Deduction Of Initial State Wavefunction By Using Green Function Method And Its Application To Photoemission

ISSN 2348 – 7968

697

Deduction of Initial State Wave function by Using Green Function

Method and Its Application to Photoemission

Lalthakimi Zadeng

Department of Physics, Mizoram University, Aizawl, Mizoram, India e-mail: kimizadeng @ gmail.com

Abstract: Initial state wavefunction is deduced by

solving one dimensional Schrödinger equation using

Green function. Kronig Penny model is used to define

the crystal potential. For the photoemission

calculations, a spatially dependent vector potential is

used. For the surface state photoemission calculation, a

properly normalized Gaussian form of the wave

function replaces the initial state wavefunction. This

model is applied to the case of W and PbS.

Keywords : Green function, Schrodinger equation, Kronig Penny model, Photoemission, photocurrent

1. Introduction

In photoemission, the evaluation of the matrix

element ψ_f H′ψ_i is of prime importance as it is

directly involved in the photocurrent density formula as

given by Fermi golden rule [1]. The calculation of the

initial state wavefunction

ψ

i is a complicated problem

and is of fundamental importance.

In this report initial state wavefunction

ψ

i in

the band states is calculated by Green function method

by solving Schrödinger equation [2]. The photocurrent

is then calculated by incorporating the spatially

dependent vector potential which is a modified form of

the dielectric model as used by Bagchi and Kar [3].

Kronig-Penny model is used to define the crystal

potential. For the surface state photoemission, the initial

state wavefunction

ψ

_i is described by the normalized

Gaussian state wavefunction whose location with

respect to the nominal surface plane is determined by

zRoR.

2. Formalism

The photocurrent density formula from golden

rule [1] approximation can be written as

∑< ′ > − − − − −

=

Ω | | | |2 ( ) ( ) ( )[1 ( )]

2 ) (

E o f E o f i E f E f E E i H f d

E

dj π _{ψ ψ δ δ ω ω}

  

(1)

where i ψ (

f

ψ ) refer to the initial (final) state

wavefunction and the perturbation

H

′

can be written as

(

A⋅p+p⋅A

)

      = ′

mc e H

2

(2)

where m is the mass of the electron, p the one-electron

momentum operator and A the vector potential of the

incident photon field.

The photon field vector formulae used in our

calculations is obtained by using the dielectric model of

Bagchi and Kar [3]. With simple modifications, we can

698

        ≥ ≤ ≤ − + − − ≤ = ) ( 0 , ) ( 1 ) ( 0 , )] ( 1 [ ) ( 1 ) ( , 1 ) ( ~ Vacuum z A Surface z a a z a A Bulk a z A z A ω ε ω ε ω ε ω (3) where i i i A θ ω ε θ ω ε θ cos ) ( 2 1 ] 2 sin ) ( [ 2 sin 1 + − −

= is a constant

depending on the dielectric function

ε

(

ω

)

, photon

energy



ω

and angle of incidence θ_i.

3. Deduction of initial state wavefunction

One dimensional Schrödinger equation is given by

) ( ) ( ) ( 2

2 _{z V z z}

i

k ψ = ψ

+ ∇ _ 



 (4)

The Green function of a free particle is defined as a

function G(z,y) which satisfies theequation for a point

source, i.e

(

∇2+k_i2

)

G(z,y)=δ(z−y) (5)

The Green function G(z,y)can be obtained from the

above equation as

i k y z i ik ie y z G 2 ) ( ) , ( − − = (6)

Solving Eqs. (4) and (5) we get

∫

∞

∞ −

= V y y G z y dy

z) ( ) ( ) ( , )

( ψ

ψ (7)

The crystal potential is defined by Kronig-Penny

δ-potential given by

∑ ∞

−∞

= −

=

n y nd

d p y

V( ) 2 δ( ) (8)

where p is the strength of the potential, d is the lattice

constant.

Hence putting the values of V(y) and G(z,y) from

above,

ψ

(

z

)

can be evaluated.

The initial state wavefunction

ψ

_i

(

z

)

can be written as

   

>

− + <

= ) ( 0 ) & ( 0 ) ( * ) ( ) ( vacuum z z Te bulk surface z z R z z i χ ψ ψ ψ (9)

The reflection coefficient R and the transmission

coefficient T are evaluated by matching the

wavefunctions and its derivative at z = 0, and are given

by         − − − + − = a i k ika e i k a i k i k a i k a i k i k ika e i k R sin cos sin cos ) 0 ( ) 0 ( * χ χ φ φ a i k ika e i k a i k i k ka i ik T sin cos sin 2 ) 0 ( χ φ − − − = where

(

ka k_ia

)

a ik e ika e a i k a i k Cp i cos cos 1 2 sin 2 ) 0 ( −       − − = φ

and C is a

normalization constant given by 12

2         = i k e m C π , and a k=±π.

The final form of ψi(z)is

(

)

(

)

{

}

(

)

(

)

            > − +         − + − + − < − − +         − + − + − = 0 cos 1 1 sin cos 2 sin 4 0 sin cos cos 1 1 sin cos 2 sin 4 ) ( z z e i k a i k a i ik e a i k i k a i k i k a i k a i k pC z z e z i k z i k i k a i k a i ik e a i k i k a i k i k a i k a i k pC z i χ χ π µ χ χ π ψ (10) where 2 2 2  i mE i

k = , ( )

2 2 2 i E o V m − = 

χ , µ=k_i−π d.

In Eq.(10), for the surface state photoemission

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699

normalized Gaussian form of the wavefunction located

at z = zRoR plane and is given by

(

₂

)

1 4 _{2 2}

( ) 2 exp ( )

i z β πa  β z zo a 

Ψ = _− − _ (11)

where β describes the width of the Gaussian and is a

dimensionless quantity, a is the surface width.

The final state wavefunction

ψ

f is the

scattering state due to the existence of a step potential at

the surface plane z = 0. This potential is defined by

) ( )

(z V z

V =− oθ , where

θ

(z) is a unit function.

Therefore

ψ

_f is given by

           >         − + − +           < − +           = ) ( 0 , 2 1 2 2 ) & ( 0 , 2 2 1 2 2 ) ( vacuum z z f ik e f k f q f q f k z f ik e f k m surface bulk z z e z f ik e f k f q f k f k m z f   π α π ψ (12) where 2 2 2  f mE f

k = and ( ₀)

2 2 2 _V f E m

q = −



and

ω



+ =E_i f

E . The factor

z

e−α is included on the bulk

to take into account the inelastic scattering of the

electrons. The photocurrent was calculated by

evaluating the matrix element in Eq.(1) and by using the

above wavefunctions in Eqs. (10), (11) and (12). The

matrix element when expanded is given by

( )

( )

i

f A z

dz d dz d z A

I ψ _ω ~_ω ψ

2 1 ~ ₊ =

∫

∫

∫

∫

∞ − − − ∞ − + + + = 0 * 0 * 0 * * ~ ~ 2 1 ~ ~ dz dz d A dz dz A d dz dz d A dz A i f a i f a i f a i f ψ ψ ψ ψ ψ ψ ψ ψ ω ω ω ω

Some of the matrix element could not be solved

analytically, hence FORTRAN programs were written

to solve it.

4. Results and Discussions:

We have applied the model developed to

calculate photocurrent from W and PbS.

In Fig. (1) a plot of photocurrent as a function

of incident photon energy (



ω

) for W for three

locations of the initial state wavefunction

ψ

_i denoted

by

z

=

z

o is shown. The Fermi level of W is ERFR=

10.25eV, the initial state energy is ERiR= 0.3eV (with

respect to ERFR), and the height of the step potential is VR0R

= 15 eV. For zRoR= - 2.645 o

A

, which lies in the surface region. It is found that as the photon energy increases,

the photocurrent also increases and reaches a maximum

value at



ω

= 20 eV. With further increase in the

photon energy, the photocurrent starts decreasing above

ω



= 20 eV and attains a minimum value at



ω

= 26

eV. The plasmon energy (



ω

_p) of W is 25.3 eVP

4

P

.

Hence, from the plot we observed that the photocurrent

is maximum below the plasmon energy and minimum at

or around the plasmon energy. For the location of the

initial state wavefunction at zRoR = 0

o

A, which is in the

vacuum region, we find that as the photon energy

increases, the photocurrent decreases and attains a

constant value beyond



ω

= 24 eV. For the other

location of the initial state wavefunction at zRoR = – 5.29

o

A

, which lies in the bulk region, we observed that as photon energy increases, photocurrent increases slightly

700

= 16 eV and a minimum at



ω

= 23 eV. The

magnitude of this maximum is only about 5% of that for

the initial state wavefunction located at the surface

region. The results obtained is compared with that of

the experimental result of Weng et. alP

5

P

and also

theoretical result of Bagchi and KarP

3

P

and we have found

that they agree reasonably well.

Figure (1): Photocurrent variation against photon energyω

for W at three different locations of the initial state

wavefunction i ψ .

Fig. (2) shows variation of photocurrent

against photon energy (



ω

) for three different

locations of the initial state wavefunction namely zRoR =

0, - 2.116 and – 5.29 Ao for PbS. We have taken the

height of the step potential VR0R = 15 eV and the initial

state energy ERiR = 0.3 eV below the Fermi level. We find

that for the location of the initial state wavefunction at

the surface region, that is at zRoR = - 2.116 o

A

, with the increase in the photon energy, the photocurrent

increases and a peak is observed at photon energy



ω

= 7 eV and further increase in photon energy shows a

decrease in photocurrent and a minimum is observed at

ω



= 14.5 eV. The electromagnetic field is minimum

near the plasmon energy, hence a minimum near the

plasmon energy in photoemission for the location of the

initial state wavefunction at the surface is expected

which is observed. The plasmon energy of PbS is 14.4

eV. For the location of the initial state wavefunction at

the vacuum region given by zRoR = 0 o

A

, the photocurrent is found to decrease with the increase in the photon

energy and does not show a maximum or a minimum in

photocurrent. Such maximum and minimum values are

also not observed for the location of the initial state

wavefunction in the bulk region given by zRoR = – 5.29

o

A.

In this case, the photocurrent decreases initially with the

increase in photon energy and attains a constant value

after around photon energy ω= 6 eV. W

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01

10 15 20 25 30 35

Photon Energy (eV)

P

h

o

to

c

u

rr

e

n

t (

a

rb

. u

n

it

)

ISSN 2348 – 7968

701

Figure (2): Plot of photocurrent variation against

photon energy



ω

for PbS at three different locations of the

initial state wavefunction

ψ

_i

5. Conclusion:

The model presented here seems to reproduce

results as obtained by other theoretical models [6,7,8].

The main drawback of this model is that the calculation

of the initial state wavefunction

ψ

_i by Green function

method was considered only for the bulk photoemission

and for the surface we have assumed a Gaussian type

wavefunction. But from the above observations we can

conclude that though our model is obtained by simple

calculations it works well and is applicable to metals as

well as semiconductors.

References:

[1] D.R. Penn, "Photoemission spectroscopy in the

presence of adsorbate-covered surfaces", Phys. Rev.

Lett. 28, 1972, pp. 1041-1044.

[2] S. G. Davison and M. Steslicka, Basic Theory of

Surface States, Oxford, 1992.

[3] A. Bagchi and N. Kar, “Refraction effects in angle-

resolved photoemission from surface states on

metals”,Phys. Rev. B18, 1978, pp. 5240-5247.

[4] J.H. Weaver, D.W. Lynch and C.G. Olsen, “Optical

Properties of Crystalline Tungsten”, Phys. Rev.

B12, 1975, pp- 1293-1297.

[5] S. I. Weng, T. Gustaffson, E. W. Plummer,

“Experimental and theoretical study of the surface

resonances on the (100) faces of W and Mo”, Phys

Rev. B18, 1978, pp.1718- 1740

.

[6] P. Das, R.K.Thapa and N.Kar, “Photoemission

calculation with a simple model for the photon field:

Application to aluminium”

,

Mod. Phys. Lett. B5,

1991, pp. 65-72.

[7] R. K. Thapa, P. Das and N. Kar, “Photoemission

calculation with Kronig-Penney model”,Mod. Phys.

Letts. B 8 1994, pp. 361-366.

[8] Zaithanzauva Pachuau, B. Zoliana, P.K. Patra, D.T.

Khathing, R. K. Thapa, :Application of Mathieu

potential to photoemission calculations: the case of

a strong potential”, Phys. Letts. A 294, 2002, pp.52-

57.

PbS

0.00E+00 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00 3.00E+00 3.50E+00

5 10 15 20 25

Photon Energy (eV)

P

hot

oc

ur

r

e

nt

(

a

r

b. unit

)

702

About Author:

Lalthakimi Zadeng completed MSc. Physics from Delhi University and received Ph.D. degree from

Assam University. She is currently working as an Assistant Professor in Physics Department, Mizoram