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Effect of a modified potential barrier in field emission
Solid State Communications,Vol. 10, PP. 1175—1179, 1972.
Pergamon Press.
Printed in Great Britain
EFFECT OF A MODIFIED POTENTIAL BARRIER IN FIELD EMISSION Tien Tsai Yang School of Engineering and Applied Science, University of California, Los Angeles, California 90024 and Jan Jan Yang Autonetics Division, North American Rockwell Corporation, Anaheim, California 92803
(Received 17 January 1972 by R Barrze
The use of a modified surface potential barrier, which includes not only classical image force and the external force but also the exchange and correlation forces on the field emission has been investigated for tungsten. The theoretical calculations of the emitted current density and total energy distribution are obtained. A comparison of the experimental results with the theoretical predictions has been made. The agreement between them is good.
INTRODUCTION THE THEORY OF field emission was developed by Fowler and Nordheim’ who used a Sommerfeld— Hartree model to describe the metal. The conduction electrons were treated as a degenerate Fermi— Dirac gas. Fowler and Nordheim made the simplifying assumption of a one dimensional barrier problem. In their original calculation they used a step function for the surface potential. Since a step function is a rather crude description, Nordheim2 replaced it by the classical image— force potential. The inclusion of the image force does indeed lead to a calculated value of the current density which differs in magnitude from the step function barrier, An investigation of the validity of the Fowler—Nordheim field emission theory has been made by Barbour et al. ~ Tney found for fields 5 x i0~ V/cm the observed currents deviate from the linear log j vs. 1/ F curve, becoming progressively less than predicted 1175
values as the field increases. For a field of be V/cm, the deviation is about 5%. A possible source of the disagreement might have been the validity of the classical image force potential model used in the Fowler and Nordheim theory 4 developed a model for the Cutler and Gibbons surface potential barrier based on the quantum mechanical calculation made by Bardeen5 on the form of the potential at the surface of a sodium like metal and the analysis of Sachs and Dexter6 on the quantum limits of the image force theory. Employing this model the emitted current and total energy distribution were calculated. The results were found to be in better agreement with the experimental data. However, a small deviation still exists between the theoretical and the experimental data. In their work on the periodic deviation in the Schottky effect, Yang and Yang7’8 proposed a surface potential model which included not only the classical image force and the external force but also the exchange and correlation forces. The physical basis of this model has been discussed in detail in reference 7 and will not be repeated here. The potential has
1176
MODIFIED POTENTIAL BARRIER IN FIELD EMISSION
the form
Vol. 10, No. 12
times the probability of penetration. We shall
V(x)
=
b
1
—
2(x \
+ ._~.....\
2~/
(x
-~.
\
use the total energy representation the energy associated with motion inand theletx w be
2
_J._~ 2w~/
direction
w —
2t’x
+ °
V(x)
=
=
—
k~ k~ = k~+ V(x) —
(1)
~
.i_\ 2~/
is the total energy of the electron and k’s are
0 for x <0
(2)
V° is the potential difference between a point outside the surface and a point in the interior. The second term of equation (1) represents the classical image force energy; the third term represents the exchange and correlation energies; the last term is the external applied energy. Wa is the summation of the work function and the difference between the Fermi energy and the bottom of conduction band, i.e., Wa = 8 + (EF— Er). (x 0 + I / 2w~)is the position of the maximum of the potential barrier. b is the coefficient of the exchange and correlation energy, depending on the properties of the metal. For V(0) = 0, it is found that ~O ~‘a — 4bw~.2 Figure 1 is the potential barrier model [expressed as equations (1) and (2)1 that will be used in the present analysis.
the wave vectors of the electron. We also define the following quantities : N(W)dW = number of . . electrons with energy ~4in dM incident on the 2 per sec. surface per cm D(W) = probability that an electron of energy W will penetrate the barrier. Then ~ dW N D H~dW 3 =
is the number of electrons with energy in the range dW that emerge from the metal per cm2 per sec. The current density is obtained from (3)
j
e
=
J D(W)N(Vi)dV~
By the WKB approximation, for W < V,~, the trans9 is mission coefficient D(l~) = exp — ~ (V(X) — 2 dx~ (5) x, where x~and x 2 are the roots of the radicand and X~< Xa and m is the mass of the electron. Usin g the modified potential of equation (1) for
f[
V(x), -
I
(4)
W)1
X2
D(U)
=
exp
-
j
f
.!!~i(Vo_ 2(x+1 ...L.) 2I~a
//////.._r.~....\\S\\\\\\
2
ELc~4~W~
DISTANCE
FiG. 1. Model of the surface potential barrier for a metal with an external applied field.
It. CALCULATION AND DISCUSSION
2~x~ +X_L)2_~J~
(x+_L)
dx) (6)
The integral in equation (6) is a Jacobean elliptic integral which can be solved by a standard method,1° Since field emission electrons come from the neighborhood of the apparent Fermi energy E~,D(~)can be expended in a Taylor series W = E~= EF — E,, — 4bWa2. Retaining only the first two terms we obtain:
(a) The current density calculation
D(H)
=
exp{
— P
— q (W
-
E~)]
(7)
where The current density can be computed by con sidering the flux of electrons incident on the barrier
P
=
—
(log D~=
(8)
Vol. 10, No. 12
q
=
MODIFIED POTENTIAL BARRIER IN FIELD EMISSION
{
(b) The total energy distribution —
~
(logD)]w
=
(9)
E.
Both parameters P and q in equations (8) and (9) depend on the value of b, the coefficient of the exchange and correlation energies. For a Fermi gas”
N(W)
=
1177
47TmKT log[1 h3
+
exp
(
—
—
E.)) ‘KT]
Similar to the expression of the current density, the total energy distribution 14
P() d
=
$
N(,W)D(W)ddW
(14)
w
N(,W)dI4d represents the number of electrons with total energy C in dE and in dW incident on 2 per sec. ~ And the barrier per cm
N(,W)
_.i~!f()
(15)
h3 (10) where By the low temperature approximation, N(H) can be expressed :
=
N(W) (W
—
0 for W>E~. E~.)for W < E-
~(~) =
e~~T+l
= —
.
(11)
is the Fermi—Dirac distribution function.
Substituting equations (7) and (11) into (4) =
J
477me e-~ (W E~.)~ h3 E~~-
Substituting equation (15) into equation (14)
d~
—
i
.
and integrating it, the total energy distribution is
(12)
then P() de
e-P ~
=
f() d
qh3 Integrating, (16) j
(0)
=
j (0) denotes j at T
q2h3
=
exp
(_
P)
(13)
o.
In reference 7 of the periodic thermionic emission study, b was determined to be 3/64 ~ for tungsten, where W~= 8 +(EF— E~)= 6.1eV which was obtained by using the band structure calculations of Mattheiss.’2 Numerical values of j were calculated for tungsten by using b = 3/64 ~ = 6.1 eV and fields up to 10e V/cm. The results are plotted in Fig. 2; the curve of Cutler and Nagy 13 as well as the calculations based on the classical image force potential are shown in the same figure. A 4.6 per cent deviation from the classical image force theory at F = 10e V/cm can be seen from Fig. 2. This close agreement between our results and the experimental data again confirms the value of b obtained in the periodic Schottky deviation of the thermionic emission study.
The calculation of the total energy distribu-
tion for tungsten based on equation (16) is plotted along with the experiment results in Fig. 3. The curve is computed for F = 3 x i0~V/cm, 8 = 4.4eV, b = 3164 Wa and Wa = 6.1eV and is normalized with respect to those of Young and Muller’4 for T = 300°K. The results based on the classical image force theory and the Cutler and Nagy’s ~ model are also included for comparison. A close agreement between the present calculation and the experiments can be seen in Fig. 3.
Although better agreement is obtained with a modified surface potential it is still a crude approximation to the real surface barrier. The results obtained in the present analysis do not imply that the modified potential is necessarily a correct and unique model to be used in treating the surface phenomena.
1178
MODIFIED POTENTIAL RARRIER IN FIELD EMISSION
Vol. 10. No. 12
IMAGE POTENTIAL
8 CUTLER& NAGY PRESENT RESULT
-,
o -J
6~
4.
3
I
1
2
3
108/fr I V/CM
FiG. 2. Comparison of the field emission current density for three different models. 1.0~
•
PRESENT RESULT
//,r
//.
EXPERIMENTAL DATA
N 6
-
CUTLER
AND NACV
MACE POTENTIAL
—
/N’ ~
N NN
NN
.4
1~
;// \~ \
/i:
‘-I.
///
/~
ENERGy leVI
FIG. 3. Comparison of the experimental total energy distribution of field emitted electrons at room temperature with the theoretical distributions for three different models.
FEFERENCES FOWLER RH. and NORDHEIM LW., Proc. R. Soc. (London) A119, 173 (1928). 2. NORDHEIM LW., Proc. R. Soc. (London) A121, 173 (1928). 3. BARBOUR J.B., DOLAN W.W., MARTIN E.E. and DYKE W.P., Phys. Rev. 92, 45 (1953). 4.
CUTLER P.H. and GIBBONS RH., Phys. Rev. 111, 394 (1958).
5.
BARDEEN
6.
SACHS R.G. and DEXTER DL., J. app!. Phys. 21, 1304 (1950).
J.,
Phys. Rev. 49, 653 (1936).
Vol. 10, No.12
MODIFIED POTENTIAL BARRIER IN FIELD EMISSION
8.
YANG J.J. and YANG T.T., Phys. Rev., Bi, 3614 (1970). YANG 1•J. and YANG T.T., Phys. Rev. Bi, 4270 (1970).
9.
GOOD RH. and MULLER E.W., Handbuch der Physik, Vol. 21, p. 181, Springer, Berlin, (1956).
7.
10.
BYRD P.F. and FRIEDMAN M.D.. Handbuch of Elliptic Integrak for Engineers and Physicists
11. 12.
p. 77, Springer, Berlin, (1954). STRATTON P., J. Phys. Chern. Solids 23, 1177 (1962). MATTHEISS L.F., Phys. Rev. 151. 450 (1966).
13.
CUTLER PH. and NAC-Y D., Surf. Sci. 3. 71(1964).
14.
YOUNG RD. and MULLER E.W., Phvs. Rev., 113. 115 (1959).
Eine modifizierte Oberflachenpotentialschwelle, die nicht nur die
klassische Spiegelkraft und die ~usserliche Kraft sondern auch die Aüstausch und Korrelationskräfte auf die Feldemission einschliesst, wurde für Wolfram untersucht. Die theoretischen Berechnungen der Emissionsstromdichte und der totalen Energieverteilung wurden ausgefurht. Die experimentellen Ergebnisse zeigen eine gute Ubereinstimmung mit den theoretischen Veraus sagen.
1179
Pergamon Press.
Printed in Great Britain
EFFECT OF A MODIFIED POTENTIAL BARRIER IN FIELD EMISSION Tien Tsai Yang School of Engineering and Applied Science, University of California, Los Angeles, California 90024 and Jan Jan Yang Autonetics Division, North American Rockwell Corporation, Anaheim, California 92803
(Received 17 January 1972 by R Barrze
The use of a modified surface potential barrier, which includes not only classical image force and the external force but also the exchange and correlation forces on the field emission has been investigated for tungsten. The theoretical calculations of the emitted current density and total energy distribution are obtained. A comparison of the experimental results with the theoretical predictions has been made. The agreement between them is good.
INTRODUCTION THE THEORY OF field emission was developed by Fowler and Nordheim’ who used a Sommerfeld— Hartree model to describe the metal. The conduction electrons were treated as a degenerate Fermi— Dirac gas. Fowler and Nordheim made the simplifying assumption of a one dimensional barrier problem. In their original calculation they used a step function for the surface potential. Since a step function is a rather crude description, Nordheim2 replaced it by the classical image— force potential. The inclusion of the image force does indeed lead to a calculated value of the current density which differs in magnitude from the step function barrier, An investigation of the validity of the Fowler—Nordheim field emission theory has been made by Barbour et al. ~ Tney found for fields
values as the field increases. For a field of be V/cm, the deviation is about 5%. A possible source of the disagreement might have been the validity of the classical image force potential model used in the Fowler and Nordheim theory 4 developed a model for the Cutler and Gibbons surface potential barrier based on the quantum mechanical calculation made by Bardeen5 on the form of the potential at the surface of a sodium like metal and the analysis of Sachs and Dexter6 on the quantum limits of the image force theory. Employing this model the emitted current and total energy distribution were calculated. The results were found to be in better agreement with the experimental data. However, a small deviation still exists between the theoretical and the experimental data. In their work on the periodic deviation in the Schottky effect, Yang and Yang7’8 proposed a surface potential model which included not only the classical image force and the external force but also the exchange and correlation forces. The physical basis of this model has been discussed in detail in reference 7 and will not be repeated here. The potential has
1176
MODIFIED POTENTIAL BARRIER IN FIELD EMISSION
the form
Vol. 10, No. 12
times the probability of penetration. We shall
V(x)
=
b
1
—
2(x \
+ ._~.....\
2~/
(x
-~.
\
use the total energy representation the energy associated with motion inand theletx w be
2
_J._~ 2w~/
direction
w —
2t’x
+ °
V(x)
=
=
—
k~ k~ = k~+ V(x) —
(1)
~
.i_\ 2~/
is the total energy of the electron and k’s are
0 for x <0
(2)
V° is the potential difference between a point outside the surface and a point in the interior. The second term of equation (1) represents the classical image force energy; the third term represents the exchange and correlation energies; the last term is the external applied energy. Wa is the summation of the work function and the difference between the Fermi energy and the bottom of conduction band, i.e., Wa = 8 + (EF— Er). (x 0 + I / 2w~)is the position of the maximum of the potential barrier. b is the coefficient of the exchange and correlation energy, depending on the properties of the metal. For V(0) = 0, it is found that ~O ~‘a — 4bw~.2 Figure 1 is the potential barrier model [expressed as equations (1) and (2)1 that will be used in the present analysis.
the wave vectors of the electron. We also define the following quantities : N(W)dW = number of . . electrons with energy ~4in dM incident on the 2 per sec. surface per cm D(W) = probability that an electron of energy W will penetrate the barrier. Then ~ dW N D H~dW 3 =
is the number of electrons with energy in the range dW that emerge from the metal per cm2 per sec. The current density is obtained from (3)
j
e
=
J D(W)N(Vi)dV~
By the WKB approximation, for W < V,~, the trans9 is mission coefficient D(l~) = exp — ~ (V(X) — 2 dx~ (5) x, where x~and x 2 are the roots of the radicand and X~< Xa and m is the mass of the electron. Usin g the modified potential of equation (1) for
f[
V(x), -
I
(4)
W)1
X2
D(U)
=
exp
-
j
f
.!!~i(Vo_ 2(x+1 ...L.) 2I~a
//////.._r.~....\\S\\\\\\
2
ELc~4~W~
DISTANCE
FiG. 1. Model of the surface potential barrier for a metal with an external applied field.
It. CALCULATION AND DISCUSSION
2~x~ +X_L)2_~J~
(x+_L)
dx) (6)
The integral in equation (6) is a Jacobean elliptic integral which can be solved by a standard method,1° Since field emission electrons come from the neighborhood of the apparent Fermi energy E~,D(~)can be expended in a Taylor series W = E~= EF — E,, — 4bWa2. Retaining only the first two terms we obtain:
(a) The current density calculation
D(H)
=
exp{
— P
— q (W
-
E~)]
(7)
where The current density can be computed by con sidering the flux of electrons incident on the barrier
P
=
—
(log D~=
(8)
Vol. 10, No. 12
q
=
MODIFIED POTENTIAL BARRIER IN FIELD EMISSION
{
(b) The total energy distribution —
~
(logD)]w
=
(9)
E.
Both parameters P and q in equations (8) and (9) depend on the value of b, the coefficient of the exchange and correlation energies. For a Fermi gas”
N(W)
=
1177
47TmKT log[1 h3
+
exp
(
—
—
E.)) ‘KT]
Similar to the expression of the current density, the total energy distribution 14
P() d
=
$
N(,W)D(W)ddW
(14)
w
N(,W)dI4d represents the number of electrons with total energy C in dE and in dW incident on 2 per sec. ~ And the barrier per cm
N(,W)
_.i~!f()
(15)
h3 (10) where By the low temperature approximation, N(H) can be expressed :
=
N(W) (W
—
0 for W>E~. E~.)for W < E-
~(~) =
e~~T+l
= —
.
(11)
is the Fermi—Dirac distribution function.
Substituting equations (7) and (11) into (4) =
J
477me e-~ (W E~.)~ h3 E~~-
Substituting equation (15) into equation (14)
d~
—
i
.
and integrating it, the total energy distribution is
(12)
then P() de
e-P ~
=
f() d
qh3 Integrating, (16) j
(0)
=
j (0) denotes j at T
q2h3
=
exp
(_
P)
(13)
o.
In reference 7 of the periodic thermionic emission study, b was determined to be 3/64 ~ for tungsten, where W~= 8 +(EF— E~)= 6.1eV which was obtained by using the band structure calculations of Mattheiss.’2 Numerical values of j were calculated for tungsten by using b = 3/64 ~ = 6.1 eV and fields up to 10e V/cm. The results are plotted in Fig. 2; the curve of Cutler and Nagy 13 as well as the calculations based on the classical image force potential are shown in the same figure. A 4.6 per cent deviation from the classical image force theory at F = 10e V/cm can be seen from Fig. 2. This close agreement between our results and the experimental data again confirms the value of b obtained in the periodic Schottky deviation of the thermionic emission study.
The calculation of the total energy distribu-
tion for tungsten based on equation (16) is plotted along with the experiment results in Fig. 3. The curve is computed for F = 3 x i0~V/cm, 8 = 4.4eV, b = 3164 Wa and Wa = 6.1eV and is normalized with respect to those of Young and Muller’4 for T = 300°K. The results based on the classical image force theory and the Cutler and Nagy’s ~ model are also included for comparison. A close agreement between the present calculation and the experiments can be seen in Fig. 3.
Although better agreement is obtained with a modified surface potential it is still a crude approximation to the real surface barrier. The results obtained in the present analysis do not imply that the modified potential is necessarily a correct and unique model to be used in treating the surface phenomena.
1178
MODIFIED POTENTIAL RARRIER IN FIELD EMISSION
Vol. 10. No. 12
IMAGE POTENTIAL
8 CUTLER& NAGY PRESENT RESULT
-,
o -J
6~
4.
3
I
1
2
3
108/fr I V/CM
FiG. 2. Comparison of the field emission current density for three different models. 1.0~
•
PRESENT RESULT
//,r
//.
EXPERIMENTAL DATA
N 6
-
CUTLER
AND NACV
MACE POTENTIAL
—
/N’ ~
N NN
NN
.4
1~
;// \~ \
/i:
‘-I.
///
/~
ENERGy leVI
FIG. 3. Comparison of the experimental total energy distribution of field emitted electrons at room temperature with the theoretical distributions for three different models.
FEFERENCES FOWLER RH. and NORDHEIM LW., Proc. R. Soc. (London) A119, 173 (1928). 2. NORDHEIM LW., Proc. R. Soc. (London) A121, 173 (1928). 3. BARBOUR J.B., DOLAN W.W., MARTIN E.E. and DYKE W.P., Phys. Rev. 92, 45 (1953). 4.
CUTLER P.H. and GIBBONS RH., Phys. Rev. 111, 394 (1958).
5.
BARDEEN
6.
SACHS R.G. and DEXTER DL., J. app!. Phys. 21, 1304 (1950).
J.,
Phys. Rev. 49, 653 (1936).
Vol. 10, No.12
MODIFIED POTENTIAL BARRIER IN FIELD EMISSION
8.
YANG J.J. and YANG T.T., Phys. Rev., Bi, 3614 (1970). YANG 1•J. and YANG T.T., Phys. Rev. Bi, 4270 (1970).
9.
GOOD RH. and MULLER E.W., Handbuch der Physik, Vol. 21, p. 181, Springer, Berlin, (1956).
7.
10.
BYRD P.F. and FRIEDMAN M.D.. Handbuch of Elliptic Integrak for Engineers and Physicists
11. 12.
p. 77, Springer, Berlin, (1954). STRATTON P., J. Phys. Chern. Solids 23, 1177 (1962). MATTHEISS L.F., Phys. Rev. 151. 450 (1966).
13.
CUTLER PH. and NAC-Y D., Surf. Sci. 3. 71(1964).
14.
YOUNG RD. and MULLER E.W., Phvs. Rev., 113. 115 (1959).
Eine modifizierte Oberflachenpotentialschwelle, die nicht nur die
klassische Spiegelkraft und die ~usserliche Kraft sondern auch die Aüstausch und Korrelationskräfte auf die Feldemission einschliesst, wurde für Wolfram untersucht. Die theoretischen Berechnungen der Emissionsstromdichte und der totalen Energieverteilung wurden ausgefurht. Die experimentellen Ergebnisse zeigen eine gute Ubereinstimmung mit den theoretischen Veraus sagen.
1179