- Email: [email protected]
On the nondirect optical transition characteristics in the photon-enhanced field emission spectra
J. Pl~y.r. Chrnr.
Solids.
1972. Vol. I?. pp. 157-16
I.
Pergamon
Press.
Printed
in Great hirain
ON THE NONDIRECT OPTICAL TRANSITION CHARACTERISTICS IN THE PHOTON-ENHANCED FIELD EMISSION SPECTRA Institute for Chemical, Technological
F. KiiRMENDI and Metallurgical
Investigations, Beograd, Yugoslavia
(Received I 5 March I97 I) Abstract-An analysis of the photon-enhanced field emission characteristics due to nondirect optical transitions is given. It is shown that the minima and maxima appearing in energy distributions which belong to levels from which and to which optical transitions occur, give information enabling one to discern nondirect transitions from direct ones. 1. INTRODUCTION
density per unit energy of emitted when optical transitions are absent:
measurements give valuable informations about energy band characteristics of the investigated materials. However this method enables one to observe directly only the upper energy levels to which optical transitions occur. The lower levels cannot be investigated so easily this way especially when nondirect optical transitions exist. This fact causes many difficulties in interpreting the obtained experimental data [ 11. In this paper a possibility for energy band investigations in metals by photon-enhanced field emission is discussed. It is shown that this method in principle enables the direct observation of the upper and lower energy levels simultaneously outside and inside the Fermi surface. The upper levels appear as maxima in the energy distributions of optically excited and emitted electrons and the lower levels as minima below the Fermi level due to created holes. Comparing the energy difference between a maximum and minimum with the energy of incident photons, one can make conclusions about the direct or nondirect character of optical transition.
THE
PHOTOEMISSION
2. PHOTON-ENHANCED
electrons
(3 Jll -=-.exp(-$).ldE(ik) &t(q) d
.do
xA,(q; w) . AL(ik; w) .L(o)
(1) In the zero temperature O/ we will use h(w)
approximation
/T +=
= e(EF-QJ)
where
IF&!&-0) =
F.
Here E,,(q) is the measured energy of emitted electrons, E(ik) the electron energy in the metal. For a non-interacting electron gas model and non-resolution limited measuring apparatus the spectral weight functions AR (q; w) and AL(ik: o) are delta functions. In ( I) AL gives the electron distribution in the zero temperature approximation [4]:
FIELD EMISSION
To derive an expression for photon-enhanced field emission from an electron gas model described in [2], we will start from Gadzuk’s equation [3] which gives the current
n(ik)
= J AL(ik; O) do
(2)
on the energy level i with wave vectork. Introducing now photon interactions with I57
F.
158
KiiRMENDl
electrons as shown schematically in Fig. I(a), the number of electrons in state (ik) due to optical transitions will be reduced. This fact is manifested in the change of A,,(ik: w) in accordance with (2). Using the relation A,‘(ik)
=-b*
ImG
suppose that in zero temperature approximation phonon emission exists only. With help of Fig. 3(a). one finds that M(ik;
W) =-
(2:)z
* z F$(ik;jk)
x Ff,-(jk;fk-kk,)
(3)
,I
we can find the number of remained electrons nr( ik) which did not suffer optical interactions through Green’s function G(ik; w). Taking ALT into (1) we get a minimum in the current density per unit energy on the level E(ik) below the Fermi level EP from which optical transitions occur as shown in Fig. I(c). 2.1 Photlotl-assisted optical transitions In calculating the imaginary part of Green’s function 1mG (ik: w) we will consider the diagrams (a) and (b) in Fig. 2, which describe two types of nondirect optical transitions.
X G,(fk-kk,;
1 --M(ik;
0) . Pdk,;
x d(h)
. dfl.
b.
c
in
the self-energy which describes the energy shift of electron levels is small and will be omitted. The imaginary part is
w) +iq 7) -+ Of
hv)
(5)
field emission. (a) Electron emission through potential barrier optical transitions. (b) Current density per unit energy distribution are absent. (c) The appearance of energy band and nondirect transition characteristics in the current distribution.
In G(ik; w) = o--E(ik)
X Wk,;
In this expression C, is a constant giving the result of summation over spin coordinates, phonon branches, etc., Fij- the electronphoton, F,- the electron-phonon coupling constant. G,,, DO and P,, are the electron, phonon and photon Green’s functions re-. spectively. The summation is over intermediate j and final .f states. The real part of
a. Fig. 1. Photon-enhanced the presence of nondirect when optical transitions optical
w+hv-SZ)
(4)
the self-energy M (ik) due to phonon-assisted optical transitions is given in Fig. 2(a). We
Im M(ik;
w) z -lT.C,C
Fi. F$. 8[w-E(ik)] jJ [Et.%) --E(ik) - &,I* (6)
PHOTON-ENHANCED
FIELD
EMISSION
SPECTRA
159
system. From Fig. 2b:
with
Im M(ik;
uy being the incident photon frequency, dthe dipole vector, y-the polarization vector. The electron-phonon coupling constant depends on the type of phonon emission. We will take from [2] Fjf=
e. a*. c. h. ]kp] cp) “* . &(kp)
w) Ffj [E(jk)-E(ik)-/Iv,]’
=n’C’z x
dw’ . 6 Lw + hv, _ w’ - E(fk
- k,) ] (8)
X Im V(k,; CO).
The integral is evaluated in[5,6] for various k, and electron systems. Using the result in [5] fork, @ kF, ImM in (6) will be
where Q(kp) is the energy of a created phonon and CI, c, p are constants. From (3), (4) and (6) the spectral weight function is obtained:
ImM=--.C2z
Ffj. F; jJ [E(jk)--E(ik) -hv,]’
Fyj. Fjf. 8[u-E(ik)] Cl.zjJ [ELM --E(ik) -hv,l* A,“(ik;
w) =
[w--E(ik)]“+[7r.
Cr. z
,~(j~~~~~,~(~~~~Z}’
a
b.
Fig. 2. (a) The diagram between (ik) - (ik) gives the self-energy for phonon-assisted nondirect optical transition processes. (ik) is the initial, (Jc) the intermediate (fk-k,,) the final state. (k,,, 0) and (k,,hu) are the phonon and photon lines. Self-energy diagram for optical transition assisted by electron-electron interaction.
2.2 Optical transitions assisted by excited electron-electron system interuction The same way as in 2. I one can calculate the spectral weight function considering the diagram b in Fig. 2., which gives the selfenergy due to interaction between optically excited electrons and the whole electron
F;‘, =-7Tc2” jJ [E(Jk)--E(/k)-hv,]” x ln (P,‘+2mw,)‘“-PPF Pj- (P~-2mo,)*l’ where w,-is pj = (jhk).
the
plasmon
and (b)
.L!L 2U”Pj
1
(9) energy
and
F. KijRMENDI
160
2.3 Current density per unit energy The current density per unit energy for electrons below the Fermi level in the presence of optical transition is given by
2 J
c
J./
[E( jk) Fii(yi)
-hv,12
ii
E,(q) = EW--k,).
(12)
In the lowest order approximation -= aje Wn(q)
(10) The integration results
for
aje Jo aE,o=d’exp
-- Il. c2 d ’
;
[E(jk)
ImM(ik;
Fil, . F; - E(zk) -hv,]*
E,(q) = E(lk).
1
Jo Em(q) -EF = d exp d
For Fe. Ff, # 0 some electrons level i and the factor
1
aE,(q)
Jo Em(q) -EF d = d exp
d ;
[EC&)
--E(ik)
-h,12
Em(q) -& d
I
Em(q) = E(fk-kk,). This result shows the resonant character of electron emission for E (jk) - E (rk) = hv,. Here we supposed that the excited electrons which do not leave the emitter scatter to lower levels giving the Lorentzian shape to peaks obtained in the energy distribution. 3. DISCUSSION
Analysing the obtained results we can compare the energy of incident photons hv, with the energy difference between a maximum and minimum in the energy distribution of emitted electrons: E(fk-kk,)
-E(ik)
= hvY-EC-,
leave the
reduces the current density from the energy level E,(q) = E( ik) and a minimum appears in the energy distribution. This minimum represents the level E(rk) from which optical transitions occur. For our simple model the current density per unit energy due to excited electrons above the Fermi level is given by
aje
* exp
(11)
In the absence of optical transitions F: . F;,= 0 and (ap)/(aE,(q)) represents the Fowler-Nordheim result, Fig. 1(b): aje aE,(q)
d
w) = const.
Em(q) -EF d
FL. F;
Jo.T’c2 -
1
E,-, gives the energy lost by excited electrons in the interaction between these electrons and the whole system. This way the main features of optical transitions can be investigated. It is important to point out that many other cases can occur in the nondirect optical transition process, like of electron-hole interaction, Auger processes and many-body effects[7]. All these interactions appear in the photon-enhanced field emission spectra characteristics and can be in principle analysed. I. BERGLUND 1030 ( 1964). 2. ABRIKOSOV
REFERENCES C. and SPICER A.,
GORKOV
W., Phys. L.
and
Rev.
136A.
DZYALO-
PHOTON-ENHANCED SHINSKII in
Statistical
I., Methods Physics.
sf
Quantum
Prentice
Field
FIELD Theory
Hall. New Jersey
(1963). 3. GADZU K J., Surface Sci. 15,466 (I 969). 4. Solid Store Physics. (Edited by F. Seitz, D. Tumbull
EMISSION
SPECTRA
and H. Ehrenreich), Vol. 23, p. New York (1969). 5. QUINN J., Phys. Reo. 126,1453 6. KAZARIAN M., Fiz. tverd. Tela 7. SPICER W., Phys. Rev. 154,385
161 30. Academic Press, (1962). 8, 1053
(1967).
(I 966).
Solids.
1972. Vol. I?. pp. 157-16
I.
Pergamon
Press.
Printed
in Great hirain
ON THE NONDIRECT OPTICAL TRANSITION CHARACTERISTICS IN THE PHOTON-ENHANCED FIELD EMISSION SPECTRA Institute for Chemical, Technological
F. KiiRMENDI and Metallurgical
Investigations, Beograd, Yugoslavia
(Received I 5 March I97 I) Abstract-An analysis of the photon-enhanced field emission characteristics due to nondirect optical transitions is given. It is shown that the minima and maxima appearing in energy distributions which belong to levels from which and to which optical transitions occur, give information enabling one to discern nondirect transitions from direct ones. 1. INTRODUCTION
density per unit energy of emitted when optical transitions are absent:
measurements give valuable informations about energy band characteristics of the investigated materials. However this method enables one to observe directly only the upper energy levels to which optical transitions occur. The lower levels cannot be investigated so easily this way especially when nondirect optical transitions exist. This fact causes many difficulties in interpreting the obtained experimental data [ 11. In this paper a possibility for energy band investigations in metals by photon-enhanced field emission is discussed. It is shown that this method in principle enables the direct observation of the upper and lower energy levels simultaneously outside and inside the Fermi surface. The upper levels appear as maxima in the energy distributions of optically excited and emitted electrons and the lower levels as minima below the Fermi level due to created holes. Comparing the energy difference between a maximum and minimum with the energy of incident photons, one can make conclusions about the direct or nondirect character of optical transition.
THE
PHOTOEMISSION
2. PHOTON-ENHANCED
electrons
(3 Jll -=-.exp(-$).ldE(ik) &t(q) d
.do
xA,(q; w) . AL(ik; w) .L(o)
(1) In the zero temperature O/ we will use h(w)
approximation
/T +=
= e(EF-QJ)
where
IF&!&-0) =
F.
Here E,,(q) is the measured energy of emitted electrons, E(ik) the electron energy in the metal. For a non-interacting electron gas model and non-resolution limited measuring apparatus the spectral weight functions AR (q; w) and AL(ik: o) are delta functions. In ( I) AL gives the electron distribution in the zero temperature approximation [4]:
FIELD EMISSION
To derive an expression for photon-enhanced field emission from an electron gas model described in [2], we will start from Gadzuk’s equation [3] which gives the current
n(ik)
= J AL(ik; O) do
(2)
on the energy level i with wave vectork. Introducing now photon interactions with I57
F.
158
KiiRMENDl
electrons as shown schematically in Fig. I(a), the number of electrons in state (ik) due to optical transitions will be reduced. This fact is manifested in the change of A,,(ik: w) in accordance with (2). Using the relation A,‘(ik)
=-b*
ImG
suppose that in zero temperature approximation phonon emission exists only. With help of Fig. 3(a). one finds that M(ik;
W) =-
(2:)z
* z F$(ik;jk)
x Ff,-(jk;fk-kk,)
(3)
,I
we can find the number of remained electrons nr( ik) which did not suffer optical interactions through Green’s function G(ik; w). Taking ALT into (1) we get a minimum in the current density per unit energy on the level E(ik) below the Fermi level EP from which optical transitions occur as shown in Fig. I(c). 2.1 Photlotl-assisted optical transitions In calculating the imaginary part of Green’s function 1mG (ik: w) we will consider the diagrams (a) and (b) in Fig. 2, which describe two types of nondirect optical transitions.
X G,(fk-kk,;
1 --M(ik;
0) . Pdk,;
x d(h)
. dfl.
b.
c
in
the self-energy which describes the energy shift of electron levels is small and will be omitted. The imaginary part is
w) +iq 7) -+ Of
hv)
(5)
field emission. (a) Electron emission through potential barrier optical transitions. (b) Current density per unit energy distribution are absent. (c) The appearance of energy band and nondirect transition characteristics in the current distribution.
In G(ik; w) = o--E(ik)
X Wk,;
In this expression C, is a constant giving the result of summation over spin coordinates, phonon branches, etc., Fij- the electronphoton, F,- the electron-phonon coupling constant. G,,, DO and P,, are the electron, phonon and photon Green’s functions re-. spectively. The summation is over intermediate j and final .f states. The real part of
a. Fig. 1. Photon-enhanced the presence of nondirect when optical transitions optical
w+hv-SZ)
(4)
the self-energy M (ik) due to phonon-assisted optical transitions is given in Fig. 2(a). We
Im M(ik;
w) z -lT.C,C
Fi. F$. 8[w-E(ik)] jJ [Et.%) --E(ik) - &,I* (6)
PHOTON-ENHANCED
FIELD
EMISSION
SPECTRA
159
system. From Fig. 2b:
with
Im M(ik;
uy being the incident photon frequency, dthe dipole vector, y-the polarization vector. The electron-phonon coupling constant depends on the type of phonon emission. We will take from [2] Fjf=
e. a*. c. h. ]kp] cp) “* . &(kp)
w) Ffj [E(jk)-E(ik)-/Iv,]’
=n’C’z x
dw’ . 6 Lw + hv, _ w’ - E(fk
- k,) ] (8)
X Im V(k,; CO).
The integral is evaluated in[5,6] for various k, and electron systems. Using the result in [5] fork, @ kF, ImM in (6) will be
where Q(kp) is the energy of a created phonon and CI, c, p are constants. From (3), (4) and (6) the spectral weight function is obtained:
ImM=--.C2z
Ffj. F; jJ [E(jk)--E(ik) -hv,]’
Fyj. Fjf. 8[u-E(ik)] Cl.zjJ [ELM --E(ik) -hv,l* A,“(ik;
w) =
[w--E(ik)]“+[7r.
Cr. z
,~(j~~~~~,~(~~~~Z}’
a
b.
Fig. 2. (a) The diagram between (ik) - (ik) gives the self-energy for phonon-assisted nondirect optical transition processes. (ik) is the initial, (Jc) the intermediate (fk-k,,) the final state. (k,,, 0) and (k,,hu) are the phonon and photon lines. Self-energy diagram for optical transition assisted by electron-electron interaction.
2.2 Optical transitions assisted by excited electron-electron system interuction The same way as in 2. I one can calculate the spectral weight function considering the diagram b in Fig. 2., which gives the selfenergy due to interaction between optically excited electrons and the whole electron
F;‘, =-7Tc2” jJ [E(Jk)--E(/k)-hv,]” x ln (P,‘+2mw,)‘“-PPF Pj- (P~-2mo,)*l’ where w,-is pj = (jhk).
the
plasmon
and (b)
.L!L 2U”Pj
1
(9) energy
and
F. KijRMENDI
160
2.3 Current density per unit energy The current density per unit energy for electrons below the Fermi level in the presence of optical transition is given by
2 J
c
J./
[E( jk) Fii(yi)
-hv,12
ii
E,(q) = EW--k,).
(12)
In the lowest order approximation -= aje Wn(q)
(10) The integration results
for
aje Jo aE,o=d’exp
-- Il. c2 d ’
;
[E(jk)
ImM(ik;
Fil, . F; - E(zk) -hv,]*
E,(q) = E(lk).
1
Jo Em(q) -EF = d exp d
For Fe. Ff, # 0 some electrons level i and the factor
1
aE,(q)
Jo Em(q) -EF d = d exp
d ;
[EC&)
--E(ik)
-h,12
Em(q) -& d
I
Em(q) = E(fk-kk,). This result shows the resonant character of electron emission for E (jk) - E (rk) = hv,. Here we supposed that the excited electrons which do not leave the emitter scatter to lower levels giving the Lorentzian shape to peaks obtained in the energy distribution. 3. DISCUSSION
Analysing the obtained results we can compare the energy of incident photons hv, with the energy difference between a maximum and minimum in the energy distribution of emitted electrons: E(fk-kk,)
-E(ik)
= hvY-EC-,
leave the
reduces the current density from the energy level E,(q) = E( ik) and a minimum appears in the energy distribution. This minimum represents the level E(rk) from which optical transitions occur. For our simple model the current density per unit energy due to excited electrons above the Fermi level is given by
aje
* exp
(11)
In the absence of optical transitions F: . F;,= 0 and (ap)/(aE,(q)) represents the Fowler-Nordheim result, Fig. 1(b): aje aE,(q)
d
w) = const.
Em(q) -EF d
FL. F;
Jo.T’c2 -
1
E,-, gives the energy lost by excited electrons in the interaction between these electrons and the whole system. This way the main features of optical transitions can be investigated. It is important to point out that many other cases can occur in the nondirect optical transition process, like of electron-hole interaction, Auger processes and many-body effects[7]. All these interactions appear in the photon-enhanced field emission spectra characteristics and can be in principle analysed. I. BERGLUND 1030 ( 1964). 2. ABRIKOSOV
REFERENCES C. and SPICER A.,
GORKOV
W., Phys. L.
and
Rev.
136A.
DZYALO-
PHOTON-ENHANCED SHINSKII in
Statistical
I., Methods Physics.
sf
Quantum
Prentice
Field
FIELD Theory
Hall. New Jersey
(1963). 3. GADZU K J., Surface Sci. 15,466 (I 969). 4. Solid Store Physics. (Edited by F. Seitz, D. Tumbull
EMISSION
SPECTRA
and H. Ehrenreich), Vol. 23, p. New York (1969). 5. QUINN J., Phys. Reo. 126,1453 6. KAZARIAN M., Fiz. tverd. Tela 7. SPICER W., Phys. Rev. 154,385
161 30. Academic Press, (1962). 8, 1053
(1967).
(I 966).