3718
J. Phys. Chem. 1984, 88, 3718-3722
Emergence of Photoelectrons from a Metal Surface into Liquid Argon: A Monte Carlo Treatment Augustine 0. Allen,+ Philip J. Kuntz, and Werner F. Schmidt* Hahn-Meitner-Znstitut fur Kernforschung Berlin, Bereich Strahlenchemie, D-1000 Berlin 39, Federal Federal Republic of Germany (Received: August 24, 1983; Zn Final Form: January 15, 1984)
Experimentally it was found by Tauchert et al. that the photoemission current from a metal cathode in liquid argon as a function of light frequency was related to the emission current in vacuo by a frequency-independent attenuation factor Y. Here we report on a Monte Carlo calculation of Y.Ten thousand electron trajectories were calculated for each chosen combination of electron energy and applied electric field strength. A constant mean free path was assumed. The escape probability of the electrons as function of electron kinetic energy was then computed. After a presumed distribution in electron energies from the illuminated cathode was taken into account, very good agreement was found with the experimental results.
Introduction Passage of electrons from a metal into a fluid or solid dielectric is a process of technological importance, yet theoretical understanding remains incomplete. Injection of electrons into simple gases or liquids has been investigated in detail by Silver and colleagues’s2and :hey explained their experimental results on the efficiency of injection by a diffusion model. Recently, their data on electron injection into gaseous hydrogen and argon have been discussed by two of the present authors in light of a Monte Carlo treatment of random scattering of electrons in a gas.3 Photoelectron emission from a metal cathode into nonpolar liquids has been studied extensively during recent years.“’ It was found that the threshold energy WllSwas shifted from its vacuum value W,, by an energy Vo,which was interpreted as the electron affinity of the liquid or as the bottom of the conduction band (cf. Figure 1 ) (1) = W“,, + Vo V, may be positive or negative. In these experiments, the current into a liquid illqat a given (hv - WlIq),light frequency v, light intensity Z, and applied electric field strength F, is considerably less than the current into the vacuum at (hv - W,,,) and Z by an attenuation factor Y = iliq/ivac (2) WIIS
which depends strongly on the nature of the liquid as well as on the experimental conditions. Though many data exist on Y values,6>’no quantitative prediction for them has hitherto succeeded. In the course of the experiments to measure Vo,Tauchert4 obtained data on Y for liquid argon at 85 K. In the present work, we calculate values of Y using our Monte Carlo trajectory model for different conditions and compare our results with Tauchert’s values.
Method of Calculation and Results The Monte Carlo trajectory model has been described in detaiL3 Here we give only a brief outline. The photoelectron emerges essentially perpendicular to the surface* ( x direction). It moves in a potential field having two components, both varying only in the x direction but in opposite senses (Figure 1 ) : the applied potential VA = -FAX (3) which tends to send the electron away from the surface, and the image potential V, = -e2/(16at0t,x) = - B / x (4) (FA is the applied electric field strength, e is the electronic charge, t Awardee of the Alexander von Humboldt-Stiftung, whose contribution to the support of this work is gratefully acknowledged.
0022-3654/84/2088-3718$01.50/0
and to and t, are the dielectric permittivity of vacuum and LAr, respectively) which tends to send it back. The potential is conventionally zero in the medium with no applied field and negative otherwise. The total potential V ( x ) acting on the electron is the sum V ( X )= -FAX - ( B / x )
(5)
At
where dV/dx = 0, V ( x ) has a maximum value V, = -2(BF,t,)’/’
(7)
Application of an electric field thus lowers the threshold energy wlq by A W = IVml
w,iq(FA)= W,,,(FA=O) - ~ ( B F A ) ’ ~ ’
(8)
A W is also called the Schottky correction. Emitted electrons are characterized by an initial excess kinetic energy E, over the barrier. In the course of their outward movement the electrons suffer numerous collisionsat each of which their total energy E,
E , = E, + V,
(9)
decreases slightly. At some point lEtl will be smaller than IWllql. If this happens at an x C x,, the electron returns to the cathode; if x > x, then the electron escapes to the anode. The main problem consists in the calculation of the fraction of electrons which escape to the anode as a function of excess kinetic energy E,. In our Monte Carlo method, we first calculate the trajectory of a single electron, as it moves in the field with “collisions” occurring at random intervals, each bringing about a new random (1) P. Smejtek, M. Silver, K. S . Dy, and D. G. Onn, J . Chem. Phys., 59, 1374 (1973). (2) P. Smejtek and M. Silver, J. Phys. Chem., 76, 3890 (1972). (3) P. J. Kuntz and W. F. Schmidt, J . Chem. Phys., 76, 1136 (1982); J. Electrostat., 12, 33 (1982). (4) W. Tauchert, H. Jungblut, and W. F. Schmidt, Can. J . Chem., 55, 1860 (1977); W. Tauchert, Thesis, Freie Universitat Berlin, 1975. ( 5 ) A. 0. Allen, Natl. Stand. ReJ Data Ser., Natl. Bur. Stand., No. US. Natl. Bur. Standards, 58 (1976). (61 R. A. Holroyd, B. K. Dietrich, and H. A. Schwarz, J . Phys. Chem., 76, 3794 (1972). (71 R. A. Holrord. S.Tames, and A. Kennedy, J . Phvs. Chem., 79, 2857 (1 975). (8) R. Williams, Semicond. Semimetals, 6 , 112 (1970)
0 1984 American Chemical Societv
The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 3719
Photoemission Current from a Metal to Liquid Argon
E
v
= I
wv
t
Ef metal
O
t
v
V I
i Cl
I liquid argon t 0.0 1
Figure 1. Energy levels at the metal/LAr interface. Upper part without applied electric field; vl, vacuum level; cl, conduction level; E", energy of an electron under vacuum; Ef,energy of the Fermi level. Lower part with applied electric field: -, total electrostatic potential acting on the electron; - - -, applied potential; ., image potential. TABLE I: Parameters Used in the Calculations 2000 WI1q9 eV 3.55 xo = B/Wlq," A 0.66 X, = ( B F ~ ) ~ 'A ~, 342 v, = 2(BF4)'lZ,eV 0.014 Ao, A 6.92 140 AI, A fractional kinetic energy 5.51 X loss ( ~ ~ / W ( A I / A O )
FA,V/cm 4000 3.55 0.66 242 0.019 7.38 140 5.17 X
6000 3.55 0.66 198 0.024 7.79 140 4.89 X
'x0 is the distance from the cathode where the electrons has the same energy as the Fermi level of the metal. It is assumed that at this distance the electron tunnels into the cathode.
direction making an angle I9 with the field. Prior to the first collision, the electron moves in a straight line in the x direction, but between collisions, it follows a C U N ~path in three dimensions. The probability that it travels a distance S without undergoing a collision and then collides within a further distance, dS, is e-slAldS/A,. The probability of its new initial direction is sin I9 dI9. At each collision, after determining the value of x (and therefore V), the computer generates two random numbers, rs and rB,lying between 0 and 1. The new path length S is -Al In (1 - rs) and the new angle I9 is c o d (2rB- 1). At each collision the electron loses on the average a fraction of its kinetic energy (Al/ho).(2rn/M)where rn and M are the mass of the electron and argon atom, respectively. Al and boare two mean free paths (see below). The new, slightly reduced value of the kinetic energy is calculated. Then, the curved path of the electron, as it moves in the field, is calculated in order to find the x coordinate at the TABLE II: Results of the Calculations FA = 2000 V/cm E;, eV F, error" clb 0.01 0.0710 f0.0050 12 f0.0048 0.02 0.0636 19 0.03 0.0504 f0.0043 22 0.04 0.0524 =t0.0044 26 0.0491 0.05 f0.0042 30 0.0418 10.0039 43 0.10 0.20 0.0382 f0.0038 56 0.0333 10.0035 64 0.30 0.0328 0.40 f0.0035 73
F, 444 676 825 960 1088 1537 2076 2537 2801
0.1163 0.0959 0.085 1 0.0843 0.0722 0.0607 0.0521 0.041 1 0.0398
1
Figure 2. Escape probability as a function of excess kinetic energy E,. Error bars give the 95% confidence interval: 0,2000 V/cm; 0,4000 V/cm; V, 6000 V/cm.
-
parameter
0.1
next collision. Eventually, if the electron in the course of its random motion has not returned to the cathode, its original excess energy E, will have been lost (total energy < V,). At this moment the computer notes which fate befell it (x > or < x,) and how many collisions were required to decide. The result is the fraction of escaped electrons p ( E , ) as a function of excess kinetic energy E,. The parameters we assumed in these calculations are given in Table I. For each of the three applied electric fields escape fractions were calculated for nine different initial excess kinetic energies E,; 10 000 trajectories were run for each condition, or 270 000 trajectories in all. The fraction of the trajectories which reach the anode (escape probability) is shown in Figure 2. Numerical details are compiled in Table 11. An "error range", which depends on the number of trajectories run, was calculated from statistical theory and gives the range in which 95% of the values should lie, if the whole procedure could be repeated many times. The escape probability is smaller the higher the initial kinetic energy. An electron that has crossed the barrier at x, on its way to the anode, and still has enough energy to surmount it, may at any time undergo a collision or series of collisions that send it back across the barrier into the cathode; the higher Ei,the longer it takes to lose its excess energy through hundreds or thousands of collisions, and the greater its chances of recrossing the barrier and eventually returning to the cathode. The propagation of the electron distribution in the absence of a potential function has been analyzed in our previous publication (p 1143 of ref 3). Given enough collisions all electrons which start from the cathode will return eventually making the escape probability zero. The effect of the applied potential is to increase the kinetic energy of the electrons. Since the energy loss in each collision is proportional to the kinetic energy fewer collisions are required at higher field strength to decide the fate of the electron (cf. Table 11, column
FA= 4000 V/cm erroro clb f0.0062 10 14 f0.0058 f0.0055 20 22 f0.0054 25 10.0051 f0.0047 34 48 *0.0044 51 f0.0039 64 10.0038
F A
czc 303 455 582 683 773 1130 1579 1925 2187
F, 0.1605 0.1328 0.1129 0.1014 0.0929 0.07 15 0.0609 0.0493 0.0461
= 6000 V/cm
error' f0.0072 f0.0067 f0.0062 f0.0059 f0.0057 f0.0051 f0.0047 f0.0042 f0.0041
clb 9 15 17 20 24 31 42 52 58
c2b ~
243 369 47 1 554 63 1 913 1332 1633 1882
O95% confidence interval. bAveragenumber of collisions required to decide the fate of these electrons which returned to the cathode. The same for those electrons which escaped to the anode.
Allen et al.
3720 The Journal of Physical Chemistry, Vol. 88, No. 17, 1984
0.2
0.1
"0
0.4
0.3
Ei [eV I Figure 5. Comparison of experimental photocurrent data of Figure 4 (2000 V/cm) and data obtained from Figure 3 with the calculated attenuation factor Y: - --, distribution function f l ; --., distribution function fi.
photon energy [eVI Figure 3. Photocurrent as a function of photon energy for a zinc cathode under vacuum at 85 K.
P
I
Figure 6. Comparison of experimental photocurrent data from Figure 4 (4000 V/cm) and the data obtained from Figure 3 with the calculated distribution attenuation factor Y: - - -, distribution function f l ; function fi. -.-e,
TABLE III: Empirical Relations between the Four Sets of Data W1iq -
.ob'0 3.4
I
I
I
I
3.6 3.8 4.0 photon energy [eVI
I
I
4.2
Figure 4. Photocurrent as a function of photon energy for a zinc cathode in LAr at 85 K: 0, 2000 V/cm; 0,4000 V/cm; V, 6000 V/cm.
c2). Those electrons not sent across the barrier in their first few collisions are likely to return to the cathode after very few collisions (cf. Table 11, column cl). The hypothesis advanced by Young and Bradburyg3lothat the fate of the electron is largely determined by the direction of its (9) L. A. Young and N. E. Bradbury, Phys. Reu., 43, 34 (1933). (10) See also M. Silver and M. P. Shaw in "Photoconductivity and Related Phenomena", J. Mort and D. M. Pai, Eds., Elsevier, Amsterdam, 1976, Chapter I.
field strength, V cm-l vacuum (LAr) 2000 (LAr) 4000 (LAr) 6000
W, eV
W",,, eV
3.763 3.590 3.560 3.540
-0.17 -0.20 -0.22
AW 0.014 0.019 0.024
Vo," eV -0.16 -0.180 -0.20
(RLd/
Rvac)OS 0.18 0.21 0.24
"Average Vo = -0.18 & 0.02.
first collision clearly does not apply in LAr.
Comparison with Experiment and Discussion In Tauchert's experiment4 a photocathode of vacuum-deposited zinc was illuminated at a number of nearly monochromatic wavelengths in the region 200-300 nm, obtained from a xenon lamp. The relative output of photons at each chosen wavelength was known, and the data consisted of values of the ratio R = i / I of the emitted electric current i to the relative photon output I; the units of R are arbitrary but absolute with respect to the different sets of data. The dependence of R on photon energy is shown in Figure 3 for emission in vacuum and in Figures 4-6
The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 3721
Photoemission Current from a Metal to Liquid Argon
I
I 0.1
I
I
0.3
0.4
1
0.2 E i [eVl
Figure 7. Comparison of experimental photocurrent data from Figure 4 (6000 V/cm) and the data obtained from Figure 3 with the calculated attenuation factor Y: - - -, distribution function fi; distribution -e--,
function fi. for emission in LAr at 85 K for three different field strengths. The solid lines were calculated by a linear regression analysis. Quantities extracted from these results are compiled in Table 111. All data follow above the threshold the general relationship between photocurrent and (hv - W) which was derived by Fowler [cf. ref 111 R = K(hv -
W)’
(10)
This allows the introduction of a frequency-independent attenuation factor Y given as the ratio or, in other words, the experimental results for LAr follow from the vacuum data if Y can be calculated. The emitted electrons at each frequency are characterized by an energy distributionf(EJ which has been found to depend on the chemical and structural properties of the photocathode and on the magnitude of (hv - W). While for a thick silver cathode a bell-shaped distribution was measured, the distribution from a thick, frosty sodium surface was more triangular, rising linearly from zero to an energy close to (hv - W)and then falling off rapidly.12 If we take into account the energy distribution of the electrons f(Ei) the attenuation factor Y is then given by Y = J‘hFwf(Ei) p ( E i ) dEi Numerical evaluation of the integral in (12) was performed with the values of p ( E i ) given in Figure 2 and with two assumed normalized distribution functions f,and f2given by
h(T)= 2 T
(13)
with T = Ei/(hv - W). In Figures 5-7 calculated photocurrents are compared with the experimental data. The calculated results fit the experiment much better than might be expected. The calculated R’/2values are closely linear in energy and the absolute values agree better than 20% over most of the photon energy range. One uncertainty in the comparison stems from the error in the determination of Wliq. There may be, (1 1) J. C. Riviere in “Solid State Surface Science”, Vol. 1, M.Green, Ed., Marcel Dekker, New York, 1969, Chapter 4. (12) G.L. Weissler, “Encyclopedia of Physics” Vol. 21, S. Fliigge, Ed., Springer-Verlag, Berlin, 1956, p 304.
however, a fundamental deficiency in the calculated escape probability, which may result in somewhat too large values at excess kinetic energies below 0.1 eV. The mean free path A, taken in our calculation stems from a theory of electron mobility in LAr developed by Lekner and C~hen.’~,’~ Lekner calculated the potentials acting on an excess electron moving in liquid argon at 85 K and succeeded in relating its scattering in the liquid to that occurring in argon gas at low pressure. His results are presented in terms of two fictitious “mean free paths” A. and AI for energy and momentum transfer, respectively. The meaning of these numbers is that the electron behaves as if it suffers a “collision” involving a random change in direction after moving freely in the applied field a random distance, averaging to AI, from the previous “collision”. It loses energy at a rate such that for each A. of its path it loses on the average a fraction of its kinetic energy 2 m / M = 2.723 X We simplified this description by assuming that at each direction-changing collision a fractional energy loss of (A, /A0)(2m/M) occurs. This way we eliminated the need for a second randomization of &. Our experiments were done at applied field strengths of 2, 4, and 6 kV/cm; Lekner’s values of A I extracted from mobility data were 1.40 X 10” cm throughout this range, but his A. was 6.7 X and 8.7 X cm at respective fields of 1 and 10 kV/cm; we used interpolated values for Ao. Lekner’s theory described reasonably well the dependence of the electron drift velocity on the electric field strength at 84-85 K. Later, mobilities as a function of temperature and density became available15and Lekner’s theory failed to predict the correct magnitude of the mobility and its variation with density, especially near the critical temperat~re.’~,”Recently, a new theoretical concept was presented by Basak and Cohen18 who applied deformation-potential theory to the excess electron mobility in LAr. A correlation between Voand electron mobility was derived which, however, predicts much too low values of the electron mobility in the region of the critical point and it underestimates the electron mobility at the triple point as well, if experimental Vovalues are u ~ e d . ’ ~It, ~seems ~ that at present no theoretical model exists which describes satisfactorily the electron mobility as a function of density. Furthermore, no attempt has been made by the various critics of the Lekner approach to improve on the theoretical description of the field strength dependence. Recently, Lekner’s calculation of the electron mean energy as a function of applied field strength could be checked vs. experimental values and surprisingly good agreement was found.21 Considering all factors and the results of the present work, we are led to conclude that Lekner’s treatment represents a good approximation at the triple point. Since our Monte Carlo trajectory calculations are very time consuming, no other set of AI, A. values was tried. From the present results it is difficult to predict how p(E,,F) would be affected by a different set of parameters. In our previous work we explored the influence of the quantity ua ua = F / ( N E , )
(15)
( N is the number density of argon atoms and u the collision cross section) on the number of collisions A2 required to bring the electron energy below V, (cf. Table 1 of ref 3). It was found that with increasing ua a smaller number of collisions was needed. (13) J. Lekner and M. H. Cohen, Phys. Reu., 158, 305 (1967). (14) J. Lekner, Phys. Reu., 158, 130 (1967). (15) J. A. Jahnke, L. Meyer, and S. A. Rice, Phys. Reu. A , 3,734 (1971). (16) J. A. Jahnke, N. A. W. Holzwarth, and S . A. Rice, Phys. Reu. A , 5 , 463 (1972). (17) J. Gryko and J. Popielawski, Phys. Rev. A , 16, 1333 (1977). (18) S. Basak and M. H. Cohen, Phys. Reu. B, 20, 3404 (1979). (19) R. Reininger, U. Asaf, I. T. Steinberger, P. Laporte, S. Bernstorff, and V. Saile, Ann. Isr. Phys. Sor., 6, 2821 (1983). (20) R. Reininger, V. Asaf, I. T. Steinberger, and S. Basak, Phys. Rev. B., in press. (21) E. M. Gushchin, A. A. Kruglov, and I. M. Obodovskii, Sou. Phys. JETP, 55, 650 (1982).
J . Phys. Chem. 1984, 88, 3722-3126
3722 TABLE I V Estimates of the Influence of A I F on Y F, V/cm 2000 4000 AI, A AIF Y(EjzO.1)
140 2.80 x 10-3 0.23
AI, A AIF Y(,q=O.l)
70
70
1.4 x 10-3
2.4 x 10-3 0.225
Ai,
A
AlF
Y(Ei=O.l)
6000
Set lo 140 5.6 x 10-3 0.28
140 8.4 x 10-3 0.32
Set 2 0.21
70 4.2 x 10-3 0.25
Set 3 280 5.6 x 10-3 0.28
290 11.2 x 10-3 0.365
290 16.8 x 10-3
0.46
’Best fit of experimental data by distributionfl. Furthermore, it could be seen that A2 depended only weakly on N . We are led to conclude that at the LAr density A2 is a function of aLAr = A I F / E only. Although we do not know the exact dependence of p(Ei,F) on A2 we may assume that a L A r is the physical quantity relating the influence of A I and F on p(Ei,F) or, in other words, p(Ei,F) is a function of “LAr only. In order
to estimate the influence of the variation of A, on p(Ei,F), we as a function of cyLAr. A straight line was plotted (RLAr/Rvac)o.S obtained Y=
= 0.185
+ 16AlF
(16)
We do not know how Y(LYLA,)will behave for small and larrge LY values, but taking the linear approximation we find the following variations given in Table IV. Although the variation of Y with A, is not very pronounced, we see that doubling the value of A, (set 3) leads to differences between measured values and estimated values especially at 6 kV/cm which are outside of the error limits. Taking a smaller AI value (set 2) leads to a reduction of Y although less pronounced. Summarizing, we found that our trajectory model allowed a direct calculation of the photoemission yield in LAr with a set of parameters, which were derived from mobility data by means of Lekner’s model. Variation of A, by more than a factor of 2 (or leads to discrepancies between the calculated and observed yields. This means that the chosen values of A, and Ao, obtained from mobility data, are applicable to Tauchert’s experiments and that large deviations in these values would lead to internal inconsistencies. Registry No. Argon, 7440-37-1.
Electron Mobility Calculations in Liquld Xenon by the Method of Partial Waves Akos Vertest Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: August 25, 1983; In Final Form: November 8, 1983)
The method of partial waves in the framework of the fluctuation model has been applied to describe the density dependence of the electron mobility in liquid xenon. The limiting factor in the transport of excess electrons in such systems can be attributed to their scattering due to density fluctuations. Instead of the usual Born approximation we handled the scattering problem with the more general method of partial waves. This made it possible to work out the average fluctuation size from the transport data. Calculations were carried out in the density range (6.7-11.5) X lo2’ atoms/cm3. Good agreement with measured mobility curves can be obtained by considering reasonable values for the average fluctuation size.
1. Introduction The relatively high mobility of excess electrons in insulating liquids such as some hydrocarbons, argon, krypton, and xenon is a fascinating challenge for theoreticians. Just as in the theory of liquid metals,’ the main question involves the existence and description of the quasifree state in view of the lack of translational symmetry in the liquid. Nevertheless insulating liquids differ remarkably from liquid metals because of their different band structure, mostly due to the presence of a band gap in the insulators. The mobility of a quasifree electron is governed by the scattering mechanisms which limit its drift velocity to a relatively small value. The search for possible scattering mechanisms made it clear that considering a single atom or molecule as a scattering center leads to a limited understanding of the conduction phenomena. In contrast to this single scatterer approximation, especially at higher densities multiple scattering must play an important role. Electron mobility calculations have been carried out in liquid hydrocarbons by Berlin, Nyikos, and Schiller with the fluctuation model2 and by Basak and Cohen in liquid argon with the deformation potential model? Both theories account for the multiple scattering of the quasifree electrons in a similar manner, namely, using a spherical potential derived from the density fluctuations ‘Address correspondence to the author at the Central Research Institute for Physics, P.O.Box 49, H-1525 Budapest, Hungary.
0022-3654/84/2088-3722$01.50/0
of the liquid. This potential can be visualized as a perturbation of the energy of the bottom of the conduction band, AVO,rather than some kind of superposition of interaction potentials from individual particles. This perturbation energy can be expanded in a series in terms of density fluctuation. Both theories then handle the scattering problem with the aid of the Born approximation and are restricted to the vicinity of the mobility maximum. In order to describe many other hydrocarbons, helium, neon, and hydrogen another electronic state has to be considered, a slowly moving localized state with a mobility comparable to that of heavy ions. This state can be understood in terms of the Springett, Jortner, and Cohen theory of electronic bubbles4 A further question arises about the mechanism of formation of such localized states in a liquid. Experimental investigation of picosecond dynamics of electron transfer5 indicates two possible stages of electron localization, first transition from the quasifree state to a bound state and then relaxation of the liquid structure around the electron. This second process has been recently investigated (1) J. M. Ziman, Phil. Mug.,6 , 1013 (1961). (2) Yu.A. Berlin, L. Nyikos, and R. Schiller, J . Chem. Phys., 69,2401 (1978). (3) S. Basak and M. H. Cohen, Phys. Rev. E , 20, 3404 (1979). (4) B. E. Springett, J. Jortner, and M. H. Cohen, J. Chem. Phys., 48,2720 (1968). (5) G. A. Kenney-Wallace, G. E. Hall, L. A. Hunt, and K. Sarantidis, J . Phys. Chem., 84, 1145 (1980).
0 1984 American Chemical Society
