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Effect of non-equilibrium distribution on energy exchange processes at a metal surface
SURFACE
SCIENCE 12 (1968) 208-220 0 North-Holland
EFFECT
OF NON-EQUILIBRIUM
EXCHANGE
PROCESSES IRENE
Department
of Physics,
Publishing
DISTRIBUTION AT A METAL
Co., Amsterdam
ON ENERGY
SURFACE*
M. ENGLE
The Pennsylvania State University, Pennsylvania 16802, U.S.A.
University
Park,
and P. H. CUTLER** Laboratory
for Electrophysics,
The Technical
University,
Lyngby,
Denmark
T-F emission current densities, total energy distributions, and Nottingham inversion temperatures are calculated for several values of the average effective replacement energy of the emitted electrons. Replacement energies on the order of several hundredths of an eV less than the Fermi energy were found to lower the calculated inversion temperatures appreciably. The effect of the non-equilibrium distribution of the conduction electrons was included in the calculations using: f = fo + fikz, where fo is the equilibrium Fermi function. Slight differences due to the inclusion of the fikz term were noted in the calculated inversion temperatures.
1. Introduction Recent experimental work on the Nottingham Effect by Swanson, Crouser and Charbonnierl) has produced results in disagreement with existing theory. In examining the sources of discrepancies, they considered the applicability of the free electron model (i.e., limiting case of Stratton’s theory for a degenerate metal of arbitrary band structure) to describe T-F emission and attendant energy-exchange processes and concluded that, expecially for the transition metal tungsten, bulk conduction processes and electron-lattice interactions) and/or non(e.g., electron-electron equilibrium effects of temperature and field should be included in any analysis. Other recent theoretical work2) has shown that the disagreement also cannot be accounted for solely on the basis of deficiencies in the models of the surface potential barrier. In the T-F emission process, energy exchanges at the cathode resulting from the difference between the average energy of the emitted electrons (E) and the average energy (E’) = E, = R of the replacement electrons supplied bythe circuit are important in determining * Research partially sponsored by the Air Force Office of Scientific Research, United States Air Force, under AFOSR Grant #213-66 ** Permanent address: Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A. 208
ENERGY
EXCHANGE
PROCESSES
209
the local temperature at the emitter surface. If (E) is less than (E’), the cathode tends to be heated during emission, while if (E’) is less than (E), the cathode tends to be cooled by the exchange. The energy exchange is known as the Nottingham effect. Earlier calculations were predicated on Nottingham’s original assumption that the average energy of the conduction electrons in the emitter was assumed to be, under equilibrium conditions, the Fermi energy [. The assumption was thought to have had some experimental verifications), but more recent work by the same investigators indicates some doubt as to its validity. In fact, it is suggested that the replacement electrons might have an energy different from the Fermi energy by an amount on the order of k,T (ref. 1,4) and as Swanson et al. point out, “ . . . near the inversion condition, a variation of E, by a few hundredths of an eV may cause the inversion temperature to depart significantly from the predicted value based on the Nottingham assumption that E,=[.” (ref. 1). In the present work, we have considered the possibility that (E’) is less than 5 and have investigated the dependence of the computed inversion temperature (at which the Nottingham effect changes from one of heating to one of cooling) upon (E’). We have also included, within a free electron formulation, the first order effects of the non-equilibrium distribution induced by the emission current density upon the energy distribution of the electrons incident from within the metal on the emitter surface, as well as the first order temperature dependence of the Fermi parameter [. The following well-known approximations and assumptions have been used in the analysis: i) A free-electron model, with no explicit band structure effects. ii) The surface barrier model is reduced to the one-dimensional approximation. The particular choice of model of the surface is a quantum-modified image potential. iii) The probability of emission from the surface of the metal of an electron of total energy E is dependent only upon the normal component of the energy W, where the total energy E of the electron in the metal has the form D-2
E = k
D2
D2
+ V(x) = ;& + ‘& + W,
(1.1)
and W=(P:/2m)+ V( x ) is the energy associated with the direction of motion normal to the surface. iv) For electrons with energy W greater than the maximum height of the barrier, it is assumed that the transmission probability is unity. v) The WKB approximation is used to calculate the transmission probability for electrons with energies W less than the potential maximum.
210
IRENE
vi) The temperature
M. ENGLE
dependence
gas and is given by the standard
AND P. H. CUTLER
of the Fermi energy is for a free electron expressions)
(l-2) where k, is Boltzmann’s constant. In the present work, numerical calculations of Nottingham inversion temperatures were performed for fields from 3 x 10’ V/cm to 9 x 10’ V/cm. The temperatures were in the range 650°K to 2200°K. In section 2, the free-electron theory of T-F emission and Nottingham effect are briefly review and in section 3, the steady state solutions of the field and temperature dependent energy distribution is developed. The method of calculation of the inversion temperatures is treated in section 4. Section 5 gives results of calculations and section 6 contains a discussion of results and conclusions. 2. The free electron theory of T-F emission The background of a free electron formulation of the problem has been discussed in a previous worklpe). Only the relevant points and approximations are reiterated here. The model of the surface potential barrier consists of terms associated with the externally applied field, the classical image force on a charge removed from but in the vicinity of the surface of a conductor and a quantum correction terms). The potential has the form V(x) = - eFx - e2/4x + r]e2/4x2.
(2.1)
The zero of energy is here taken to be an electron at rest at infinity and at zero applied field; - W, is the mean value of the potential experienced by an electron within the metal. The T-F emission current density is a combination of the emission of the electrons from a metal by means of tunneling through the surface barrier and pure thermionic emission; note that even if the phenomenon of thermionic emission did not exist, the emission should be enchanced by higher temperature, since the higher energy levels are more densely populated at higher temperatures, thereby increasing the tunnel emission. Using the Good and Miiller formalism7), we take N( W, E) to be the number of electrons of energy E in the interval dE with x-components of energy W in the interval d W incident on the plane x=0 (i.e., “surface” or metal-vacuum boundary) per unit area per second. N(W, E) is the supply function. Following Young’s formalisms), we have for a free electron gas at
ENERGY
equilibrium
obeying
EXCHANGE
211
PROCESSES
Fermi statistics
(2.2) where h is Planck’s
constant
andf,
is the Fermi function; 1
fcl =~ 1 + ev[I(E- WWI’
(2.3)
In this study, the equilibrium Fermi function,f, will be replaced by one which contains the first order non-equilibrium dependence upon the field and temperature within the emitter, i.e.
fo -+f”on-eq. To justify
this transposition,
we note that the expression
is independent of the explicit form of the function f, provided symmetric function of W (see Appendix). Therefore
f( W, E) is a
4nm
(2.4) The form of fno_,used in the calculations is discussed in section 3. Let D(W) be the probability of penetration of the barrier by an electron with x-component of energy W. In this study, the WKB approximation for D(W) has been used. It is given by x2
D(W)=exp
-
;
[
SJI
V’(x) -
WI}dx] ;
(2.5)
x1
x1 and x2 are the roots of the equation V(x)- W=O. The product N( W, E) D(W) d W dE is the number of electrons of energy E in the interval dE with x-component of energy Win the interval d W which are emitted from the surface per unit area per second. The expression E
[N(W, E) D(W)dW]
P(E) dE =
s
dE
(2.6)
is the total number of electrons of energy E in the interval dE which are emitted from the surface per unit area per second. P(E) is the rate at which electrons of energy E are emitted from the surface.
IRENE
212
M. ENGLE
AND P. H. CUTLER
The rate at which heat is lost or gained by electrons of energy E in the interval dE is P(E) (E-R) dE. The rate HI (heating) at which heat is given up by the surface region to emitted electrons with energies below E’ is given by the following expression : R
H, =
The rate H, (cooling)
P(E)(E s -W.Y
- R)dE.
(2.7)
at which heat is removed from levels above R is given by cc
H,=
P(E)(E-R)dE.
(2.8)
s
R
The sum H= HI + H, is the net rate of removal of heat from the surface. If HO, the effect is that of cooling. At low temperatures, the higher energy levels become filled and contribute preferentially to the emission current, since the probability of emission is greater for higher temperatures. The Nottingham inversion temperature Ti is defined as that temperature at which H=O. The inversion temperature tends to be a stable one for the emission, since as T rises, H will increase and the emitter will tend to be cooled down to the inversion temperature, and conversely, if the temperature falls below Ti, H will also fall below zero and the surface tends to be heated until it has reached the inversion temperature again. 3. The non-equilibrium
supply function
In a treatment of the non-equilibrium distribution function on electron emission, K. Lehovecs) introduced an fnonleq and expanded the function f(E) in Legendre polynomials with the field directions as the axis; retaining terms only to first order, he obtains: f(E)non-eq = fo (E) + fi (E) 4 3
(3.1)
where k is the electron wave vector and k, the component of the wave vector in the x-direction. Assuming a completely free non-interacting electron gas k,
=
px= in;
)
h where u, is x-component of the velocity. For sufficiently low fields within the metal, Lehovec standard expressions) for the non-equilibrium correction
(3.2)
uses the following to the distribution
ENERGY
EXCHANGE
PROCESSES
213
function; (3.3) z is a characteristic collision parameter. F’ is the effective field within the metal emitter. Since the externally applied field is rapidly damped within the metal, F’ can be taken to be the normal emission current density times the resistivity of the metal: F’ = Jp,
(3.4)
where it is to be noted that p=p(F’, T); having used the low field approximation, we assume p =p(T) only. The relaxation time, which describes phenomenologically the electron lattice interaction, is a complicated function of the electron wave vector, energy, and local temperature. Wilsonlo) using a nearly free electron model, has derived the following expression for the parameter: (3.5) 8 is the Debye temperature and m * the effective mass of the electron. We have assumed m * = m, the free electron mass. Although cyclotron resonance and De Haas Van-Alphen measurements of the effective mass in tungsten yield average values smaller than the free electron massrl), the difference is not significant within the order of approximation assumed in this calculation. _4 and A are defined by WilsoniO): (3.6) a is the lattice constant, M the ion mass and C is the electron lattice coupling constant, on the order of <. It is defined as (3.7) where u, are the Bloch functions and Y’ is the lattice potential. Since accurate calculations of the interaction constant for transition metals are not available, a reasonable estimate of C was inferred from calculated and experimentally determined values for other metals. From table (5.1) of Pinesla) and table IX 1 of WilsonrJ), it is seen that (CGheory and (CiQ+,. are of the order of unity, so Cx[ and ranges between l-10 eV. In selecting C and other values of the electronic para-
IRENE
214
M.ENGLE
AND P. H. CUTLER
meters, we have ignored all multi-band effects, consistent the free electron approximation. To compute the correction term in eq. (3.1) we use
with assumption
,,=)w- Y)=i(w+WJ, since V here is taken to be the average potential
of
(3.8)
inside the metal. Also
(3.9) Collecting
and substituting
in (3.1)-(3.3) (3.10)
where B=------
2h2e
(+m*)*
4. Calculation
A
e (3.11)
A k,’
of the inversion temperatures
For the calculations in this work we assume 4 =4.5 eV, which represents an average work function for a clean tungsten metal surface. W,= 10.2 eV and n =0.069 A0 were used in order to facilitate comparison with earlier calculations 2,10) and with experiment. The inversion temperatures were calculated by means of a trial and error method, i.e., seeking H(T,) =O. A low estimate of the inversion temperature is assumed for a particular value of the field. When using the non-equilibrium distribution, an initial value of the current density J is also introduced. The additional parameters introduced in the non-equilibrium case were chosen to be appropriate for tungsten, using tabulated or experimental values where possible. They are as follows a=3.16 A (ref. 13), C=[= 5.7 eV determined by W, and 4, the work function tungsten, 8=315”K (ref. 14), M=mass of tungsten atom.
for
The rates HI and If2 at which electrons with energies above or below, respectively, the average replacement energy R were calculated separately. The numerical calculation of HI and of J proceeds exactly as described in
ENERGY
EXCHANGE
PROCESSES
section 4.1 of ref. 2. D(W) was computed
using
215
the WKB
12
- logD(W)
=
8m
1’
S[ A2
approximation
(V(x) + Wa) dx.
XI
(4.1)
x1 and x2 are given by 10) cos(-@) + I W1/3eF - A cos [$(0 + rr)] + IWV3eF
A
(4.2) (4.3)
where
(4.4) and 8 = cos-1
>I’
3
l W12/3eF2
H2 is divided into two parts, infinite interval [R, co]
- e/4F
Hz,+ HzT to facilitate
computation
(4.5)
over the
a, HzF=
P(E)(E-R)dE
(4.6)
s R
and is calculated by a procedure identical to that used to calculate H. Since HzT gives a very small contribution to H in the given field and temperature range, it is sufficient to calculate it using a modified RichardsonDushman equation: m Hi,=:gJexp(-%)(E-R)dE.
(4.7)
0
Hz= = - R
>O
since
R
(4.8)
When the non-equilibrium distribution is used, values of the total emission current density are also calculated for each value of the trial T using the computed P(E), and D(W), and N( W, E) arrays: R
J = IeJ
s -W,
m
P(E)dE+lel
P(E)dE+ I
R
+
‘
(k,T - V,,,) exp
(4.9)
:
216
IRENE M. ENGLE
AND P. H. CUTLER
The sum H= H1 + Hz,+ HZT is computed and tested for algebraic sign. If H-CO, the temperature is known to be too low and the initial temperature is raised by a given amount for the next try; if H>O, Tis lowered for the next attempt. By a halving process, the inversion temperature can be calculated to within any degree of precision desired. In this work, the calculated Ti were determined to within 5 “K.
5. Results of the calculations Table 1 lists the calculated inversion temperatures using the equilibrium distribution for several assumed values of the replacement energy. We note first that the inclusion of the temperature dependence of the Fermi parameter as given by eq. (1.2) was found to have little effect upon the computed inversion temperatures, but the assumption that the average replacement energy R was a few hundredths of an eV less than zero yields significant lowering of the inversion temperatures. TABLE 1
Computed inversion temperatures
using an equilibrium supply function for different values of replacement energy
F (10-7
Ti R=
-4.50eV
3 4 5 6 7 8 9
R=
-4.52eV
R=
i =5(T)
V/cm) 762 1004 1238 1470 1701 1932 2180
715 955 1195 1420 1650 1880 2120
-4.52eV
i=i@) 710 951 1187 1420 1650 1880 2120
R=
-454eV i = i(T) 670 880 1130 1365 1590 1820 2057
R= -4.56eV c = L’(T) 583 826 1068 1300 1530 1760 1994
Negligible lowering of computed inversion temperatures was found when the non-equilibrium distribution given by eq. (3.9) was used. However, in the course of calculating the inversion temperatures using the non-equilibrium supply function, it was interesting to study the magnitude of the change in f as a function of field as fO+,fnon_eq. Since f,k, is energy dependent, the shifts were observed at one energy. Because afo/aE behaves somewhat like a delta function of width k,T about E=[, it was chosen as the energy at which to evaluate the change in .f for different values of externally applied fields. fnon_eq can be expressed fnon-eq = f(l
+ %=W=<,
217
ENERGYEiXCHANGEPROCESSES
where
p=p(T) was obtained by interpolating experimental values for tungsten15). The J used is the theoretically calculated value of the emission current density. It was obtained in the course of calculating q, using the method described in ref. 2. The T is the calculated value of inversion temperature for that particular external fiefd using the equilibrium supply function. The calculated values of S are listed in table 2. It is seen that in spite of appreciable changes in the distribution function, the effect is wholly inadequate to make the computed values of inversion temperature be in agreement with experiment. TABLET F
(Kk7V/cm) 3 4 .5 6 7 8 9
1.5 4.3 4.2 2.2 7.9 2.1
46 x 104 x 105 x 106 x 107 x 107 x 108
s
J
P
(A/cm)
(michrohmem) 17.5 25
32 39 47 54 63
762 1004 1238 1470 1701 $932 2180
5.5 1.5 3.5 3.0 1.4 4.7
x 10-3 x IO-4 x x0-3 x IO-2 x IO-’ x 10-l 1.1
The most recently computed inversion temperatures are shown with experimentally measured values in fig. 1. We may note that there is still considerable quantitative discrepancy between the theoretical values computed in the present work and the iatest experimental measurement, although the magnitude of the discrepancy has been halved. However, there also seems to be disagreement between experimental results carried out at different times by the same groupr*s). This is indicative of the difficulty involved in obtaining accurate and reproducible measurements of inversion temperatures; it also suggests the need to try to refine techniques, if possible, and/or make additional measurement so that at least random errors may be minimized in the analysis. It is to be stressed that the several corrections to the extant free-electron formulation introduced by the present treatment, i.e., (i) average energy of the replacement electronsin the conduction band different than Fermi energy,
218
IRENE
M. ENGLE
AND P. H. CUTLER
t TPK 1 R=(
to)=-4.Sev
1600--
1400--
1200-CHARBONNIER, STRAYER,SWANSON, AND MARTIN
lOOO--
800--
‘/ /*
600 t 400
SWANSON, CROUSER, AND CHARBONNIER
LX-
I
I
I
II
2
3
4
5 F*lO’(v/cm
6
I
I
7
8
9
I --c
Fig. 1. Comparison of experimental and computed inversion temperatures for tungsten and for different values of the average replacement energy R.The experimental points are those of refs. 1 and 3.
(ii) use of an non-equilibrium distribution function and (iii) use of a temperature dependent Fermi energy are purely phenomenological in character and of lowest order. Nevertheless, within these limitations, one can note that the calculated 6’s are large enough to reasonably suggest that a combination of the use of an fnon_eq and the assumption R < [ might produce significantly lower calculated inversion temperatures. Of preferential interest is the tentative conclusion that the basic assumption of the Nottingham effect, namely, that the average energy of the replacement electrons in the conduction band is equal to the Fermi energy is not valid at least for transition metals. Swanson et al. have, in the analysis and interpretation of their experimental data, emphasized the importance of this as a possible source of the disagreement. They have considered the relationship between R= (E’) and the absolute thermoelectric power of S of the metal, R = (E’) and have suggested
further
= E, - eTS,
theoretical
and experimental
(6.1) investigations
to
ENERCIY
EXCHANGE
219
PROCESSES
determine magnitude and tem~rature dependence of deviation of (E’) from the Fermi energy. That this procedure is probably feasible only experimentally becomes evident when we consider the expression for S (ref. 16) (for the simple isotropic case) S =;
~~~~~~~~=~,
(6.3
where the conductivity o(E) is given by
It is seen that the thermoelectric power depends critically on the energy dependence of the mean free path and effective mass of the electron, and the complicated band structure of tungsten woutd make any theoretical analysis of dubious value quantitativelylr). Thus, even though experiment and the present heuristic treatment compel belief, until the assumption R = (E’) < 5 is conctusively supported experimentally, rigorous theoretical analysis seems unwarranted at present. Acknowledgements The authors woufd Iike to express their appreciation for the cooperation of the staff of the Computation Center at the ~ennsy~~a~ia State University. Appendix We have generalized Young’s treatment71 for a God-equilibrium disiribution functionf( W, E). Using this notation, we have 101cos e
N(a,E)adE=noI-dndE=n(E)dE~sinBdBd~
Iv1COS8
where N(SZ, E) is the number of electrons of energy E which have X-cornponents of velocity lying between v cos 8 and o cos 8 + v cos (0 + de)
n( W, E) is the electron energy distribution with respect to E=O where the electron is at rest an infinite distance outside the metal; dW= mvXdx f dV.
220
IRENE
Since
V is taken
M. ENGLE
to be a constant,
AND P. H. CUTLER
- W,, inside
the metal,
0, = v cos 0 = &(2/m)
(E -
dV=O
V)] cos 8 )
Idu,j = v sin 8 de, ldW/
=
N(Q,E)dQdE=
mv2
cos
6
sin 8 dd ,
n(W,E)dE -4n
dW JPm
Also, since N(W,
E) =
s
(E -
VI
N(Q, E) d4,
0
we
have n(W,E)dEdW N(W,E)dWdE=p------m-c-lXJm(E - VII
4;3mf(W,E)dWdE.
References 1) L. W. Swanson, L. C. Crouser and F. M. Charbonnier, 2) I. Engle and P. H. Cutler, Surface Sci. 8 (1967) 288.
Phys. Rev. 151 (1966) 327.
3) F. M. Charbonnier, R. W. Strayer, L. W. Swanson and E. E. Martin, Phys. Rev. Letters 13 (1964) 397. 4) Erwin W. Miiller, Field Emission in: Annual Review of Physical Chemistry, Vol. 18 (1967) p. 35. 5) Frederick Seitz, Modern Theory of Solids (McGraw-Hill, New York, 1940) p. 149. 6) P. H. Cutler and J. J. Gibbons, Phys. 92 (1958) 1140. 7) R. H. Good and E. W. Miiller, Handbuch derphysik, Vol. 21, Ed. S. Fliigge (SpringerVerlag, Berlin, 1956) p. 181. 8) R. D. Young, Phys. Rev. 113 (1959) 110. 9) Kurt Lehovec, Phys. Rev. 96 (1954) 921. 10) A. H. Wilson, The Theory of Metals (Cambridge Univ. Press, New York, 1965) sections 9.3 and 9.5. 11) D. M. Sparlin and J. A. Marcus, Phys. Rev. 144 (1966) 484. 12) D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963) section 5.7. 13) Charles Kittel, Introduction ro Solid State Physics (Wiley, New York 1962) p. 41. 14) A. H. Wilson, The Theory ofMetals (Cambridge Univ. Press, New York, 1954) p. 143. 15) Handbook of Chemistry and Physics, 43rd ed. (Chemical Rubber Publ. Co., Cleveland, Ohio, 1961) p. 3044. 16) N. F. Mott and H. Jones, Theory of Mefals and Alloys (Dover Publ., New York, 1958) section 15.2. 17) J. M. Ziman, Theory of Phonons (Oxford Univ. Press, Oxford, 1960) section 9.11.
SCIENCE 12 (1968) 208-220 0 North-Holland
EFFECT
OF NON-EQUILIBRIUM
EXCHANGE
PROCESSES IRENE
Department
of Physics,
Publishing
DISTRIBUTION AT A METAL
Co., Amsterdam
ON ENERGY
SURFACE*
M. ENGLE
The Pennsylvania State University, Pennsylvania 16802, U.S.A.
University
Park,
and P. H. CUTLER** Laboratory
for Electrophysics,
The Technical
University,
Lyngby,
Denmark
T-F emission current densities, total energy distributions, and Nottingham inversion temperatures are calculated for several values of the average effective replacement energy of the emitted electrons. Replacement energies on the order of several hundredths of an eV less than the Fermi energy were found to lower the calculated inversion temperatures appreciably. The effect of the non-equilibrium distribution of the conduction electrons was included in the calculations using: f = fo + fikz, where fo is the equilibrium Fermi function. Slight differences due to the inclusion of the fikz term were noted in the calculated inversion temperatures.
1. Introduction Recent experimental work on the Nottingham Effect by Swanson, Crouser and Charbonnierl) has produced results in disagreement with existing theory. In examining the sources of discrepancies, they considered the applicability of the free electron model (i.e., limiting case of Stratton’s theory for a degenerate metal of arbitrary band structure) to describe T-F emission and attendant energy-exchange processes and concluded that, expecially for the transition metal tungsten, bulk conduction processes and electron-lattice interactions) and/or non(e.g., electron-electron equilibrium effects of temperature and field should be included in any analysis. Other recent theoretical work2) has shown that the disagreement also cannot be accounted for solely on the basis of deficiencies in the models of the surface potential barrier. In the T-F emission process, energy exchanges at the cathode resulting from the difference between the average energy of the emitted electrons (E) and the average energy (E’) = E, = R of the replacement electrons supplied bythe circuit are important in determining * Research partially sponsored by the Air Force Office of Scientific Research, United States Air Force, under AFOSR Grant #213-66 ** Permanent address: Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A. 208
ENERGY
EXCHANGE
PROCESSES
209
the local temperature at the emitter surface. If (E) is less than (E’), the cathode tends to be heated during emission, while if (E’) is less than (E), the cathode tends to be cooled by the exchange. The energy exchange is known as the Nottingham effect. Earlier calculations were predicated on Nottingham’s original assumption that the average energy of the conduction electrons in the emitter was assumed to be, under equilibrium conditions, the Fermi energy [. The assumption was thought to have had some experimental verifications), but more recent work by the same investigators indicates some doubt as to its validity. In fact, it is suggested that the replacement electrons might have an energy different from the Fermi energy by an amount on the order of k,T (ref. 1,4) and as Swanson et al. point out, “ . . . near the inversion condition, a variation of E, by a few hundredths of an eV may cause the inversion temperature to depart significantly from the predicted value based on the Nottingham assumption that E,=[.” (ref. 1). In the present work, we have considered the possibility that (E’) is less than 5 and have investigated the dependence of the computed inversion temperature (at which the Nottingham effect changes from one of heating to one of cooling) upon (E’). We have also included, within a free electron formulation, the first order effects of the non-equilibrium distribution induced by the emission current density upon the energy distribution of the electrons incident from within the metal on the emitter surface, as well as the first order temperature dependence of the Fermi parameter [. The following well-known approximations and assumptions have been used in the analysis: i) A free-electron model, with no explicit band structure effects. ii) The surface barrier model is reduced to the one-dimensional approximation. The particular choice of model of the surface is a quantum-modified image potential. iii) The probability of emission from the surface of the metal of an electron of total energy E is dependent only upon the normal component of the energy W, where the total energy E of the electron in the metal has the form D-2
E = k
D2
D2
+ V(x) = ;& + ‘& + W,
(1.1)
and W=(P:/2m)+ V( x ) is the energy associated with the direction of motion normal to the surface. iv) For electrons with energy W greater than the maximum height of the barrier, it is assumed that the transmission probability is unity. v) The WKB approximation is used to calculate the transmission probability for electrons with energies W less than the potential maximum.
210
IRENE
vi) The temperature
M. ENGLE
dependence
gas and is given by the standard
AND P. H. CUTLER
of the Fermi energy is for a free electron expressions)
(l-2) where k, is Boltzmann’s constant. In the present work, numerical calculations of Nottingham inversion temperatures were performed for fields from 3 x 10’ V/cm to 9 x 10’ V/cm. The temperatures were in the range 650°K to 2200°K. In section 2, the free-electron theory of T-F emission and Nottingham effect are briefly review and in section 3, the steady state solutions of the field and temperature dependent energy distribution is developed. The method of calculation of the inversion temperatures is treated in section 4. Section 5 gives results of calculations and section 6 contains a discussion of results and conclusions. 2. The free electron theory of T-F emission The background of a free electron formulation of the problem has been discussed in a previous worklpe). Only the relevant points and approximations are reiterated here. The model of the surface potential barrier consists of terms associated with the externally applied field, the classical image force on a charge removed from but in the vicinity of the surface of a conductor and a quantum correction terms). The potential has the form V(x) = - eFx - e2/4x + r]e2/4x2.
(2.1)
The zero of energy is here taken to be an electron at rest at infinity and at zero applied field; - W, is the mean value of the potential experienced by an electron within the metal. The T-F emission current density is a combination of the emission of the electrons from a metal by means of tunneling through the surface barrier and pure thermionic emission; note that even if the phenomenon of thermionic emission did not exist, the emission should be enchanced by higher temperature, since the higher energy levels are more densely populated at higher temperatures, thereby increasing the tunnel emission. Using the Good and Miiller formalism7), we take N( W, E) to be the number of electrons of energy E in the interval dE with x-components of energy W in the interval d W incident on the plane x=0 (i.e., “surface” or metal-vacuum boundary) per unit area per second. N(W, E) is the supply function. Following Young’s formalisms), we have for a free electron gas at
ENERGY
equilibrium
obeying
EXCHANGE
211
PROCESSES
Fermi statistics
(2.2) where h is Planck’s
constant
andf,
is the Fermi function; 1
fcl =~ 1 + ev[I(E- WWI’
(2.3)
In this study, the equilibrium Fermi function,f, will be replaced by one which contains the first order non-equilibrium dependence upon the field and temperature within the emitter, i.e.
fo -+f”on-eq. To justify
this transposition,
we note that the expression
is independent of the explicit form of the function f, provided symmetric function of W (see Appendix). Therefore
f( W, E) is a
4nm
(2.4) The form of fno_,used in the calculations is discussed in section 3. Let D(W) be the probability of penetration of the barrier by an electron with x-component of energy W. In this study, the WKB approximation for D(W) has been used. It is given by x2
D(W)=exp
-
;
[
SJI
V’(x) -
WI}dx] ;
(2.5)
x1
x1 and x2 are the roots of the equation V(x)- W=O. The product N( W, E) D(W) d W dE is the number of electrons of energy E in the interval dE with x-component of energy Win the interval d W which are emitted from the surface per unit area per second. The expression E
[N(W, E) D(W)dW]
P(E) dE =
s
dE
(2.6)
is the total number of electrons of energy E in the interval dE which are emitted from the surface per unit area per second. P(E) is the rate at which electrons of energy E are emitted from the surface.
IRENE
212
M. ENGLE
AND P. H. CUTLER
The rate at which heat is lost or gained by electrons of energy E in the interval dE is P(E) (E-R) dE. The rate HI (heating) at which heat is given up by the surface region to emitted electrons with energies below E’ is given by the following expression : R
H, =
The rate H, (cooling)
P(E)(E s -W.Y
- R)dE.
(2.7)
at which heat is removed from levels above R is given by cc
H,=
P(E)(E-R)dE.
(2.8)
s
R
The sum H= HI + H, is the net rate of removal of heat from the surface. If H
supply function
In a treatment of the non-equilibrium distribution function on electron emission, K. Lehovecs) introduced an fnonleq and expanded the function f(E) in Legendre polynomials with the field directions as the axis; retaining terms only to first order, he obtains: f(E)non-eq = fo (E) + fi (E) 4 3
(3.1)
where k is the electron wave vector and k, the component of the wave vector in the x-direction. Assuming a completely free non-interacting electron gas k,
=
px= in;
)
h where u, is x-component of the velocity. For sufficiently low fields within the metal, Lehovec standard expressions) for the non-equilibrium correction
(3.2)
uses the following to the distribution
ENERGY
EXCHANGE
PROCESSES
213
function; (3.3) z is a characteristic collision parameter. F’ is the effective field within the metal emitter. Since the externally applied field is rapidly damped within the metal, F’ can be taken to be the normal emission current density times the resistivity of the metal: F’ = Jp,
(3.4)
where it is to be noted that p=p(F’, T); having used the low field approximation, we assume p =p(T) only. The relaxation time, which describes phenomenologically the electron lattice interaction, is a complicated function of the electron wave vector, energy, and local temperature. Wilsonlo) using a nearly free electron model, has derived the following expression for the parameter: (3.5) 8 is the Debye temperature and m * the effective mass of the electron. We have assumed m * = m, the free electron mass. Although cyclotron resonance and De Haas Van-Alphen measurements of the effective mass in tungsten yield average values smaller than the free electron massrl), the difference is not significant within the order of approximation assumed in this calculation. _4 and A are defined by WilsoniO): (3.6) a is the lattice constant, M the ion mass and C is the electron lattice coupling constant, on the order of <. It is defined as (3.7) where u, are the Bloch functions and Y’ is the lattice potential. Since accurate calculations of the interaction constant for transition metals are not available, a reasonable estimate of C was inferred from calculated and experimentally determined values for other metals. From table (5.1) of Pinesla) and table IX 1 of WilsonrJ), it is seen that (CGheory and (CiQ+,. are of the order of unity, so Cx[ and ranges between l-10 eV. In selecting C and other values of the electronic para-
IRENE
214
M.ENGLE
AND P. H. CUTLER
meters, we have ignored all multi-band effects, consistent the free electron approximation. To compute the correction term in eq. (3.1) we use
with assumption
,,=)w- Y)=i(w+WJ, since V here is taken to be the average potential
of
(3.8)
inside the metal. Also
(3.9) Collecting
and substituting
in (3.1)-(3.3) (3.10)
where B=------
2h2e
(+m*)*
4. Calculation
A
e (3.11)
A k,’
of the inversion temperatures
For the calculations in this work we assume 4 =4.5 eV, which represents an average work function for a clean tungsten metal surface. W,= 10.2 eV and n =0.069 A0 were used in order to facilitate comparison with earlier calculations 2,10) and with experiment. The inversion temperatures were calculated by means of a trial and error method, i.e., seeking H(T,) =O. A low estimate of the inversion temperature is assumed for a particular value of the field. When using the non-equilibrium distribution, an initial value of the current density J is also introduced. The additional parameters introduced in the non-equilibrium case were chosen to be appropriate for tungsten, using tabulated or experimental values where possible. They are as follows a=3.16 A (ref. 13), C=[= 5.7 eV determined by W, and 4, the work function tungsten, 8=315”K (ref. 14), M=mass of tungsten atom.
for
The rates HI and If2 at which electrons with energies above or below, respectively, the average replacement energy R were calculated separately. The numerical calculation of HI and of J proceeds exactly as described in
ENERGY
EXCHANGE
PROCESSES
section 4.1 of ref. 2. D(W) was computed
using
215
the WKB
12
- logD(W)
=
8m
1’
S[ A2
approximation
(V(x) + Wa) dx.
XI
(4.1)
x1 and x2 are given by 10) cos(-@) + I W1/3eF - A cos [$(0 + rr)] + IWV3eF
A
(4.2) (4.3)
where
(4.4) and 8 = cos-1
>I’
3
l W12/3eF2
H2 is divided into two parts, infinite interval [R, co]
- e/4F
Hz,+ HzT to facilitate
computation
(4.5)
over the
a, HzF=
P(E)(E-R)dE
(4.6)
s R
and is calculated by a procedure identical to that used to calculate H. Since HzT gives a very small contribution to H in the given field and temperature range, it is sufficient to calculate it using a modified RichardsonDushman equation: m Hi,=:gJexp(-%)(E-R)dE.
(4.7)
0
Hz= = - R
>O
since
R
(4.8)
When the non-equilibrium distribution is used, values of the total emission current density are also calculated for each value of the trial T using the computed P(E), and D(W), and N( W, E) arrays: R
J = IeJ
s -W,
m
P(E)dE+lel
P(E)dE+ I
R
+
‘
(k,T - V,,,) exp
(4.9)
:
216
IRENE M. ENGLE
AND P. H. CUTLER
The sum H= H1 + Hz,+ HZT is computed and tested for algebraic sign. If H-CO, the temperature is known to be too low and the initial temperature is raised by a given amount for the next try; if H>O, Tis lowered for the next attempt. By a halving process, the inversion temperature can be calculated to within any degree of precision desired. In this work, the calculated Ti were determined to within 5 “K.
5. Results of the calculations Table 1 lists the calculated inversion temperatures using the equilibrium distribution for several assumed values of the replacement energy. We note first that the inclusion of the temperature dependence of the Fermi parameter as given by eq. (1.2) was found to have little effect upon the computed inversion temperatures, but the assumption that the average replacement energy R was a few hundredths of an eV less than zero yields significant lowering of the inversion temperatures. TABLE 1
Computed inversion temperatures
using an equilibrium supply function for different values of replacement energy
F (10-7
Ti R=
-4.50eV
3 4 5 6 7 8 9
R=
-4.52eV
R=
i =5(T)
V/cm) 762 1004 1238 1470 1701 1932 2180
715 955 1195 1420 1650 1880 2120
-4.52eV
i=i@) 710 951 1187 1420 1650 1880 2120
R=
-454eV i = i(T) 670 880 1130 1365 1590 1820 2057
R= -4.56eV c = L’(T) 583 826 1068 1300 1530 1760 1994
Negligible lowering of computed inversion temperatures was found when the non-equilibrium distribution given by eq. (3.9) was used. However, in the course of calculating the inversion temperatures using the non-equilibrium supply function, it was interesting to study the magnitude of the change in f as a function of field as fO+,fnon_eq. Since f,k, is energy dependent, the shifts were observed at one energy. Because afo/aE behaves somewhat like a delta function of width k,T about E=[, it was chosen as the energy at which to evaluate the change in .f for different values of externally applied fields. fnon_eq can be expressed fnon-eq = f(l
+ %=W=<,
217
ENERGYEiXCHANGEPROCESSES
where
p=p(T) was obtained by interpolating experimental values for tungsten15). The J used is the theoretically calculated value of the emission current density. It was obtained in the course of calculating q, using the method described in ref. 2. The T is the calculated value of inversion temperature for that particular external fiefd using the equilibrium supply function. The calculated values of S are listed in table 2. It is seen that in spite of appreciable changes in the distribution function, the effect is wholly inadequate to make the computed values of inversion temperature be in agreement with experiment. TABLET F
(Kk7V/cm) 3 4 .5 6 7 8 9
1.5 4.3 4.2 2.2 7.9 2.1
46 x 104 x 105 x 106 x 107 x 107 x 108
s
J
P
(A/cm)
(michrohmem) 17.5 25
32 39 47 54 63
762 1004 1238 1470 1701 $932 2180
5.5 1.5 3.5 3.0 1.4 4.7
x 10-3 x IO-4 x x0-3 x IO-2 x IO-’ x 10-l 1.1
The most recently computed inversion temperatures are shown with experimentally measured values in fig. 1. We may note that there is still considerable quantitative discrepancy between the theoretical values computed in the present work and the iatest experimental measurement, although the magnitude of the discrepancy has been halved. However, there also seems to be disagreement between experimental results carried out at different times by the same groupr*s). This is indicative of the difficulty involved in obtaining accurate and reproducible measurements of inversion temperatures; it also suggests the need to try to refine techniques, if possible, and/or make additional measurement so that at least random errors may be minimized in the analysis. It is to be stressed that the several corrections to the extant free-electron formulation introduced by the present treatment, i.e., (i) average energy of the replacement electronsin the conduction band different than Fermi energy,
218
IRENE
M. ENGLE
AND P. H. CUTLER
t TPK 1 R=(
to)=-4.Sev
1600--
1400--
1200-CHARBONNIER, STRAYER,SWANSON, AND MARTIN
lOOO--
800--
‘/ /*
600 t 400
SWANSON, CROUSER, AND CHARBONNIER
LX-
I
I
I
II
2
3
4
5 F*lO’(v/cm
6
I
I
7
8
9
I --c
Fig. 1. Comparison of experimental and computed inversion temperatures for tungsten and for different values of the average replacement energy R.The experimental points are those of refs. 1 and 3.
(ii) use of an non-equilibrium distribution function and (iii) use of a temperature dependent Fermi energy are purely phenomenological in character and of lowest order. Nevertheless, within these limitations, one can note that the calculated 6’s are large enough to reasonably suggest that a combination of the use of an fnon_eq and the assumption R < [ might produce significantly lower calculated inversion temperatures. Of preferential interest is the tentative conclusion that the basic assumption of the Nottingham effect, namely, that the average energy of the replacement electrons in the conduction band is equal to the Fermi energy is not valid at least for transition metals. Swanson et al. have, in the analysis and interpretation of their experimental data, emphasized the importance of this as a possible source of the disagreement. They have considered the relationship between R= (E’) and the absolute thermoelectric power of S of the metal, R = (E’) and have suggested
further
= E, - eTS,
theoretical
and experimental
(6.1) investigations
to
ENERCIY
EXCHANGE
219
PROCESSES
determine magnitude and tem~rature dependence of deviation of (E’) from the Fermi energy. That this procedure is probably feasible only experimentally becomes evident when we consider the expression for S (ref. 16) (for the simple isotropic case) S =;
~~~~~~~~=~,
(6.3
where the conductivity o(E) is given by
It is seen that the thermoelectric power depends critically on the energy dependence of the mean free path and effective mass of the electron, and the complicated band structure of tungsten woutd make any theoretical analysis of dubious value quantitativelylr). Thus, even though experiment and the present heuristic treatment compel belief, until the assumption R = (E’) < 5 is conctusively supported experimentally, rigorous theoretical analysis seems unwarranted at present. Acknowledgements The authors woufd Iike to express their appreciation for the cooperation of the staff of the Computation Center at the ~ennsy~~a~ia State University. Appendix We have generalized Young’s treatment71 for a God-equilibrium disiribution functionf( W, E). Using this notation, we have 101cos e
N(a,E)adE=noI-dndE=n(E)dE~sinBdBd~
Iv1COS8
where N(SZ, E) is the number of electrons of energy E which have X-cornponents of velocity lying between v cos 8 and o cos 8 + v cos (0 + de)
n( W, E) is the electron energy distribution with respect to E=O where the electron is at rest an infinite distance outside the metal; dW= mvXdx f dV.
220
IRENE
Since
V is taken
M. ENGLE
to be a constant,
AND P. H. CUTLER
- W,, inside
the metal,
0, = v cos 0 = &(2/m)
(E -
dV=O
V)] cos 8 )
Idu,j = v sin 8 de, ldW/
=
N(Q,E)dQdE=
mv2
cos
6
sin 8 dd ,
n(W,E)dE -4n
dW JPm
Also, since N(W,
E) =
s
(E -
VI
N(Q, E) d4,
0
we
have n(W,E)dEdW N(W,E)dWdE=p------m-c-lXJm(E - VII
4;3mf(W,E)dWdE.
References 1) L. W. Swanson, L. C. Crouser and F. M. Charbonnier, 2) I. Engle and P. H. Cutler, Surface Sci. 8 (1967) 288.
Phys. Rev. 151 (1966) 327.
3) F. M. Charbonnier, R. W. Strayer, L. W. Swanson and E. E. Martin, Phys. Rev. Letters 13 (1964) 397. 4) Erwin W. Miiller, Field Emission in: Annual Review of Physical Chemistry, Vol. 18 (1967) p. 35. 5) Frederick Seitz, Modern Theory of Solids (McGraw-Hill, New York, 1940) p. 149. 6) P. H. Cutler and J. J. Gibbons, Phys. 92 (1958) 1140. 7) R. H. Good and E. W. Miiller, Handbuch derphysik, Vol. 21, Ed. S. Fliigge (SpringerVerlag, Berlin, 1956) p. 181. 8) R. D. Young, Phys. Rev. 113 (1959) 110. 9) Kurt Lehovec, Phys. Rev. 96 (1954) 921. 10) A. H. Wilson, The Theory of Metals (Cambridge Univ. Press, New York, 1965) sections 9.3 and 9.5. 11) D. M. Sparlin and J. A. Marcus, Phys. Rev. 144 (1966) 484. 12) D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963) section 5.7. 13) Charles Kittel, Introduction ro Solid State Physics (Wiley, New York 1962) p. 41. 14) A. H. Wilson, The Theory ofMetals (Cambridge Univ. Press, New York, 1954) p. 143. 15) Handbook of Chemistry and Physics, 43rd ed. (Chemical Rubber Publ. Co., Cleveland, Ohio, 1961) p. 3044. 16) N. F. Mott and H. Jones, Theory of Mefals and Alloys (Dover Publ., New York, 1958) section 15.2. 17) J. M. Ziman, Theory of Phonons (Oxford Univ. Press, Oxford, 1960) section 9.11.