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Photostimulated field emission — triangular barrier model
Surface Science 95 (1980) L243-L248 0 North-Holland Publishing Company
SURFACE SCIENCE LETTERS PHOTOSTIMULATED
FIELD EMISSION - TRIANGULAR
BARRIER MODEL *
Carey SCHWARTZ and Milton W. COLE Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802. USA
Received 18 January 1980; accepted for publication 7 March 1980
Photostimulated field emission (PSE) should prove to be an extremely useful probe of electronic states. We report on a generalization and extension of a previous theory to the case of a simple triangular barrier model of the surface potential. Results for both dj/de, the differential current per unit energy, and j/F2 are presented, where F is the applied electric field. These differ from the approximate results of Bagchi which are not valid at high field. The dependence of j on the transmission coefficient is shown to arise in a natural way.
Photoemission (PE) and field emission (FE) have become valuable tools for analyzing the electronic properties of metal surfaces, both with and without adsorbates [ 1,2]. Appreciation has developed recently of the merit of combining these techniques to do photostimulated field emission [3-71. In this experiment the potential barrier depends on the applied field, permitting study of electronic states at energies & < rZF t Aw which may be inaccessible in PE or low temperature FE experiments. Such states are of particular interest because of their sensitivity to the form of the potential energy at the surface. Here w is the frequency of the incident radiation, assumed to be monochromatic. This paper reports results for the photostimulated current, j, and its energy distribution, @/de, derived from a simple one-dimensional triangular barrier model of the one electron potential energy (shown in fig. 1). That is, V(z) = (V, - eFz) e(z), where F is the electric field at the surface and e(z) is the unit step function. Study of more realistic surface potentials is in progress. Our work generalizes and extends earlier calculations by Bagchi [S] . He finds the electron-photon interaction in the dipole approximation to be
(1)
* Prepared for the United States Department of Energy under contract DE-AS02-79ER10454. L243
L244
C. Schwartz, hf. W. Cole / PSE - triangular barrier model
-
Or
PFE
\
0
Fig. 1. Schematic depiction of potential energy and electronic energy levels contributing to field emission (FE) and photostimulated field emission (PFE) currents.
y = - (ieli/m)(2ntz/wS2)“*
(6 1 i) ,
(2)
where n is the radiation field normalization volume, t5 is the polarization, a is a photon annihilation operator, and z^the unit vector normal to the surface. Let the solid be a cube of side L, the electron’s position component parallel to the surface be p, and &= E + h2K2/2m. Then the unperturbed electronic states are of the form 9e,k(PZ)=~E(Z)eXp(~‘p)L-3’2,
(3)
&(z) = exp(ikz) + R exp(-ikz) = T(E) [ Ai 5(z)=-&[z
- i Bi@)]
+ (E -
ZGO,
(4a)
z>o,
(4b)
V&F1 ,
(5)
where k* = 2mE/A*, kz = 2meF/h*, and Ai and Bi are Airy functions nuity of the wave function and its derivative yields T = 2{ [Ai&,) - i Bi(Eo)] + (ik,/k)
[Ai’
- i Bi’(Eo)]}-’
.
[9]. Conti-
(6)
A prime means differentiation with respect to g and &, = i(O). The transmission coefficient at energy E is readily found to be D(E) = k, ITl*/lrk
.
(7)
Following Bag&i [8], we write the perturbed electronic state as J, + = J, Ed + \I,I, where the excited state contribution is proportional to the square root of the mean photon occupation number n, 312 $1
= h(z)
w-M-
p)L-
(84
,
$1 (z) = “/n I’* .I- dz G&Q+hw(Z,z’)(a&/az’)
.
@b)
The Green function GE+hw can be expressed [8] in terms of the solutions U(Z) and u(z) of the onedimensional Schrijdinger equation at energy E + hw, which are out-
C. Schwartz, M. W. Cole / PSE - triangular barrier model
L245
going waves at z = +oo:
z+ - ,
u(z) = Ai(.$+) - i Bi(t+) ,
(9)
u(z) = exp(-ik+z) ( Sr ME+)
-
i WE+)1+& [A@+)+ i W+)l
ZGO,
(lOa)
z>O,
(lob)
where the subscript plus denotes substitution of the promoted energy E t hw for E in the previous expressions. The coefficients Si are found by requiring continuity of u and its derivative at z = 0. In the limit of large z required for evaluating the current, we find from eq. (8) @l(Z)
-
1’2
2wn
u(z)[h30
W(u,
II)]-’
dz’ u(z’) &(z’)(dV/dz’)
j
(lla)
_m =rU(z),
(1 lb)
where W(U, u) is the Wronskian of the two solutions, which we find to be W(U, u) = -2ti+/T(E
+ tzw) .
(12)
Froms eqs. (8), (1 l), and (12) we find the outgoing probability the initial state (&, k) to be .Z,(C, k) = tzk,IIY/m7r The evaluation
current arising from
.
(13)
of the expression (11) for r is facilitated
by realizing that
1.
u& = @/2mw)[dB’(u,
LMz
Denoting the integral eFe(z), we obtain
occurring
in eq. (11) by Z and noting
dv/dz = V&(z) -
Z = V,, u(O) &(O) - eF _f dz’ u(.‘) ~,d..‘) 0
(14) The integral and resulting term proportional to F were omitted by Bag&i. Typical values for initial states, GE, near the Fermi level (VO -E N 2-5 eV, F - 0.1-0.5 V/A) yield .&,- 5-10, so that an asymptotic expansion [9] of the Airy functions is permissible. This leads to &(0)/~#&0) Inserting
= -[2m(V,,
- E)/A’] 1’2
and
l#~(O)l~ = 4E/vc, .
these results into eqs. (14) and (13) yields
X{ 1 - (eF/moVo)
[2m(Vo - E)]1’2},
(15)
L246
C. Schwartz, M. W. Cole / PSE - triangular barrier model
ignoring terms of order F’. This result agrees with that of Bag&i [8], apart from the factor in curly brackets; the correction is linear in F and of order 0.1. For a more general potential than the triangular barrier, our results (to be reported elsewhere) will not be so simple. The advantage of computational simplicity is lost in the latter case, of course. The current density is then found by multiplying J, by the charge of the electron and the probability that the initial state (&,K) is occupied and by then summing over the set {C, K}, djlde = -2e I c~ K] f(e - fiw) J,(E, K) S(e - Ao - A2K2/2m - E),
(16)
where f(e) is the Fermi occupation function and the factor 2 arises from spin. The total current j is then found by integrating eq. (16) over all energies E. In fig. 2 we show dj/de at F = 0.5 V/A. As expected from eq. (15), there is a significant decrease in the current from those states near the Fermi level. However,
9-
a
E
(eV)
Fig. 2. Typical d//de calculated from eq. (16) using the following parameters; Vo = 10.8 eV, &F = 6.30 eV, fiw = 3.5 eV, F = 0.5 V/A, and T = 71 K. Solid curve is Bagchi’s result, broken curve is the present work.
L247
C. Schwartz, M. W. Cole / PSE - triangular barrier model
I
8 F-’
Fig. 3. j/F2 versus l/F using the parameters present work.
(ii/V)
of fig. 2. Solid curve from Bagchi, broken curve
when plotting j/F2 it is only at the highest applied fields that this result is apparent, This is clearly seen in fig. 3 by the non-negligible decrease in j/F2 in the present calculation from that of Bag&i. Once agai this effect is due to the correct calculation of the integral I. We are pleased to acknowledge S.S. Sidhu and M.J.G. Lee.
the many valuable discussions
with R.H. Good,
References [l] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487; T.E. Feuchtwang, P.H. Cutler and J. Schmit, Surface Sci. 75 (1978) 401. [2] M.L. Glasser and A. Bagchi, Progr. Surface Sci. 7 (1975) 113. [3] B.I. Lundqvist, K. Mountfield and J.W. Wilkins, Solid State Commun. 10 (1972) 383.
L248
C. Schwartz, M. W. Cole / PSE - triangular barrier model
[4] M.J.G. Lee, Phys. Rev. Letters 30 (1973) 1193.
[5] [6] [7] [8] [9]
M.J.G. Lee and R. Reifenberger, Surface Sci. 70 (1978) 114. Ch. Klient and T. Radon, Surface Sci. 70 (1978) 131. Y. Teisseyre, R. Haug and R. Coelho, Surface Sci. 87 (1979) 549. A. Bagchi, Phys. Rev. BlO (1974) 542. M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions York, 1965) ch. 10.
(Dover, New
SURFACE SCIENCE LETTERS PHOTOSTIMULATED
FIELD EMISSION - TRIANGULAR
BARRIER MODEL *
Carey SCHWARTZ and Milton W. COLE Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802. USA
Received 18 January 1980; accepted for publication 7 March 1980
Photostimulated field emission (PSE) should prove to be an extremely useful probe of electronic states. We report on a generalization and extension of a previous theory to the case of a simple triangular barrier model of the surface potential. Results for both dj/de, the differential current per unit energy, and j/F2 are presented, where F is the applied electric field. These differ from the approximate results of Bagchi which are not valid at high field. The dependence of j on the transmission coefficient is shown to arise in a natural way.
Photoemission (PE) and field emission (FE) have become valuable tools for analyzing the electronic properties of metal surfaces, both with and without adsorbates [ 1,2]. Appreciation has developed recently of the merit of combining these techniques to do photostimulated field emission [3-71. In this experiment the potential barrier depends on the applied field, permitting study of electronic states at energies & < rZF t Aw which may be inaccessible in PE or low temperature FE experiments. Such states are of particular interest because of their sensitivity to the form of the potential energy at the surface. Here w is the frequency of the incident radiation, assumed to be monochromatic. This paper reports results for the photostimulated current, j, and its energy distribution, @/de, derived from a simple one-dimensional triangular barrier model of the one electron potential energy (shown in fig. 1). That is, V(z) = (V, - eFz) e(z), where F is the electric field at the surface and e(z) is the unit step function. Study of more realistic surface potentials is in progress. Our work generalizes and extends earlier calculations by Bagchi [S] . He finds the electron-photon interaction in the dipole approximation to be
(1)
* Prepared for the United States Department of Energy under contract DE-AS02-79ER10454. L243
L244
C. Schwartz, hf. W. Cole / PSE - triangular barrier model
-
Or
PFE
\
0
Fig. 1. Schematic depiction of potential energy and electronic energy levels contributing to field emission (FE) and photostimulated field emission (PFE) currents.
y = - (ieli/m)(2ntz/wS2)“*
(6 1 i) ,
(2)
where n is the radiation field normalization volume, t5 is the polarization, a is a photon annihilation operator, and z^the unit vector normal to the surface. Let the solid be a cube of side L, the electron’s position component parallel to the surface be p, and &= E + h2K2/2m. Then the unperturbed electronic states are of the form 9e,k(PZ)=~E(Z)eXp(~‘p)L-3’2,
(3)
&(z) = exp(ikz) + R exp(-ikz) = T(E) [ Ai 5(z)=-&[z
- i Bi@)]
+ (E -
ZGO,
(4a)
z>o,
(4b)
V&F1 ,
(5)
where k* = 2mE/A*, kz = 2meF/h*, and Ai and Bi are Airy functions nuity of the wave function and its derivative yields T = 2{ [Ai&,) - i Bi(Eo)] + (ik,/k)
[Ai’
- i Bi’(Eo)]}-’
.
[9]. Conti-
(6)
A prime means differentiation with respect to g and &, = i(O). The transmission coefficient at energy E is readily found to be D(E) = k, ITl*/lrk
.
(7)
Following Bag&i [8], we write the perturbed electronic state as J, + = J, Ed + \I,I, where the excited state contribution is proportional to the square root of the mean photon occupation number n, 312 $1
= h(z)
w-M-
p)L-
(84
,
$1 (z) = “/n I’* .I- dz G&Q+hw(Z,z’)(a&/az’)
.
@b)
The Green function GE+hw can be expressed [8] in terms of the solutions U(Z) and u(z) of the onedimensional Schrijdinger equation at energy E + hw, which are out-
C. Schwartz, M. W. Cole / PSE - triangular barrier model
L245
going waves at z = +oo:
z+ - ,
u(z) = Ai(.$+) - i Bi(t+) ,
(9)
u(z) = exp(-ik+z) ( Sr ME+)
-
i WE+)1+& [A@+)+ i W+)l
ZGO,
(lOa)
z>O,
(lob)
where the subscript plus denotes substitution of the promoted energy E t hw for E in the previous expressions. The coefficients Si are found by requiring continuity of u and its derivative at z = 0. In the limit of large z required for evaluating the current, we find from eq. (8) @l(Z)
-
1’2
2wn
u(z)[h30
W(u,
II)]-’
dz’ u(z’) &(z’)(dV/dz’)
j
(lla)
_m =rU(z),
(1 lb)
where W(U, u) is the Wronskian of the two solutions, which we find to be W(U, u) = -2ti+/T(E
+ tzw) .
(12)
Froms eqs. (8), (1 l), and (12) we find the outgoing probability the initial state (&, k) to be .Z,(C, k) = tzk,IIY/m7r The evaluation
current arising from
.
(13)
of the expression (11) for r is facilitated
by realizing that
1.
u& = @/2mw)[dB’(u,
LMz
Denoting the integral eFe(z), we obtain
occurring
in eq. (11) by Z and noting
dv/dz = V&(z) -
Z = V,, u(O) &(O) - eF _f dz’ u(.‘) ~,d..‘) 0
(14) The integral and resulting term proportional to F were omitted by Bag&i. Typical values for initial states, GE, near the Fermi level (VO -E N 2-5 eV, F - 0.1-0.5 V/A) yield .&,- 5-10, so that an asymptotic expansion [9] of the Airy functions is permissible. This leads to &(0)/~#&0) Inserting
= -[2m(V,,
- E)/A’] 1’2
and
l#~(O)l~ = 4E/vc, .
these results into eqs. (14) and (13) yields
X{ 1 - (eF/moVo)
[2m(Vo - E)]1’2},
(15)
L246
C. Schwartz, M. W. Cole / PSE - triangular barrier model
ignoring terms of order F’. This result agrees with that of Bag&i [8], apart from the factor in curly brackets; the correction is linear in F and of order 0.1. For a more general potential than the triangular barrier, our results (to be reported elsewhere) will not be so simple. The advantage of computational simplicity is lost in the latter case, of course. The current density is then found by multiplying J, by the charge of the electron and the probability that the initial state (&,K) is occupied and by then summing over the set {C, K}, djlde = -2e I c~ K] f(e - fiw) J,(E, K) S(e - Ao - A2K2/2m - E),
(16)
where f(e) is the Fermi occupation function and the factor 2 arises from spin. The total current j is then found by integrating eq. (16) over all energies E. In fig. 2 we show dj/de at F = 0.5 V/A. As expected from eq. (15), there is a significant decrease in the current from those states near the Fermi level. However,
9-
a
E
(eV)
Fig. 2. Typical d//de calculated from eq. (16) using the following parameters; Vo = 10.8 eV, &F = 6.30 eV, fiw = 3.5 eV, F = 0.5 V/A, and T = 71 K. Solid curve is Bagchi’s result, broken curve is the present work.
L247
C. Schwartz, M. W. Cole / PSE - triangular barrier model
I
8 F-’
Fig. 3. j/F2 versus l/F using the parameters present work.
(ii/V)
of fig. 2. Solid curve from Bagchi, broken curve
when plotting j/F2 it is only at the highest applied fields that this result is apparent, This is clearly seen in fig. 3 by the non-negligible decrease in j/F2 in the present calculation from that of Bag&i. Once agai this effect is due to the correct calculation of the integral I. We are pleased to acknowledge S.S. Sidhu and M.J.G. Lee.
the many valuable discussions
with R.H. Good,
References [l] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487; T.E. Feuchtwang, P.H. Cutler and J. Schmit, Surface Sci. 75 (1978) 401. [2] M.L. Glasser and A. Bagchi, Progr. Surface Sci. 7 (1975) 113. [3] B.I. Lundqvist, K. Mountfield and J.W. Wilkins, Solid State Commun. 10 (1972) 383.
L248
C. Schwartz, M. W. Cole / PSE - triangular barrier model
[4] M.J.G. Lee, Phys. Rev. Letters 30 (1973) 1193.
[5] [6] [7] [8] [9]
M.J.G. Lee and R. Reifenberger, Surface Sci. 70 (1978) 114. Ch. Klient and T. Radon, Surface Sci. 70 (1978) 131. Y. Teisseyre, R. Haug and R. Coelho, Surface Sci. 87 (1979) 549. A. Bagchi, Phys. Rev. BlO (1974) 542. M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions York, 1965) ch. 10.
(Dover, New