the continuity
equations should
1. Conserve the total charge inside the device, as well as that
entering and leaving.
2. Respect the local positive definite nature of carrier density.
Negative density is
3. unphysical.
4. Respect monotonicity of the solution (i.e. it should not introduce
spurious space oscillations).
Conservative schemes are usually achieved by subdivision of the
computational domain into patches (boxes) surrounding the mesh points.
The currents are then defined on the boundaries of these elements, thus
enforcing conservation (the current exiting one element side is exactly
equal to the current entering the neighboring element through the side in
common). For example, on a uniform 2-D grid with mesh size ∆, the
electron continuity equation could be discretized in an explicit form as
follows
(18)
This simple approach has certainly practical limitations, but is
sufficient to exemplify the idea behind the conservative scheme. With the
present convention for positive and negative components, it is easy to see
that in the absence of generation-recombination terms, the only
contributions to the overall device current arise from the contacts.
Remember that, since electrons have negative charge, the particle flux is
opposite to the current flux. The actual determination of the current
densities appearing in (18) will be discussed later.
When the equations are discretized, using finite differences for
instance, there are limitations on the choice of mesh size and time step:
1.
The mesh size ∆x is limited by the Debye length.
2.
The time step is limited by the dielectric relaxation time.
A mesh size must be smaller than the Debye length
(19)
where N is the doping. In GaAs and Si, at room temperature the
Debye length is approximately 400 ˚A when N ≈ 1016 cm−3 and decreases
to about only 50 ˚A when N ≈ 1018 cm−3.
The dielectric relaxation time may be estimated using
(20)
To see the importance of respecting the limitations related to the
dielectric relaxation time, imagine to have a space fluctuation of carrier
concentration which is going to relax to equilibrium with a law
(21)
A finite difference discretization of this equation gives at the firsttime step
(22)
Clearly, if ∆t > tdr, the value of ∆n(∆t) is negative, which means
that the actual concentration is evaluated to be less than the equilibrium
value, and it is easy to see that the solution at higher time steps will decay
oscillating between positive and negative values of ∆n. An excessively
large ∆t may lead, therefore, to nonphysical results. In the case of high
mobility, the dielectric relaxation time can be very small. For instance, a
sample of GaAs with a mobility of 6, 000 cm2/V-s and doping 1018 cm-3 has
approximately tdr ≈ 10-15 s, and in a simulation the time step ∆t should be
chosen to be considerably smaller.
3. Numerical methods for solving the Equations
Direct numerical methods for device simulation have been limited
by the complexity of the equation, which in the complete 3-D timedependent form requires seven independent variables for time, space and
momentum. To-date, most semiconductor applications have been based on
stochastic solution methods (Monte Carlo), by which a possible history of
a single particle is stochastitally simulated. When simulation is carried on
for a sufficient time, the history of the particle is a correct representation
of the unknown function.
1.4. Newton’s Method
Newton’s method is a coupled procedure which solves the
equations simultaneously, through a generalization of the NewtonRaphson method for determining the roots of an equation. For example
(23)
Starting from an initial guess Vo, no, and po, the corrections V, ∆n,
and ∆p are calculated from the Jacobian system
(24)
which is obtained by Taylor expansion. The solutions are then
updated according to the scheme
(25)
where k indicates the iteration number. In practice, a relaxation approach
is also applied to avoid excessive variations of the solutions at each
iteration step.
The system (25) has 3 equations for each mesh point on the grid.
On the other hand, convergence is usually fast for the Newton method,
provided that the initial condition is reasonably close to the solution, and
is in the neighborhood where the solution is unique. There are several
viable approaches to alleviate the computational requirements of the
Newton’s method.
In the Newton-Richardson approach, the Jacobian matrix is
updated only when the norm of the error does not decrease according to a
preset criterion. In general, the Jacobian matrix is not symmetric positive
definite, and fairly expensive solvers are necessary. Iterative schemes have
been proposed to solve each step of Newton’s method by reformulating
(25) as
(25)
Since the matrix on the left-hand side is lower triangular, one may
solve decoupling into three systems of equations solved in sequence.
(26)
and the result is used in the next block of equations
(27)
Similarly, for the third block
(28)
1.5. Scharfetter-Gummel approximation
To discretize continuity equations, determining currents at midpoints of mesh lines is crucial. Solutions are available only at grid nodes,
requiring interpolation for current determination. For consistency with
Poisson's equation, linear potential variation between neighboring nodes is
assumed, resulting in a constant field along mesh lines. Mid-point fields
are obtained using centered finite differences. Estimating carrier density at
mid-points involves assuming linear variation through the arithmetic
average between neighboring nodes. This approach is accurate for small
potential variations and exact when the field between nodes is zero,
indicating identical carrier density at both points.
In order to illustrate this, let’s consider a 1-D mesh where we want
to discretize the electron current
(29)
and its finite centered-difference approximation
(30)
In the simple approach indicated above, the carrier density ni+1/2 is
expressed as
(31)
In the Scharfetter and Gummel quasi-variational approach the
current in the interval [xi; xi+1] is expressed as a truncated expansion about
the value at the mid-point xi+½
(32)
From (32) we obtain a first order differential equation for Jn which
can be solved to provide n(x) in the mesh interval, using as boundary
conditions the values of carrier density ni and ni+1. We obtain
(33)
where g(x, φ ) is the growth function
(34)
The result in (33) can be used to evaluate n (xi+1/2) for the
discretization of the current in (30). It is easy to see that only when Ψ (i +
i) - Ψ (i) = 0 we have
(35)
The continuity equation can be easily discretized on rectangular
uniform and nonuniform meshes using the above results for the currents,
because the mesh lines are aligned exactly.
4. Generation and recombination:
4.1. The Shockly-Reed-Hall model.
The Shockley-Read-Hall (SRH) model is a mathematical model
that describes the generation and recombination of charge carriers
(electrons and holes) in semiconductors. It is based on the assumption
that recombination occurs through a two-step process involving a trap
level.
In the SRH model, a trap level is an energy level in the band gap
of a semiconductor material that is located between the conduction band
and the valence band. When an electron from the conduction band falls
into a trap level, it leaves behind a hole in the valence band. This process
is called electron capture. The electron can then either remain in the trap
level or be released back into the conduction band. If the electron is
released back into the conduction band, it becomes a free electron again.
This process is called electron emission.
Recombination occurs when an electron from the conduction band
falls into a hole in the valence band. This process is called radiative
recombination if the energy released is emitted as a photon of light. If the
energy is not emitted as light, then it is instead transferred to the lattice of
the semiconductor material, which causes the material to heat up. This
process is called non-radiative recombination.
The Shockley-Reed-Hall model is very often used for the
generation-recombination term due to trap levels
(36)
where Et is the trap energy level involved and τn and τp are the
electron and hole lifetimes. Surface rates may be included with a similar
formula, in which the lifetimes are substituted by s 1/sn,p where sn,p is the
surface recombination velocity.
4.2. Auger recombination
The Auger recombination may be accounted for by using the
formula
(37)
where Cn and Cp are appropriate constants. The Auger effect is for
instance very relevant in the modeling of highly doped emitter regions in
bipolar transistors.
The generation process due to impact ionization can be included
using the field-dependent rate
(38)
5. Time-dependent simulation
The time-dependent form of drift-diffusion equations is versatile
for both steady-state and transient simulations. Steady-state analysis
involves evolving the numerical system from an initial guess until a
stationary solution is reached within tolerance limits. While rarely used due
to the availability of robust steady-state simulators, it is beginner-friendly
for simple applications. Explicit schemes in steady-state analysis avoid
matrix solutions but often require extremely small timesteps for stability.
Simulating transients necessitates a physically meaningful initial
condition, obtainable from a steady-state calculation. Time-dependent
numerical approaches are suitable, but careful consideration of boundary
conditions is crucial due to displacement current during transients.
Neglecting displacement current in a transient simulation implies ideal
voltage generators, leading to the shortest switching times. In practical
scenarios with external loads and parasitics, switching times may be
longer. Neglecting displacement current effects in simulation can be useful
for assessing the ultimate speed limits of a device structure.
When dealing with unipolar devices, as often used in many
microwave applications, it is possible to formulate very simple timedependent drift-diffusion models, which can be solved with
straightforward finite difference techniques and are suitable for small
student projects. If we can neglect the generation-recombination effects,
the 1-D unipolar drift-diffusion model is reduced to the following system
of equations
(41)
(42)
The system (41) and (42) can be used for both transient or steady
state conditions if the simulation is run ∂n / ∂t = 0. A basic simple algorithm
might consist of the following steps
1. Guess the carrier distribution n(x).
2. Solve Poisson’s equation to obtain the field distribution.
3. Compute one iteration of the discretized continuity equation with
time step ∆t. v(E) and D(E) are updated according to the local field value.
4. Check for convergence. If convergence is obtained, stop.
Otherwise, go back to step (2)
A simple discretization scheme could employ an explicit finite
difference approach
(43)
(44)
There are of course many other possible explicit and implicit
discretizations. Such simple finite difference approaches are in general a
compromise which cannot provide at one time an optimal treatment of both
advective and diffusive components.
Because of spatially varying drift velocity, spurious diffusion and
dispersion are present. This could be mitigated by using a nonuniform grid
discretization, where the mesh size is locally adapted to achieve vd = ∆x/∆t
everywhere, which would involve interpolation to the new grid-points.
The discretization for a diffusive process is better behaved with a fully
implicit scheme. On the other hand, the fully implicit algorithm for
advection is not conservative. From these conflicting requirements, it
emerges that it would be beneficial to split the drift and diffusion processes,
and apply an optimal solution procedure to each. In analogy with this, the
1-D continuity equation can be solved in two steps, for instance
(45)
where again a simple explicit upwinding scheme is used for the drift,
while a fully implicit scheme is used for the diffusion.
6. The drift-diffusion model in low electric fields
Drift refers to the movement of charge carriers under the influence
of an electric field. The force exerted on a charge carrier by an electric field
is given by the equation:
F = qE
The drift velocity of a charge carrier is the velocity it acquires due
to the electric field. It is given by the equation:
vd = μE
where: μ is the mobility of the charge carrier
The mobility of a charge carrier is a measure of its ability to move
in an electric field. It is a material-dependent property that is typically
measured in cm²/V·s.
Diffusion refers to the movement of charge carriers from an area of
high concentration to an area of low concentration. This is caused by the
random thermal motion of the charge carriers. The diffusion flux of a
charge carrier is given by the equation:
Jd = -D∇n
or
Jd = -D∇p
where:
•
Jd is the diffusion current density
•
n is the concentration of electrons
•
p is the concentration of holes
•
D is the diffusion coefficient of the charge carrier
4.3. Low Electric Field Approximation
In the low electric field approximation, the electric field is assumed
to be small enough that the mobility of the charge carriers is independent
of the electric field. This means that the mobility can be approximated as a
constant.
Under this approximation, the drift-diffusion equations can be
simplified to:
•
Continuity equation:
∂n/∂t + ∇·(μn∗nE) = 0
or
∂p/∂t + ∇·(μp∗pE) = 0
•
Poisson's equation:
∇·E = (n + p)/ε
where:
These simplified equations are often used to analyze the
performance of semiconductor devices, such as diodes and transistors.
7. The drift-diffusion model in high electric fields.
When the electric field becomes high enough, the mobility of the
charge carriers can no longer be assumed to be constant. This is because
the electric field can cause the scattering mechanisms that determine carrier
mobility to become saturated. As a result, the drift velocity of the charge
carriers begins to saturate, and the diffusion coefficients can also decrease.
To account for these effects, the drift-diffusion equations need to
be modified. One common approach is to use the Scharfetter-Gummel
discretization scheme, which is a numerical method that can accurately
capture the behavior of charge carriers in the presence of strong electric
fields.
In addition to mobility saturation, other effects need to be
considered at high electric fields. These include:
•
Impact ionization: This occurs when an energetic charge carrier
collides with a lattice atom and knocks off an electron, creating an
electron-hole pair.
•
Recombination: This occurs when an electron and a hole collide and
annihilate each other, releasing energy in the form of phonons or
light.
•
Trapping: This occurs when a charge carrier is temporarily trapped
in a lattice defect. This can reduce the mobility of the charge carrier
and affect the current flow.
The combination of these effects can make it difficult to accurately
model charge transport in semiconductors at high electric fields. However,
there are a number of advanced numerical techniques that can be used to
overcome these challenges.
Here are the updated equations for the high electric field case:
Continuity equation for electrons:
∂n/∂t + ∇·(μn∗nE) = -Gn + Rn
where:
•
Gn is the generation rate of electrons
•
Rn is the recombination rate of electrons
Continuity equation for holes:
∂n/∂t + ∇·(μp∗nE) = -Gp + Rp
where:
•
Gp is the generation rate of holes
•
Rp is the recombination rate of holes
Poisson's equation:
∇·E = (qn + qp - q)/ε
where:
•
E is the electric field
•
qn is the charge density of electrons
•
qp is the charge density of holes
•
q is the charge of the electron
•
ε is the permittivity of the semiconductor