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Mathematical model for semiconductors

Fabrizio Bonani

Dipartimento di ElettronicaPolitecnico di Torino

High speed electron devices (PoliTo) Mathematical model for semiconductors 1 / 37

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Contents

1 Charge transport

Microscopic Ohms law

Semiconductor out of equilibrium

Diffusion current

Drift current

Injection levelGeneration and recombination phenomena

2 Mathematical model for semiconductors

Continuity equation

Poisson equationMathematical model

3 Quasi-Fermi levels


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Electrical conduction

Free carriers in the material are accelerated by any electric field E

Moving charges correspond to electrical current

The (average) drift velocity v of carriers is proportional to E

through the corresponding mobility [cm2 V1 s1]

vn = nE, vp = pE (n, p> 0)

since negative charges move in the opposite direction with respect

to E, positive charges in the same direction


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Scattering phenomena

During motion, free carriers are scattered by interactions with any

perturbation to the spatial periodicity of the lattice potential energy lattice vibrations due to the atoms thermal energy (phonons) impurity atoms in the lattice (i.e., elements different from the

semiconductor) lattice defects

Mobility is not constant, due to the random scattering events; ingeneral

is constant for low electric field: the value for E 0 is calledlow-field mobility 0

For large electric field, v becomes constant, the saturation velocity,whose value is around 107 cm/s

The v(E) curve may be monotonic (Si, Ge) or show negativedifferential mobility d = d|v|/dE regions (compoundsemiconductors, electrons only)


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Velocityfield curve


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Mobilitydoping curve


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Mobilitytemperature curve


Beyond ambient

temperature (300 K),

the low-field mobility is

a decreasing function

of T

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Microscopic Ohms law


Let us consider an ndoped sample,

with p 0

Applying a uniform electric field E,current Iflows:

I=dQ

dt=

dQ

ds

ds

dt=

dQ

dVAvn = qnAvn = qnAnE

The current density J= I/A [A/cm2] is given by J= E, where isthe electric conductivity due to free electrons with mobility n:

= qnn

In a sample where pholes per unit volume are also present (pmobility):

J= Jel + Jhol = qnnE + qppE

therefore = qnn+ qpp

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Resistivitydoping curve


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Semiconductor out of equilibrium

For a physical system, thermal equilibrium corresponds to the

absence of any energy exchange with the environment

Any electron device works out of equilbrium, since it transformselectrical signals

Out of equilibrium, carrier concentrations are varied from the

equilibrium values


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Concentration variations

Carrier concentrations depend in general on position and time

n= n(x, t), p= p(x, t) 1D case

Their variations are due to carrier motion due to diffusion: diffusion current carrier motion due to drift (if E is present): drift current dielectric displacement current: only for fast (i.e., high frequency)

time-varying electric field

generation and recombination (GR) phenomena


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Carrier diffusion


Diffusion is the natural tendency of agas to make the particle

concentration spatially uniform

Diffusion intensity is proportional to

the concentration gradient (first

derivative), movement goes in theopposite direction

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Diffusion current


Since particles in a semiconductorcarry a charge, a current density is

associated to their diffusion

Electrons and holes have opposite

charge, thus diffusion results in

currents with opposite direction

Jn,diff = qDnn

xJp,diff = qDp

p

x

C

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Diffusion Coefficients

Dn and Dp are electron and hole diffusion coefficients or

diffusivities [cm2/s]

At thermal equilibrium (and, approximately, near equilibrium) the

Einstein relation holds

Dn = VTn, Dp = VTp

where VT = kBT/q is the electrical equivalent of temperature

VT = 26 mV at 300 K


D if

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Drift current

Drift current corresponds to carrier motion induced by an electric

field E

Microscopic Ohms law holds

Jn,dr = qnnE Jp,dr = qppE

Neglecting the displacement current, total current is

J= Jn,diff + Jn,dr Jn+ Jp,diff + Jp,dr Jp


S i d t t f ilib i l t

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Semiconductor out of equilibrium: nomenclature

Symbology:

if n type: nn,pn; if p type: np,pp thermal equilibrium: nn0,pn0, np0,pp0 intrinsic concentration: ni = pi

We define the excess concentrations:nn = nn nn0

pn = pn pn0

np = np np0

pp = pp pp0

If n,p > 0 injection takes place, if n, p < 0 we have depletionWithin the quasi neutrality assumption, n p


I j ti l l

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Injection level


ndoped Si sample with ND = 1016

cm3

Low injection level: if |nn|, |pn| ND;

minority carriers only feel

concentration variations, while

nn ND

High injection level: if |nn|, |pn| ND;

both carriers feel concentration

variations

G ti d bi ti

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Generation and recombination


These phenomena correspond tocreation and destruction events for

free carriers

direct mechanisms: band-bandtransitions

indirect mechanisms: transitionsassisted by recombination centers

Generation and recombination: parameters

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Generation and recombination: parameters

GR phenomena are characterized by

generation rate G, i.e. the number of carriers generated per unit

time and volume recombination rate R, i.e. the number of carriers recombinated per

unit time and volume

We define the net recombination rate:

Un = Rn Gn Up = Rp GpIn thermal equilibrium, Un = Up = 0 must hold

There are many expressions for U, depending on the physical

mechanism for the GR

A first order approximation is

Un n n0n

=n

nUp

p p0p

=p

p

where n and p are the electron and hole lifetime. For this reason,this is called lifetime approximation


Contents

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Contents

1 Charge transport

Microscopic Ohms lawSemiconductor out of equilibrium

Diffusion current

Drift current

Injection level

Generation and recombination phenomena


Continuity equation

Poisson equation

Mathematical model



Continuity equation

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Continuity equation

An equation describing the time and space evolution of charge

concentrations can be derived based on the charge conservationprinciple

Let us consider electrons crossing a volume dV = Adx. The timevariation of the total number of electrons within the volume is:

nt

dV = nt

Adx

x

x+@x

Jnx+@x

Jnx

A


Continuity equation: charge flux balance

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Continuity equation: charge flux balance

Such variation is due to 4 contributions:

(a) electrons entering the volume per unit time

(b) electrons exiting the volume per unit time(c) electrons generated in the volume per unit time(d) electrons recombinated in the volume per unit time

therefore, being electrons negative charges:

n

tAdx=

Jn(x)

q A (a)

Jn(x+ dx)

q A (b)

+ GnAdx (c)

RnAdx (d)

Using the first order approximation

Jn(x+ dx) Jn(x) +

Jnx dx

and letting dx 0, we get the electron continuity equation

n

t=

1

q

Jnx

Un


Continuity equation: lifetime approximation

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Continuity equation: lifetime approximation

Similarly, the hole continuity equation is found

p

t =

1

q

Jpx Up

In the lifetime approximation:

n

t

=1

q

Jn

x

n n0

np

t=

1

q

Jpx

p p0p

Current densities are expressed by the drift-diffusion model:

Jn = qnnE + qDnn

x

Jp = qppE qDpp

x


Poisson equation

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Poisson equation

Continuity equations involve three unknowns: n, pand E

A third equation is required to close the model, Poisson equation:

E

x=

, E =

x

where is the semiconductor dielectric constant [F/cm], and [Ccm3] is the positive net charge density:

= qp n+ N+D NA


Mathematical model: equations

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Mathematical model: equations

The two continuity equations and Poisson equation form themathematical model for semiconductors (1D case)

n

t=

1

q

Jnx

Un Jn = qnnE + qDnn

x

p

t=

1

q

Jpx

Up Jp = qppE qDpp

x

2

x2=

= qp n+ N

+

D NA

where E = /x


Mathematical model: approximations

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Mathematical model: approximations

A simplified analysis of electron devices calls to approximate the

mathematical model equations to allow for an analytical solution

electron and hole mobility is assumed constant and equal to the low

field value diffusivity is approximated by Einstein relation (D= VT) generation and recombination phenomena are treated within the

lifetime approximation dopant atom ionization is complete


Mathematical model: approximate equations

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Mathematical model: approximate equations

Simplified equations read

n

t= n

(nE)

x+ Dn

2n

x2n n0n

pt

= p(pE)x

+ Dp2p

x2 p p0

p

2

x2=

q

(p n+ ND NA)

where E = /x


Example: neutral region

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Example: neutral region

A semiconductor region is called quasi-neutral if = 0In many practical cases, quasi-neutrality is associated to E = 0,thus implying J

dr =0

The mathematical model reduces to

n

t= Dn

2n

x2n n0n

p

t = Dp2p

x2 p p

0p

An omogeneous sample in thermal equilibrium is neutral,

therefore

p0 n0 = NA ND

Out of equilibrium, quasi-neutrality implies

= p0 + p n0 n

+ ND NA = 0

therefore

n = p


Example: stationary regime

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Example: stationary regime

Stationary regime corresponds to a time-constant solution of themathematical model (/t 0)

0 = nd(nE)

dx+ Dn

d2n

dx2n n0n

0 = pd(pE)

dx+ Dp

d2p

dx2p p0p

d2

dx2

= q

(p n+ ND NA)


Contents

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1 Charge transport

Microscopic Ohms lawSemiconductor out of equilibrium

Diffusion current

Drift current

Injection level

Generation and recombination phenomena


Continuity equation

Poisson equation

Mathematical model



Quasi-Fermi levels: definition

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They are defined postulating Shockley equations out ofequilibrium:

n= ni expEFn EFi

kBT p= ni exp

EFi EFp

kBT

EFn is the electron quasi-Fermi level, EFp the hole one

In general, EF{n,p} = EF{n,p}(x, t), and EFi = EFi(x)


Quasi-Fermi levels: use

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Since the electrostatic potential is related to the electron potential

energy

EFi(x) = q(x) = E(x) =

x=

1

q

EFix

From the quasi-Fermi level definition

n

x=

n

kBT

EFnx

EFix

Because of Einstein relation D= VT = k

BT/q, diffusion current

reads

Jn,diff = qDnn

x= nn

EFnx

EFix



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Substituting into the drift-diffusion current expressions

Jn = qnnE + qDnn

x= nn

EFnx

Jp = qppE qDpp

x = ppEFpx

Multiplying the definitions

np=n2

iexpEFn EFpkBT



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In thermal equilibrium

EFn = EFp = EF

In presence of injection:

np> n2i = EFn> EFp

In presence of depletion:

np< n2i = EFn< EFp

Quasi-Fermi levels are functions of x admitting first derivative, andtherefore continuous. In fact:

Jn = nnEFnx

, Jp = ppEFpx

1D case


Quasi-Fermi levels: example

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In neutral regions with low injection,

majority carriers have their

equilibrium value

nn ND

pp NA

Majority carrier quasi-Fermi levels

coincide with the equilibrium value of

the Fermi level


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The distance between the quasi-Fermi levels at the borders of the

depletion region is

EFn(xn) EFp(xp) =

[EFn(xn) Ec(xn)]

kBT ln(Nc/ND)+ [Ec(xn) Ec(xp)]

q(Vbi V)+ [Ec(xp) Ev(xp)]

Eg

+Ev(xp) EFp(xp)

kBT ln(Nv/NA)

Taking into account

qVbi = Eg kBT ln(Nv/NA) kBT ln(Nc/ND)

We find

EFn(xn) EFp(xp) = qV



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The distance between the majority carrier quasi-Fermi levels in

the two neutral regions is equal to the the applied bias

The costancy of the majority carrier quasi-Fermi levels in the two

neutral sides corresponds to a negligible majority carrier current


02 model

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