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TRANSFER MATRIX 3

(2)

(3)

TRANSFER MATRIX

V

x1 x2 x3 X 圖(1) 任何形式的一維位能障

where

。 ，選擇 為方程式(A2)的一個解

12 1

)

0

" + ψ =

ψ k V

) ( ) ( ) ( )

2 (

1 1

2 1 2

1 m E V x V x V x V x

k =

= h

) 2 (

2

x mV

 h

0 ) ( 1

1 x =

V

) (x1 U E>

(A1)

(A2)

0 )

1(x

V ψ1

1 1' 1

1

ψ

(A3)

) ( )

( 1 1

1

1 x B x

A ψ ψ

ψ = +

) ( )

( 2 2

2

2 x B x

A ψ ψ

ψ = +

=

=

22 21

12 11 2

2 1

1

θ θ

θ θ θ

θ B A B

A

(A4)

(A5)

(A6)

(4)

The transfer matrix from x1 to x2 is θl

It is easily related to θ：

By equation (A12)，the transmission coefficient of the current is the same from the left to the right and versa.

ψ ψ

2 2 2

2ψ Aψ

B + B1ψ1+A1ψ1



 

= 



 

2 2

1 1

A B A

B θ (A7)



 

= 

11 12

12 11

θ θ

θ

θ θ (A8)

=

~) exp(

) cosh(

~) exp(

) sinh(

~) exp(

) sinh(

) exp(

) cosh(

2 1

2 1

α β

β θ α

i S

i S

i S

i S

k

k (A9)

=

) cos(

) ) exp(

tan(

~ ) exp(

) tan(

~ ) ) exp(

cosh(

) exp(

2 1

1 2

h h i

i

h h i

i

k k

β α α β

θ (A10)



 

= 



 

1 1

2 2

A B A

B

θl (A11)

=

11 12

12 11

2 1

θ θ

θ θ θ

k k

l (A12)

(5)

If in some vicinity of x1 , then , By equation (A1) and (A3), is a plain wave. There

Elsewhere the difference between and this plain wave is, in virtue of equation (A13), of order of .Generally the best choice for x1,x2 are the points where V(x) has local minimum. In the vicinity of x1

If at x>x2 there is no barrier and thus there is only a transmitted wave B2=0,then

The current transmission t and reflection r coefficients equal

If a barrier is symmetric (and, in particular, ),then the Schrödinger equation allows for odd and even (with respect to the middle of the barrier) solutions. This leads to

in equation (A9). Thus characterizes the asymmetry of the barrier. To elucidate the physical meaning of the quantities S,α,β in equation (A9), consider several special cases.

In a quasiclassical case, the comparison of equation (A16a) to the formulas 31(a1) for τ,ñ leads to

∫

+ +

= ξ ξ ξ ξ ξ ψ ξ ξ

ξ

ψ 0 1

2 1 1

1

1( ) 1 ik ( )[V ( ) k ] ( )d (A13)

0 ) 0

1 (

1 =

= x x V

ξ (A14)

1 ≡0

V ψ1

)]

( exp[

)

( 1

1 x = ik x x

ψ (A15a)

)

1( x ψ

ξ ξ ξ

ξ 0V1( )d

4 1 1

1

1(x) exp[ ik (x x )] ~ (x x )

ψ (A15b)

11 21 1

1 11

1

2 1

θ ρ θ

τ = θ =

A B A

A (A16a)

1 2 2 2

1 1

2 2 2

2 2

1 2 1

k k A

k A t k

A r B

τ ρ

=

=

=

=

(A16b)

k k k1= 2

2

~ π

β = (A17)

β π β~

2

(6)

Thus,žS is the absolute value of the classically unavailable phase area (action).

The straightforward calculations in the case of a rectangular barrier, V(x)=0 at x<x1 , at x1<x<x2 and at x>x2 lead to θ with

A δ-function potential energy,

allows us to choose x1 = x0 - 0 , x2 = x0 + 0 , and leads to

2 1

2 1

)]

( 2 )

[(

) (

) ( )

2 (

) ( )

(

) (

1

1

2

1 1

1

2

1 1

1

x V m E

x k

dx x k dx

x k

dx x k dx

x k

dx x k S

S

x x

x x x

x

x x x

x

=

− +

+

=

∫

′′

′′

h β π

α

(A18)

(A19)

(A20)

m x k

V 2

) ) (

(

2 2 2

1 +κ h

V(x)(k22 +κ2)

d k

x x d k

k k k

k k

k d i k

k d d k

k i i

S

k d i k

k d d k

ik i

S

0

1 2 1

2 2

1 0

2 1 1

2

2 1 1

2 0

~

2

) sinh(

)]

( ) [(

) cosh(

) 1 ( 2 )

exp( 1 2 ) 1 exp( ~ ) sinh(

) sinh(

) ( ) [(

) cosh(

) 1 ( ) 2exp(

) 1 exp( ~ ) cosh(

+

=

=

=

= +

+

=

+

=

α α

κ κ κ κ

β

κ κ κ κ

α (A21)

(A22)

(A23)

(A24)

) ( )

(x v x x0

V = δ

2

~ 1 2

) exp(

)

cosh( α = β =π

k i iv

S

(A25)

(A26)

(7)

= 11

22 M

M M21=M12

k M k0 det =

計算穿隧機率的方法

V(z)

0 a a+b/2 a+b L z

double-barrier quantum well 的 potential 示意圖 z < 0

z > L

) ( )

) (

( )

( 0 0

L z ik L

z ik

z ik z

ik

e B e

B z

e A e

A z

+

+

+

= Ψ

+

= Ψ





=





+

+

A A M B

B









=

0 0

1 ) 1

0 , ( 1

1 2 1

ik ik

L S k

i k

i M

k M

k k

A

k T B

2 22

0

0 2

2

=

=

+ +

) (

)

( 11 0 22 21 0 12

22 k S

k i S k

S S k

M = + +

21 2 12

0 12 2

0 11

0

) (

) (

4

k S S

k k

S S k

k k T

+

+

=

(8)

In region

In region

In region

Et：transverse energy in a GaAs layer

=

=

=

a k a

k k

a k k

a k S

E E z

eV k

L V z V

z V

p p

p

p p

p

t p

a

cosh sinh

sinh cosh

) ( )

(

2

0

γ

γ γη γ









= −

=



 

 +  −



 

 − +

=

cos 2 sin 2

sin 2 cos 2

) ( 0 2

) (

1 1

1

1 1

1 1

2 1

b b

b b

S

E z

eV E

L z V z z

V

p p

p

p p

p

t p

a

κ κ

κ

κ κ

κ η

κ

ε σ

{ }









=

=

 +

 

 − +

=

cos 2 sin 2

sin 2 cos 2

) ( 0 2

) (

2 2

2

2 1

2 2

2 2

b b

b b

S

E z

eV E

L z V z z

V

p p

p

p p

p

t p

a

κ κ

κ

κ κ

κ η

κ

ε σ

(9)

m*：effective mass S = SSSS where

h

GaAs GaAs

m m

z z m

*

*

*

2

) (

=

= η γ

2

* 2

2

* 2

0

2 2

h h

eV a

E k m

E k m

= +

=

(10)

結 果 與 討 論

number of barrier : 2 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-20 -15 -10 -5 0 5

lnT

Energy(eV)

(11)

number of barrier : 3 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-30 -25 -20 -15 -10 -5 0 5

lnT

Energy(eV)

(12)

number of barrier : 4 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-40 -30 -20 -10 0

lnT

Energy(eV)

(13)

number of barrier : 2 well width : 50Å

barrier width : 20Å barrier height : 0.5eV bias voltage : 0V

doped consistence : 0 #/cm2

0.0 0.1 0.2 0.3 0.4 0.5

-14 -12 -10 -8 -6 -4 -2 0 2

lnT

Energy(eV)

(14)

number of barrier : 3 well width : 50Å

barrier width : 20Å barrier height : 0.5eV bias voltage : 0V

doped consistence : 0 #/cm2

0.0 0.1 0.2 0.3 0.4 0.5

-20 -15 -10 -5 0 5

lnT

Energy(eV)

(15)

number of barrier : 2 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0.02V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-15 -10 -5 0

lnT

Energy(eV)

(16)

number of barrier : 2 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0.04V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-14 -12 -10 -8 -6 -4 -2 0 2

lnT

Energy(eV)

(17)

number of barrier : 2 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0.06V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25

-12 -10 -8 -6 -4 -2 0 2

lnT

Energy(eV)

(18)

number of barrier : 2 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0.08V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25

-14 -12 -10 -8 -6 -4 -2 0

lnT

Energy(eV)

(19)

number of barrier : 2 well width : 100Å barrier width : 40Å barrier height : 0.3eV bias voltage : 0.10V

doped consistence : 0 #/cm2

0.00 0.05 0.10 0.15 0.20 0.25

-14 -12 -10 -8 -6 -4 -2 0

lnT

Energy(eV)

(20)

= 300meV、Nd = 0cm-2(量子井中心摻雜正離子的濃度)、bias voltage = 0.00V(black)、0.02V(read)、0.04V(green)、0.06V(blue)、0.08V(gyan)、

0.10V(magenta)的條件下，穿隧機率跟入射粒子能量的關係圖。從圖 中可以發現 bias voltage 越大 peak 數目會減少且向左移動。

0.0 0.1 0.2 0.3

- 2 0 - 1 5 - 1 0 -5 0 5

lnT

Energy(eV)

(21)

0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0

- 1 5 - 1 0 -5 0

lnT

Energy(eV)

(22)

參 考 資 料

1. R. Tsu and L. Eskai, Appl. Phys. Lett. 22, 562 (1973)

2. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958), p. 63.

3. B. Jogai and K. L. Wang, Appl. Phys. Lett. 46 (2) (1985) 4. M. Ya. Azbel*, Physical Review B, 22, 8 (1983)

5. M. O. Vassel, Johnson Lee, and H. F. Lockwood, J. Appl. Phys. 54, 5026 (1983)

6. E. O. Kane, in Tunneling Phenomena in Solids, edited by E. Burstein and S. Lundqvist (Plenum, New York, 1969), Chap. 1.

(Ⅱ) specific binding theory is a more accurate explanation of the mechanism of action of local anaesthetics. The evidence to support this theory is strong. Different isomers of

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