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3.4.4. Derivation of the Metal-Semiconductor Junction Current
3.4.4.1.Derivation of the diffusion current
We start from the expression for the total current and then integrate it over the width of the
depletion region:
J n = q( µ n n E + Dn
(3.4.11)
dn
)
dx
which can be rewritten by using E = -dφ/dx and multiplying both sides of the equation with exp(φ/Vt), yielding:
J n exp( −
φ
n dφ dn
φ
d 
φ 
) = qDn ( −
+ ) exp( − ) = qDn
n exp( − ) 

Vt
Vt dx dx
Vt
dx 
Vt 
(3.4.12)
Integration of both sides of the equation over the depletion region yields:
xd
Jn =
φ
qDn n exp( − )
Vt 0
φ
∫0 exp(− Vt ) dx
xd
=

φ 
V
qDn N c exp( − B ) exp( a ) − 1
Vt 
Vt

(3.4.13)
φ*
∫0 exp(− Vt ) dx
xd
Where the following values were used for the electron density and the potential:
x
n(x)
0
Nc exp(-φ B/Vt)
xd
Nd = Nc exp(-φ B/Vt) exp(φ i/Vt)
φ (x)
-φ i
+ Va
0
and φ * = φ + φ i - Va . The integral in the denominator can be solved using the potential obtained
from the full depletion approximation solution, or:
qN d
( x − xd ) 2
2ε s
(3.4.14)
qN d
qN d
x
2x
x ( xd − ) ≅
x xd = (φ i − Va )
εs
2
εs
xd
(3.4.15)
φ =−
so that φ * can be written as:
φ* =
where the second term is dropped since the linear term is dominant if x << x d . Using this
approximation one can solve the integral as:
xd
∫ exp( −
0
φ*
Vt
) dx ≅ xd
Vt
2(φ i − Va )
(3.4.16)
for (φ i – Va ) > Vt. This yields the final expression for the current due to diffusion:
Jn =
qDn N c
Vt
2q (φ i − Va ) N d
φ
V
exp( − B )[exp( a ) − 1]
εs
Vt
Vt
(3.4.17)
This expression indicates that the current depends exponentially on the applied voltage, Va , and
the barrier height, φ B. The prefactor can be understood physically if one rewrites that term as a
function of the electric field at the metal-semiconductor interface, Emax:
E max =
2q (φ i − Va ) N d
εs
(3.4.18)
yielding:
J n = qµ n Emax N c exp( −
V
φB
)[exp( a ) − 1]
Vt
Vt
(3.4.19)
so that the prefactor equals the drift current at the metal-semiconductor interface, which for zero
applied voltage exactly balances the diffusion current.
3.4.4.2.Derivation of the thermionic emission current
The thermionic emission theory2 assumes that electrons, which have an energy larger than the
top of the barrier will cross the barrier, provided they move towards the barrier. The actual
shape of the barrier is hereby ignored. The current can be expressed as:
J right− left =
∫
Ec ( x =∞) + qφ n
qv x
dn
dE
dE
(3.4.20)
For non-degenerately doped material, the density of electrons between E and E + dE is given by:
(using (2.4.7) and assuming EF,n < Ec - 3kT)
E − E F ,n
dn
4π ( 2m * ) 3 / 2
= g c ( E) F ( E ) =
E − Ec exp[−(
)]
3
dE
kT
h
(3.4.21)
Assuming a parabolic conduction band (with constant effective mass m* ), the carrier energy, E,
can be related to its velocity, v, by:
2 see also S.M. Sze "Physics of Semiconductor Devices", Wiley and Sons, second edition, p. 255
E − Ec =
m *v 2
, dE = m* vdv , and
2
E − Ec = v
m*
2
(3.4.22)
Combining (3.4.21) with (3.4.22) yields:
Ec ( x = ∞) − E F , n
dn
m*
m*v 2
dE = 2( ) 3 exp[ −
] exp[−
]4πv 2 dv
dE
h
kT
2kT
(3.4.23)
when replacing v 2 by v x2 + v y2 + v z2 and 4π v 2 dv by dv xdv ydv z the current becomes:
J r →l
∞
m*v 2y
m* 3 ∞
m*v z2
= 2( ) ∫ exp[ −
]dv y ∫ exp[ −
]dv z
h −∞
2 kT
2kT
−∞
−v0 x
∫
qv x exp[ −
−∞
= 2 q(
(3.4.24)
m*v 2x
E ( x = 0) − E F .n
]dv x exp[ − c
]
2 kT
kT
m * 3 2πkT
Ec ( x = 0) − EF . n
m*v 02x kT
)
exp[
−
]
exp[
−
]
h
m*
kT
2 kT m*
using
∞
m *v 2y
∞
m*v z2
∫ exp[ − 2kT ]dv y = −∫∞exp[− 2kT ]dv z =
−∞
2πkT
m*
(3.4.25)
The velocity v ox is obtained by setting the kinetic energy equal to the potential across the n-type
region:
m*v ox2
= qφn
2
(3.4.26)
so that v ox is the minimal velocity of an electron in the quasi-neutral n-type region, needed to
cross the barrier. Using
1
φ n = φi − Va = φ B − [ Ec ( x = ∞) − E F , n ] − Va
q
(3.4.27)
which is valid for a metal-semiconductor junction, one obtains:
J MS = A*T 2 exp( −
φB
V
)[exp( a ) − 1]
Vt
Vt
(3.4.28)
4π qm* k 2
where A =
is the Richardson constant and φ B is the Schottky barrier height which
h3
equals the difference between the Fermi level in the metal, EF,M and the conduction band edge,
Ec, evaluated at the interface between the metal and the semiconductor. The -1 term is added to
*
account for the current flowing from right to left3 . The current flow from right to left is
independent of the applied voltage since the barrier is independent of the band bending4 in the
semiconductor and equal to φ B. Therefore it can be evaluated at any voltage. For Va = 0 the total
current must be zero, yielding the -1 term.
The expression for the current due to thermionic emission can also be written as a function of the
average velocity with which the electrons at the interface approach the barrier. This velocity is
referred to as the Richardson velocity given by:
vR =
kT
2π m
(3.4.29)
V
φB
)[exp( a ) − 1]
Vt
Vt
(3.4.30)
So that the current density becomes:
J n = qvR N c exp( −
3.4.4.3.Derivation of the tunneling current
To derive the tunnel current, we start from the time independent Schrödinger equation:
−
h 2 d 2Ψ
+ V ( x ) Ψ = EΨ
2 m* dx 2
(3.4.31)
which can be rewritten as
d 2 Ψ 2 m* (V − E )
=
Ψ
dx 2
h2
(3.4.32)
Assuming that V(x) - E is independent of position in a section between x and x+dx this equation
can be solved yielding:
Ψ ( x + dx ) = Ψ ( x ) exp( − k dx) with k =
2m*[V ( x ) − E ]
h
(3.4.33)
The minus sign is chosen since we assume that the particle moves from left to right. For a
slowly varying potential the amplitude of the wave function at x = L can be related to the wave
function at x = 0 :
 L 2m*[V ( x) − E ] 
Ψ ( L ) = Ψ( 0) exp  − ∫
dx 
 0

h


3 This method assumes that the effective mass of the carriers is the same on both sides of the barrier.
4 ignoring the Schottky barrier lowering due to image charges
(3.4.34)
This equation is referred to as the WKB approximation5 . From this the tunneling probability, Θ,
x
can be calculated for a triangular barrier for which V(x)-E = qφ B (1- )
L
Θ=
 L 2 m*
Ψ ( L) Ψ * ( L)
− 2
=
exp
 ∫0 h
Ψ (0) Ψ * (0)


x
qφ B (1 − ) dx 

L

(3.4.35)
The tunneling probability then becomes6 :
 4 2 qm* φ 3 / 2 
B

Θ = exp  −
 3 h

E


(3.4.36)
where the electric field equals E = φ B/L.
The tunneling current is obtained from the product of the carrier charge, velocity and density.
The velocity equals the Richardson velocity, the velocity with which on average the carriers
approach the barrier while the carrier density equals the density of available electrons multiplied
with the tunneling probability, yielding:
J n = q vR n Θ
(3.4.37)
The tunneling current therefore depends exponentially on the barrier height to the 3/2 power.
5 Named after Wigner, Kramers and Brillouin
L
6 Using
∫
0
1−
x
2L
dx =
L
3