The WKB Approximation
* General theory revisited
⇒
⇒⇒
⇒Classical turning points
* Determination of bound-state energies
⇒
⇒⇒
⇒Triangular well
* Tunneling
⇒
⇒⇒
⇒Schottky-barrier tunneling
⇒
⇒⇒
⇒MOS Capacitors Tunneling
⇒
⇒⇒
⇒Connection Formulas
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General Theory - Quantum Mechanics Engineers | EEE 434, Exams of Quantum Mechanics
Material Type: Exam; Class: Quantum Mechanics Engineers; Subject: Electrical Engineering; University: Arizona State University - Tempe; Term: Unknown 1989;
Typology: Exams
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The WKB Approximation
* General theory revisited
Classical turning points
* Determination of bound-state energies
Triangular well
* Tunneling
Schottky-barrier tunneling
MOS Capacitors Tunneling
Connection Formulas
•^
In many situations one is interested in solving the Schrödinger equation in situations where the potential energy is a
SLOWLY-VARYING
function of position
- In this situation conventional approaches such as the use of
T
-matrices
CANNOT
be
applied and the
WKB APPROXIMATION
is often used instead
- To develop this approach we begin by writing the Schrödinger equation as* Motivated by our knowledge of the form of the wavefunction for a particle that
moves in the presence of a
CONSTANT
potential we write the wavefunction as
General Theory
2 2
2
x E x x V x m
^
h
) (^2). (^22) (
) (^
) (^ x i e
x
χ
=
FOR
FREELY-MOVING
PARTICLES
WE REMEMBER THAT THE
WAVEFUNCTION VARIES AS
ψψψψ (x) = e
ikx
•^
The approximations of Eqs. 22.4 & 22.5 will be
VALID
provided
- This inequality requires that the change in
k
per wavelength of oscillation should be
much
LESS
than the wavelength itself
- To obtain the
WAVEFUNCTION
from these approximations we rewrite Eq. 22.3 as
Taking the square root and making a binomial expansion then yields
(^
2
2 '
''
k
dk dx
k
k
dk^ dx
χ
χ
(^
'
2
''
2
2
'^
x
ik
x k x i x k x
χ
χ
) (^8).
22 (
) ( 2
) ( ) ( ) (
) ( 1 ) ( ) (
'
' 2
'
x k
x
ik
x k
x
k
x
ik
x k
x
- ± ≈ + ± ≈ χ
General Theory
•^
This last expression can be integrated to yield
- Eq. 22.10 is the
WKB APPROXIMATION
and the prefactor of 1/
k
( x
) helps to
CONSERVE
probability current for propagating states
For
NEGATIVE
kinetic energies associated with
TUNNELING
Eq. 22.10 becomes
For the calculation of
BOUND
states in potential wells the complex exponential in
Eq. 22.10 is replaced by sine or cosine terms
) (^9).
(^22) (
) (
ln 2
) (
) (^
x k
i
dx x k
x
±
∫
∫^
) (^10). 22 (
)'
)' (
exp( ) 1 (
) (^
x^
dx x k i x k x
±
=
∴
∫^
exp(
(^
2
E
x
V
m
dx
x
x
k
x
x^
h
κ
κ
ψ
General Theory
•^
The treatment of the wavefunction in the region of the turning points is complicated and we simply reproduce the results here
- Consider a particle that approaches a barrier with its classical turning point located at
x
x
L ⇒⇒⇒⇒
The wavefunctions in the vicinity of this turning point are then given as
General Theory^ ∫
) (^13). (^22) (
,
4 '
)' (
cos ) 2 (
) (^
L
x x^
x
x
dx x k
x k
x
L
>
−
≈
∫ [^
]^
) (^14).
(^22) (
, '
)' (
exp ) 1 (
) (^
L
x x^
x
x
dx x
x
x
L^
<
−
≈
-^ A PARTICLE WITH ENERGY E IMPINGES ON A POTENTIAL BARRIER THAT
VARIES
WITH POSITION
-^ THE CLASSICAL
TURNING POINT
IS LOCATED AT
x
L
WHERE
V(x
) = EL
•^
The important features of Eq. 22.13 are the factor of
TWO
in the prefactor and the
PHASE SHIFT
- The factor of two accounts for the fact that as we can see below the oscillating wave
clearly has a
LARGER
amplitude than the maximum value of the exponential
- The factor of
/4 arises from the fact that the cosine term has to start with a
POSITIVE
slope at
x
L^
which means that its phase must lie somewhere between –
/2 and 0
In fact its value is seen from Eq. 22.13 to lie exactly in the
MIDDLE
of this range
General Theory
∫^
cos
(^
L
x x^
x
x
dx
x
k
x
k
x
L
^
π
ψ
•^
For a potential well with soft walls the quantization condition of Eq. 22.14 is
MODIFIED
according to
- The factor of –
/2 arises here due to the two phase changes of –
/4 at
x
L^
x
R
In the case where only
ONE
of the walls is soft and the other is infinitely steep the
factor of ½ is replaced by
in Eq. 22.
- As a
TEST
of the WKB method we use it to determine the energy levels in a
TRIANGULAR
potential well
We assume that the boundary located at
x
L^
= 0 is
INFINITELY
hard as in our
discussion of and so determine the bound state energies from
Determination of Bound-State Energies
) (^15). (^22) (
, 3 , 2 , 1
,
1 2
) (^
K
=
^
−
=
∫^
n
n
dx x RkL x x
π
(^
K
^
∫^
n
n
dx
x
Rk
x xL
•^
For
x
> 0 the potential energy varies as
eE
xs and the right-hand turning point is therefore
just
- Eq. 22.16 can therefore be rewritten as
Note here that we have introduced the
CHANGE OF VARIABLES
s
x
/ x
R
xeE
/ s
E
The integral in Eq. 22.16 is
EASILY
solved to yield
so that by setting Eq. 22.
to be equal to (
n
we obtain
Determination of Bound-State Energies
∫^
1 0
(^2) / 1 2
/^0
(^2) / 1
2
ds s
E eE
mE
dx
x eE E m
s
eE E
s
s
^
h
h
s
R^
E eE
x^
) (^19). (^22) (
2
)
(
1 4
3 2
(^3) / 1 2
(^3) / 2
^
−
=
m eE
n
E
s
n
h
π
REMEMBER THAT THESE ENERGIES
ARE ONLY
APPROXIMATE
!
•^
Another important
APPLICATION
of the WKB method is the determination of
TUNNELING
probabilities for barriers of
ARBITRARY
shape
- Since we have seen that the tunneling probability is proportional to the
SQUARE
of the
wavefunction we use Eq. 22.11 to write the transmission probability through a barrieras
We have assumed here that the edges of the barrier are located at
x
L^
and
x
R
This expression is
REMINISCENT
of that obtained previously for tunneling
through a
UNIFORM
barrier
As an example of the application of the WKB method to tunneling we consider theproblem of electron tunneling into the
SCHOTTKY BARRIER
between a metal and
a semiconductor
Tunneling
∫^
) (^21).
22 (
)
) (
2
exp(
xR L x^
dx x
T
κ
−
≈
W
e
k
T
κ
κ
2
2 2
~
−
•^
It is well known that when a metal and semiconductor are brought into contact the TRANSFER
of charge between them can result in the formation of a
TUNNEL
barrier
- The height of this barrier takes the
MAXIMAL
value
V
b^
at the interface but
DECREASES
as we move further into the semiconductor
Solving the Poisson equation with a
CONSTANT
interface charge density shows
that that the form of this potential variation is
PARABOLIC
in the position
Tunneling
) (^22). (^22) (
2
,) ) / ( (^1) (
) (^
2
2
D b o r
b^
N e
V d d x V x V ε ε
≡
−
=
-^
A SCHEMATIC ILLUSTRATION OF THE
SCHOTTKY
BARRIER
THAT FORMS AT THE SURFACE OF A METAL
-SEMICONDUCTOR JUNCTION •^ THE
TRANSFER
OF CHARGE BETWEEN THE TWO
MATERIALS IS THE ORIGIN OF THE BARRIER WHICH DECAYS
TO AN EFFECTIVE HEIGHT OF ZERO OVER A
DISTANCE d •^ FOR ELECTRONS TO TRANSFER FROM THE METAL TO THE SEMICONDUCTOR THEY MUST THEREFORE TUNNEL
THROUGH A BARRIER OF HEIGHT V
b^ AND
THICKNESS d
The
diffusion theory
assumes that the driving force is
distributed over the length of the depletion layer.
The
thermionic emission theory
on the other hand
postulates that only energetic carriers, those, whichhave an energy equal to or larger than the conductionband energy at the metal-semiconductor interface,contribute to the current flow.
Quantum-mechanical tunneling
through the barrier
takes into account the wave-nature of the electrons,allowing them to penetrate through thin barriers. In agiven junction, a combination of all three mechanismscould exist. However, typically one finds that only onecurrent mechanism dominates.
The analysis reveals that the diffusion and thermionicemission currents can be written in the following form:^ This expression states that the current is the product of theelectronic charge,
q
, a velocity,
v
, and the density of
available carriers in the semiconductor located next to theinterface.The velocity equals the mobility multiplied with the field atthe interface for the diffusion current and the Richardsonvelocity for the thermionic emission current.
Gate Leakage
gate leakage
tunnelling current