ENE 311 Lecture 2. Diffusion Process The drift current is the transport of carriers when an electric...
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Transcript of ENE 311 Lecture 2. Diffusion Process The drift current is the transport of carriers when an electric...
ENE 311 Lecture 2
Diffusion Process
• The drift current is the transport of carriers when an electric field is applied.
• There is another important carrier current called “diffusion current”.
• This diffusion current happens as there is a variation of carrier concentration in the material.
• The carriers will move from a region of high concentration to a region of low concentration.
• This kind of movement is called “diffusion process”.
Diffusion Process
• An electron density varies in the x-direction under uniform temperature.
• The average thermal energy of electrons will not vary with x, but the electron density n(x)
• The electrons have an average thermal velocity vth and a mean free path l.
Diffusion Process• The electron at x = -l
have an equal chances of moving left or right.
• the average of electron flow per unit area F1 of electron crossing plane x = 0 from the left is expressed as
1
0.5 ( )0.5 ( ) th
c
n l lF n l v
Diffusion Process
• Likewise, the average of electron flow per unit area F2 of electron at x = l crossing plan at x = 0 from the right is
• Then, the net rate of electron flow from left to right is given by
2 0.5 ( ) thF n l v
1 2 0.5 [ ( ) ( )]
= 0.5 (0) (0)
=
th
th
th
F F F v n l n l
dn dnv n l n l
dx dx
dnv l
dx
Diffusion Process• Therefore, the net rate F can be written as
where Dn = vth.l = the diffusion coefficient (diffusivity)
• A current density caused by the electron diffusion is given by
(1)
n
dnF D
dx
e n
dnJ eF eD
dx
Diffusion Process
• Ex. An n-type semiconductor at T = 300 K, the electron concentration varies linearly from 1 x 108 to 7 x 107 cm-3 over a distance of 0.1 cm. Calculate the diffusion current density if the electron diffusion coefficient Dn is 22.5 cm2/s.
,
8 1719
2
1 10 7 10(1.6 10 )(22.5)
0.1
10.8 A/cm
e diff n n
dn nJ eD eD
dx x
Soln
Diffusion Process
Einstein Relation• From (1) and the equipartition of energy for one-
dimensional case
we can write
21 1
2 2thmv kT
2
n th
th th c
e eth
e e
e
D v l
v v
mv
e
kT m
m e
Einstein Relation
• Therefore,
• This is called “Einstein relation” since it relates the two constants that describe diffusion and drift transport of carriers, diffusivity and mobility, respectively.
• This Einstein relation can be used with holes as well.
n e
kTD
e
Value of diffusivities
Diffusion Process
Ex. Minority carriers (holes) are injected into a homogeneous n-type semiconductor sample at one point. An electric field of 50 V/cm is applied across the sample, and the field moves these minority carriers a distance of 1 cm in 100 μs. Find the drift velocity and the diffusivity of the minority carriers.
Diffusion Process
46
42
2
1 cm10 cm/s
100 10
10200 cm / V-s
50
0.0259 200 5.18 cm /s
h
hh
p h
vs
v
EkT
De
Soln
Diffusion Process
• For conclusion, when an electric field is applied in addition to a concentration gradient, both drift current and diffusion current will flow.
• The total current density due to electron movement can be written as
where n = electron density
e e n
dnJ e nE D
dx
Diffusion Process
The total conduction current density is given by
where Jh is the hole current =
p = hole density
total e hJ J J
h h h
dpJ e pE eD
dx
Diffusion Process• At a very high electric field, the drift velocity will sat
urated where it approaches the thermal velocity.
Electron as a wave
L. de Broglie said electrons of momentum p exhibits wavelike properties that are characterized by wavelength λ.
where h = Planck’s constant = 6.62 x 10-34 J.s
m = electron mass
v = speed of electron
h h
p mv
Electron as a wave
Ex. An electron is accelerated to a kinetic energy of 10 eV. Find
• velocity v• wavelength λ• accelerating voltage
Electron as a waveSoln
(a)
This implies that electron is not inside a solid since
v > vth (~105 m/s)
19
2
19
31
6
1 eV 1.6 10 J
110 eV
2
2 10 1.6 10
9.1 10
1.88 10 m/s
mv
v
v
Electron as a wave
Soln
(b)
34
31 6
10
6.62 10
9.1 10 1.88 10
3.87 10 m
h
mv
Electron as a wave
Soln
(c)
10 eV means an electron is accelerated by a voltage of 10 volts.
Therefore, accelerating voltage is 10 V.
Schrödinger’s equation• Kinetic energy + Potential energy = Total energy
• Schrodinger’s equation describes a wave equation in 3 dimensions is written as
where ħ = h/2
= Laplacian operator in rectangular coordinate
=
= wave function
V = potential term
22 ( , ) ( ) ( , ) ( , )
2r t V r r t i r t
m t
22 2 2
2 2 2x y z
Schrödinger’s equation
To solve the equation, we assume (r,t) = (r).(t) and substitute it in Schrödinger’s equation. We have
Divide both sides by (r).(t), we have
22 ( )
( ) ( ) ( ) ( ) ( ) ( )2
d tt r V r r t i r
m dt
2 2 ( ) 1 ( )( )
2 ( ) ( )
r d tV r i
m r t dt
Schrödinger’s equation
Both can be equal only if they are separately equal to a constant E as
• Time dependent case:
• Position dependent: (Time independent)
( )( )
d ti E t
dt
22 ( ) ( ) ( ) ( )
2r V r r E r
m
Schrödinger’s equation
Consider time dependent part, we can solve the equation as
where E = h = h/2 = ħ
Therefore
( ) expiE
t t
( ) expt i t
Schrödinger’s equation
For position dependent, to make it simple, assume that electrons can move in only one dimension (x-direction).
(2)
where is probability of findng electron.
2 2
2
( )( ) ( ) ( )
2
d xV x x E x
m dx
2
2 2
( ) 2( ) ( ) 0
d x mE V x x
dx
2( )x
Schrödinger’s equation
Now let us consider the 4 different cases for V(x)
Case 1: The electron as a free particle (V = 0)Equation (2) becomes
General solution of this equation is
2
2 2
( ) 2( ) 0
d x mE x
dx
( ) ikx ikxx Ae Be
Schrödinger’s equation
From (r,t) = (r).(t), the general solution of Schrodinger’s equation in this case is
where A and B are amplitudes of forward and backward propagating waves and k is related to E by
2 221
2 2
kE mv
m
tiikxikx eBAetx ),(
Schrödinger’s equation
This equation is called “de Broglie’s relation”.
2 2 21
2 2
k p
m m
p k
2 221
2 2
kE mv
m
Schrödinger’s equation
Case 2: Step potential barrier (1 dimensional)
x
Schrödinger’s equation
For region 1, the solution is already known as
(3)
For region 2, an equation (2) becomes
1 11( ) ik x ik xx Ae Be
21 2
2mEk
2
22 22 2
( ) 2( ) 0
d x mE V x
dx
Schrödinger’s equation
General solution:
(4)
Since there is no wave incident from region 2, D must be zero.
2 22 ( ) ik x ik xx Ce De
2 22 2
2 ( )m E Vk
Schrödinger’s equation
By applying a boundary condition, solutions must be continuous at x = 0 with
From (3) and (4), we have
1 2
1 2
(0) (0)
(0) )
(0d d
dx dx
1 2( )
A B C
ik A B ik C
1 2
1 2
1
1 2
2
B k k
A k k
C k
A k k
Schrödinger’s equation
Now let us consider 2 cases:
1. E > V2: In this case k2 is real and k2 and k1 are different leading B/A finite. This means an electron could be seen in both region 1 and 2. It goes over the barrier and solution is oscillatory in space in both regions.
2. E < V2: In this case k2 is imaginary. The solution shows that an electron decays exponentially in region 2. An electron may penetrate the potential barrier.
Schrödinger’s equation
Case 3: Finite width potential barrier (1 dimensional)
Schrödinger’s equation
We are interested in the case of E < V2. In this case, solutions for region 1 and region 2 are the same as previous case. Now we turn our interest to region 3. At region 3,
33( ) ik xx Fe
23 2
2mEk
Schrödinger’s equation
Consider transmittance, T is a ratio of energy in transmitted wave in region 3 to energy in incident wave in region 1.
This is called “Tunneling probability”.
22
2
22
2exp
1region in aveincident win Energy
3region in wavettedin transmiEnergy
EVm
wA
FT
T
Schrödinger’s equation
• k3 is real, so that is not zero.
• Thus, there is a probability that electron crosses the barrier and appear at other side.
• Since the particle does not go over the barrier due to E < V2, this mechanism that particle penetrates the barrier is called “tunneling”.
2
3
Schrödinger’s equation
Case 4: Finite potential well (1 dimensional)
x
Schrödinger’s equation
Consider 2 cases
1. E > V1: Solutions in all regions are similar to those previous cases that is particle travels oscillatory everywhere.
2. E < V1: Region 1 ( x < 0, V = V1) ; Solution must be exponentially decaying
(5)
11( ) k xx Ae
21 2
2 ( )m V Ek
Schrödinger’s equation
Region 2 (V = 0); Free particle case
(6)
2 2 2( ) sin( ) cos( )x B k x C k x
22 2
2mEk
Schrödinger’s equation
Region 3 (x > 0, V3 = V1); Solution is again decaying.
13( ) k xx De
2 23 12
2 ( )m V Ek k
Schrödinger’s equation
Applying boundary conditions:
at x = -L/2 at x = L/2
1 2
1 2
( ) ( )
( ) ( )
x x
d x d x
dx dx
2 3
2 3
( ) ( )
( ) ( )
x x
d x d x
dx dx
Schrödinger’s equation
It is enough to solve only at x = L/2 due to its symmetrical.
(7)
By solving (5), this leads to
(8)
3
3
2 2
2 23
cos2
sin2
k L
k L
k LC De
k LCk Dk e
22 3tan
2
k Lk k
Schrödinger’s equation
Substitute (8) into (5) and (6), we have
2
12tan
2
mL EE V E
Schrödinger’s equation
If we consider in the case of V or infinite potential well
Schrödinger’s equation
There is impossible that a particle can penetrate an infinite barrier. This brings (x) = 0 at x =0 and x = L. Applying the boundary conditions, we first have B = 0 and
(9)
We can solve for energy E by putting (3) = (6) and we get
2 ; where n = integerk L n
2
nk
L
2 2
28
n hE
mL
The uncertainty relationship
If we know the and use
where <A> is momentum or position, we can find the average and RMS values for both electron’s position and momentum.
*
2V
d
V
A dVA
dV
The uncertainty relationship
where x = uncertainty in position ofelectrons p = uncertainty in momentum of electrons
The uncertainty relationship also includes
x p h
E t h
Ex . Find the allowed energy levels for an elect -ron that is trapped in the infinite one dimen
sional potential well as in the figure below.
V(x)
-L/2 0 L/2
x