9, No. 10/October 1992. Photoelectric effect from a metal surface: a revisited theoretical model. R. Daniele. Istituto di Fisica, Via Archirafi 36, 90123 Palermo, Italy.
1916
Daniele et al.
J. Opt. Soc. Am. B/Vol. 9, No. 10/October 1992
Photoelectric effect from a metal surface: theoretical model
a revisited
R. Daniele Istituto di Fisica, Via Archirafi 36, 90123 Palermo, Italy
G. Ferrante Dipartimentodi Energetica ed Applicazioni di Fisica, Viale delle Scienze, 90128 Palermo, Italy
E. Fiordilino Istituto di Fisica, Via Archirafi 36, 90123 Palermo, Italy
S. Varr6 Central Research Institute for Physics, 1525 Budapest 114, RO. Box 49, Hungary Received September 25, 1991; revised manuscript received March 20, 1992 The Sommerfeld model extended to include radiation-electron interaction in the regime of highly intense fields is taken as the basis for studying theoretically the laser multiphoton photoelectric effect from a metal surface. Numerical analysis is carried out without approximations other than those inherent in the model itself; the study of the multiphoton aspect of the problem is based on a scheme that is nonperturbative in an essential way. The numerical analysis facilitates insight into the potential and the limits of the model in the interpretation of recent experiments and into the similarities and differences between the metal multiphoton effect and atomic multiphoton ionization. The results obtained and the importance of the competing processes not considered here clearly call for more realistic models to account for recent laboratory observations.
1.
INTRODUCTION
Analyzing the photoelectric effect from metal surfaces is one of the most enduring problems in physics. In spite of the fact that this phenomenon has played a crucial role in the birth and development of quantum mechanics, it is not yet completely understood. Experiments with it still produce surprises, and it deserves much attention because the theory is not well established at any frequency range. For example, in the x-ray regime both primary electrons (i.e., those directly ionized by the photons) and secondary electrons are observed; the total electron yield is strongly dependent on the polarization of the incident radiation and, unexpectedly, is maximum when a large component of the electric field is parallel to the metal surface.' Although the effect is not yet fully understood, 2 it has recently been used in designing and building a stellar x-ray polarimeter.' At lower photon energies and higher field intensities the observed flux of electrons from a solid surface irradiated by electromagnetic radiation results from the superposition of two different and coexisting effects: (a) the standard photoelectric effect in which the electrons absorb one or more photons and (b) thermionic emission caused by heating of the metal surface.4 5 The relative importance of the two effects depends on the intensity, frequency, and pulsational nature of the laser. Disentangling the two effects is one of the experimental tasks in this area of research. 0740-3224/92/101916-06$05.00
Recently, richly varied experimental observations showing the electron energy distribution induced by laser light illuminating a metal surface have been reported6 9 ; in all the experiments the photon frequency was less than the work function of the metal. This means that absorption of more than one photon was required for electron extraction if the extraction resulted solely from the absorption of photons. In all the experiments energetic electrons were observed at laser intensities ranging from 107 to 1010 W/cm 2; at these low intensities atoms and molecules can absorb no more than the minimum number of photons required to achieve ionization. The energy spectrum of the outgoing electrons given in Ref. 6 seems to show multiphoton peaks. In the experiment a pulsed Nd:YAG laser (hiw = 1.17 eV) with effective intensity I = 1.5 X 103 W/cm 2 was used, and the maximum kinetic energy of the electrons was observed to be 10 eV Furthermore, it was noticed that the electron yield depended on polarization, indicating that only the component of the laser field orthogonal to the metal surface was relevant in the experiment (see also Ref. 10). In the experiment reported in Ref. 7 the laser parame2 ters were hw = 1.17 eV and I= 2 x 1010 W/cm . Electrons with energies up to 600 eV were seen, but there was no evidence of multiphoton peaks owing to the low energy resolution of the instrument. However, multiphoton peaks are reported in Ref. 8. In the experiment described in Ref. 9, which used a
2 9 7 Nd:YAG laser with an intensity range of 10 -10 W/cm ,
© 1992 Optical Society of America
Daniele et al.
Vol. 9 No. 10/October 1992/J. Opt. Soc. Am. B
1.0 0.9
0.8 0.7 =
0.6
*;
0.5
.mP Ci
0.4 0.3 0.2 0.1 0.0
Electron Energy (eV)
(a) 1.10
0.00
0
200
400
600
800
Retarding Potential (V)
(b) Fig. 1. Experimental spectra of surface-metal photoelectron emission obtained from (a) Ref. 6 and (b) Ref. 7.
rather energetic electrons (up to 100 eV) were observed, but no evidence of peaks of multiphoton absorption was reported. Besides, the recorded spectra have a clear thermal character with temperatures up to several hundreds of thousands of kelvins, and this suggests that laser heating of the electron gas in the metal is likely to have taken place. Figure 1 shows the experimental electron energy spectra obtained from Refs. 6 and 7. Many acceleration mechanisms may be advocated as being capable of yielding energetic electrons: (a) the electrons, when impinging on the metal surfaces, undergo an acceleration and are ionized by conventional multiphoton absorption; (b) the laser field penetrates inside the metal bulk for several atomic layers, and this permits the escaping electrons to absorb a large number of photons through several collisions with the surrounding atoms"; and (c) the ionization is induced by a two-step mechanism in which the electrons first absorb the minimum number of photons for ionization and then, as a second step, absorb further energy through collisions with the atoms of the metal surface when they escape with a velocity almost parallel to it. Other factors can affect the final energy distribution of the electrons: a ponderomotive force acts on the electrons and works to push them outside of the laser field'2 ; standing waves may be present because of the superposition of ingoing and reflected electromagnetic fields, and
1917
they are known to exert a force on the electrons 13 14 ; the band structure of the energy levels of the metal broadens the final energy of the electrons; and the presence of an image charge is known to provide a large contribution to the work function of a metal,' 5 and this may induce the electrons to absorb additional photons. To the above primary causes we may add other, secondary charge-acceleration mechanisms, such as spacecharge effects and the resulting Coulomb repulsion between the outgoing electrons. Although in the experiments of Refs. 7 and 8 efforts were made to eliminate the space-charge effects, there is no mention of such an effort in Ref. 6. Reference 16 provides a critical review of the experiments of Refs. 6-8 from the point of view of the Coulomb repulsion between the extracted electrons and a model based on these mechanisms. The variety of the experimental conditions and observations and the lack of a well-defined physical description for unifying them clearly call for new experimental theoretical efforts. An additional source of ambiguity in the theoretical efforts is the fact that frequently, in investigations using similar models, different physical quantities are calculated with different degrees of approximation. Although the results involving the physical contents of the proposed models are roughly convergent, precisely evaluating the limits and merits of the various models is not straightforward. In this perspective we take here a simple but necessary step, namely, we address the problem of rigorously reconsidering a model that enables us to derive analytically the quantities of interest without any approximations other than those inherent in the model itself. The model presented below is a generalization of the Sommerfeld model and has been popular in the past, but generally it has been restricted to weak-field contexts10 1"7 "8 in which it has been used to obtain analytic estimates of the pertinent physical quantities and their asymptotic dependence on the field parameters. To the best of our knowledge the numerical calculation reported in Ref. 18 is an exception; in that work currents are plotted as a function of the laser intensity. The recent work by Mishra and Gersten 9 deserves separate mention. By using the same model considered here they developed for photoelectron emission from a metal surface a perturbative treatment extended to arbitrary orders and numerically calculated the electron yield versus the photon frequency from sodium and potassium surfaces for two different laser intensities. A comment on this calculation is given in Section 3. In seeking new and more comprehensive theoretical models, it is reasonable to first explore the potential of models that have served as the initial basis for the interpretation of the multiphoton photoelectric effect from a metal. Care is exercised concerning aspects of gauge consistency insofar as it is possible in a unidimensional context. 2.
MODEL
The metal is described as a unidimensional potential step function with depth -V0 and discontinuity at x = 0 that is assumed to fill all the half-space in the negative part of the x axis. A monochromatic laser field of frequency is
Daniele et al.
J. Opt. Soc. Am. B/Vol. 9, No. 10/October 1992
1918
present in the other half-space and does not penetrate the metal bulk; a discussion of the experimental aspects of this last feature can be found in Ref. 6. In the A *P gauge the full Hamiltonian is given by
i
A1
HA(t)
=
-ih-
c
a
ax
e + -A(x,t)
12
C
2 1h[a2 M1-2 2 + -2.2(X, cA aX 2ML
+ V(x)
'hie
t)
-A(x, -c
a
A(x, t) = O(x)Ao cos cot
(2)
(x) is the step function (3)
0
cot.
MhC2 T exp Liqx- -I est - 4mc -
(4) X
The matching conditions of the wave function at the discontinuity are20
2
II (0 t),
t)+ i Ao cos(cut)qiII(0, t), hC ax
Ix t)= a(0
Ax
(5)
where ift(x, t) and qP(x, t) denote the wave function inside and outside the metal, respectively. We note that Eq. (4) implies a nonzero divergence of the laser electric field at x = 0 and a consequent sheet of charge at the metal surface. This is a rather unphysical feature, and it is not clear whether it further limits the validity of the model. If we had worked in the E r gauge the Hamiltonian would have been 11 HE(x,t)
h~j2
-
2m x
+ V(X)
+
eEx
t) = II
(0, t),
aqpI (0,t)
t) a4II(0, 1
ax
ax
-G)
0,
2m
(7)
G=(EA)
(8)
2
2
2
e AO
.ex i (x T3 exp iqsx - -,Eet + i-A(t) -
(12)
ie
M__2
t
J A(t')dt' 1J,
~
q
ie2
(13)
where we have written I(E)(x, t) in terms of the vector potential A(t) by using the relation E = (-1/c)(aA/at)(x > 0) to make it easier to compare the equations for the wave function in the two different gauges in Eqs. (11) and (13). Here the term containing xA, which is absent in the A *P gauge, is the so-called Goeppert-Mayer factor; it plays the same role as the extra term in the matching conditions of Eqs. (5) when iffIf) is inserted into Eqs. (7). Then the two gauges give exactly the same equations and the same results. Since the E r gauge is well defined only in the dipole approximation 22 and our vector potential has a discontinuity at x = 0 for the sake of internal consistency, we choose to work in the A P gauge. The unknowns of our problem are the reflection coefficient R,, and the transmission coefficient T,. They are determined by substituting the wave functions HIj and qII .
can be found exactly in both gauges. In the metal the wave function is a superposition of ordinary plane waves:
q
(11)
4MC2 +4c 2m +
4mc
X j A 2 (t')dt' --
It must be stressed here that the solution of the Schrodinger equation (i at
-
f A(t')dt' ], 2
2
e2AO + s ho +MC2
(6)
and the matching conditions would have been the usual ones, i.e., continuity of the wave function and of its derivative: I (0
2
q
where the last equality is meant to define q, qo is the momentum of the electron in the absence of the laser, and e2A, 2 /4mc2 gives a shift of the threshold extraction energy that, with the field values that we use in our calculations, is generally small. A similar matching procedure has been recently applied 2' for a different physical situation in which the interaction with the light field is not limited to the region x > 0. In the E r gauge the wave function outside the metal is I
a2
f cos(2cot')dt' -
h qO
E = t)
ie2A, 2
i II(x,t) =
HI1(0,
(10)
In the summation of Eq. (9) the term with n = 0 is explicitly included and Iko is the initial momentum of the electron inside the metal. In this model the light-electron interaction that produces ionization is assumed to take place at the metal surface (x = 0), where the electrons undergo the maximum acceleration. During the process the electrons can absorb or emit any number of photons; accordingly their wave function inside the laser field becomes a superposition of Volkov states that, in the A *P gauge, assumes the form
With this form we have
ax
(9)
- V0 + noi.
= 2k
Wh
(1)
ax
aA(x, t) = 8(x)Ao cos
+Wot)
R,, exp(-ikix -- Wt ,
+
t) -ax
where
and
- exp(-ikox -
where
A(x, t)I + V (x),
---
= exp(ikox - +Wot)
Daniele et al.
Vol. 9, No. 10/October 1992/J. Opt. Soc. Am. B
-10
1919
point and at the same instant, we must set
-15 -20
s =
i2 ko 2
., > A
-25 -30 1-
_m
2m
_
=
1iqo2 2m
2m
2, 2
+ 4
4MC2
(16)
With this constraint the equation for the unknown are
33 -35-
x
Rn =
E
So-
Ts Dn,,s(as, P)
-50-
(17)
>nO E TsB.,(a.,/),
=
-55
s=-
-60 0
5
10
15
20
25
30
35
40
45
50
with
Output Energy (eV)
(a) -13-
Dn,(aspFB) =
2
Bn,.Jtsj1) =
>
-19
2
+ q,)Jn-s-2,k(a,)
+ IiyJn-s-2k+l(as)Jk(P)],
-o1
0
1
0
2
0
3
0
4 =
-31-
-34-37-
10
5
0
15
20
25
30
35
40
45
Output Energy (eV)
(b) Fig. 2. IT,!2 (on a logarithmic scale) as a function of the kinetic energy of the outgoing electron for different values of photon energy. The photon energies are as follows: curve 1, 0.585 eV; curve 2, 1.17 eV; curve 3, 2.34 eV; curve 4, 3.51 e The initial electron energy is -10 eV, 2and the laser intensities are 2 (a) 10ll W/cm and (b) 1012 W/cm . The calculated discrete values of IT,12 are connected by continuous lines to aid visualization. in the equations for the matching conditions, Eqs. (5). obtain R, exp -W't) A
S=-x
X
>
=
>
Ts xp - -
We
E,
AE
Jn(a)Jm(f3)exp[-i(n + 2m)wt],
7
2ko exp(--Wot)
Rnk
(
exp/-Wnt
TsJn(as)Jm(f3)(qs exp[-i(n + 2m)wt]
s,n,m--
+ -A{exp[-i(n + 2m + 1)cot] 2Iic + exp[-i(n + 2m Xexp -
i
-
1)ct]})
t
I(14)
(
where -y Since
J()[(kn
+ iyJn-s-2 k-l(a,)Jk(13)
-22-
S=-x
x1 2
(18)
k=-X 2qO
-16-
-25-
Jn-s-2 k(as)Jk(13),
Ifrg and
eqAo
eA_
MCO' /3- simctn hm ' mcrbt
(15)
(15)
1Pmdescribe the same electron at the same
eA0 ic
(19) (20)
In Section 3 we report numerical calculations based on the exact evaluation of the Eqs. (17) and yielding Rn and T8 . In particular, for different regimes of the parameters of the process we calculate IT,12,which is a direct measure of the probability of electron emission after multiphoton absorption.
3.
RESULTS AND FINAL REMARKS Figure 2(a) shows IT,!2 (on a logarithmic scale) as a function of the energy of the outgoing electron for a laser field intensity I = 1011 W/cm 2 , an initial electron energy of -10 eV, and a potential well depth of V0 = 15.5 eV The behavior of the curves for the various photon energies is what one would expect: the transmission coefficients T, increase sharply with the photon energy. Figure 2(b) shows the same kind of calculations with a field intensity of 1012 W/cm 2 . Here the curve for smallest photon frequency (w = 0.585 eV) exhibits oscillations and is higher than the curves for photon frequencies of 1.17 and 2.34 eV The mathematical explanation for this feature may be traced back to the values of the arguments of the Bessel functions a, and , of Eqs. (15). Large values of a, and , imply oscillating Bessel functions, and these are reflected in the behavior of logITI2. This amounts to saying that the physical process is entering a highly nonlinear regime and shows a strong nonperturbative behavior with respect to field intensity. This is particularly the case for curve 1. The detailed and precise physical meaning of the curve is, however, not well defined because the results become dependent on the mathematical model that describes the radiation field. Remembering the meaning of IT! 2, we may say either that at high intensities the electron-emission probability increases with large numbers of absorbed photons or (more realistically) that some saturation regime is being entered in which groups of emission channels have comparable probabilities. Finally, because this behavior takes place at high
Daniele et al.
J. Opt. Soc. Am. B/Vol. 9, No. 10/October 1992
1920
-29 -34 -38
-43 -4.8
-562 -61 -651 -70 0
2
4
12 10 8 6 Number of Above Threshold Photons
14
16
Fig. 3. IT,,2 as a function of the number of photons above the extraction threshold for different values of laser intensity. The photon energy is 1.17 eV, and the initial electron energy 2 is -10 eV curve 1, 107 W/cm2 ; curve 2, The laser intensities are as follows: 2 2 108 W/cm ; curve 3, 109 W/cm ; curve 4, 1010 W/cm ; curve 5, 2 1011 W/cm .
-2
-5
Figure 6 is a plot of IT,!2 as a function of s for different values of photon energy. Our results may be summarized as follows. (a) For those aspects that are comparable, multiphoton emission from a metal surface exhibits a feature that is different from features of multiphoton ionization of atoms. At the intensities and frequencies used here (especially at hI) = 0.585 and I = 1012 W/cm 2 ) a free atom may absorb additional photons above the minimum number required for electron ejection, and several of these additional ionization channels have comparable probabilities. In multiphoton electron emission from a metal surface we see instead that as a rule emission channels with increasingly higher numbers of absorbed photons have sharply decreasing probabilities. Because we consider only linear polarization and grazing incidence here we cannot draw conclusions about other aspects concerning possible similarities or differences between atom multiphoton ionization and multiphoton surface-electron emission. With the present level of understanding of the process of electron emission from metal surfaces in the presence of strong laser fields, it is perhaps premature to dwell on establishing clear-cut analogies and differences between atom multiphoton ionization and surface multiphoton
.-8
1-5
-2
-15
-5 -87
-21 -24
-141--
-27-30-
N -16
-12
-8
4
8
8
4
12
16
Photon Number
Fig. 4. IT,12 as a function of the number of exchange photons s for different values of laser intensity. Iio = 1.17 eV, and the iniThe laser intensities2 are as foltial electron energy is -10 eV 2 7 W/cm ; curve 2, 108 W/cm ; curve 3, 1, 10 curve lows: 2 2 2 9 10 W/cm ; curve 4, 1010 W/cm ; curve 5, 10ll W/cm .
/
-20 -23-/
/
-26-
/
-29
-35
-20
-16
-12
-8
-4
0
4
8
12
16
20
Photon Number
field intensities and low frequencies, we may view it as the onset of a quasi-statistical tunneling regime.
Figure 3 shows IT,!2 as a function of the number of absorbed photons above the extraction threshold for different laser electric-field intensities. Notice that Ts12
2 as a function of s for different values of initial elecFig. 5. IT,1 tron energy. I 1011l W/CM 2, and io)= 1.17 eV The initial electron energies are as follows: continuous curve, -5 eV dashed curve, -10 eV dotted-dashed curve, -12 eV 2
decreases dramatically with s. The experimental data de-
scribed in Section 1 can be compared with these curves. For example, the data from Ref. 6 that are shown in Fig. 1(a) should match curve 2 in Fig. 3. Although there is no resemblance between the two spectra, the plots as a whole do not show any exotic behavior. Figure 4 shows IT,12 as a function of s for different values of laser intensity; here, for a photon energy of 1.17 eV and an initial energy of the active electrons (in the metal) of -10 eV, the minimum number of photons for electron emission is Sm = 9.
2
Our calculation also shows that ITs1 changes with the binding energy Wo of the electrons. Figure 5 is a plot of IT5!2 for different values of Wo. We see that the behavior
is somewhat irregular; the absorption probability increases
with -Wo because a free electron does not absorb photons. Of course, a different number of photons must be absorbed for any binding energy shown for ionization to occur.
-12
-14/ -17-/
~0-20-
/
-25 -28-31
-36 -20
-16
-12
8
4
0
4
8
12
16
2
Photon Number
Fig. 6.
IT! 2 as
a function of s for different values of photon en-
ergy. I = 10"1 W/cin 2, and the initial electron energy is -10 eV The photon energies are as follows: continuous curve, 0.585 eV, short-dashed curve, 1.17 eV, dotted-dashed curve, 2.34 eV-, longand-short-dashed curve, 3.51 eV
Daniele et al. electron emission. Generally, similarities in results should reflect similar physical situations. In the multiphoton photoelectric effect in metals, one has electrons that must pass through a dense, continuous energy region to be ejected from the metal. This situation is not typical of multiphoton ionization of atoms, in which electrons cross a discrete energy spectrum that only in the upper excited states begins to become dense and quasi-continuous. These considerations suggest that qualitative similarities could exist between the process studied here and multiphoton ionization of atoms from highly excited states. Undoubtedly, it is of interest to understand to what extent the Sommerfeld model may be thought of as equivalent to that of an atom and, in such a case, to what kind of atom. (b) The transmission coefficient is drastically changed and increased by increasing the intensity; this results from the opening of a large number of ionization channels. (c) In spite of the significant increase of the number of ionization channels it appears that the model, at least in its present form, is not able to account for the energy spectra observed recently, even at the lower laser intensities.6 ` (d) New theoretical models, possibly based on different physical mechanisms (see Section 1), are clearly required along with treatments allowing for the interplay among them. A last comment concerns the results reported in Ref. 19. Although Ref.,19 uses essentially the same Sommerfeld model as is used in the present work, different quantities are calculated with a perturbative treatment extended to arbitrary order in the electron-radiation interaction, and this makes it difficult to compare our results with those of Ref. 19. However, our calculations and those of Ref. 19 have a qualitative feature in common: the calculated physical quantities are found to be sharply decreasing functions of the field parameters. This is useful information in relation to available measurements because it helps to rule out particular physical mechanisms as being responsible for the observations. A second point of theoretical interest is the question of whether our nonperturbative treatment correctly yields results based on perturbative treatments. We hope to address this question in a separate analysis.
ACKNOWLEDGMENTS We thank the University of Palermo Computation Center for the computer time generously provided. We also ac-
Vol. 9, No. 10/October 1992/J. Opt. Soc. Am. B
1921
knowledge the referees' remarks, which helped to focus some aspects of the work. This work was supported by the Italian Ministry of University and Scientific Research, the National Group of Structure of Matter, and the Sicilian Regional Committee for Nuclear and Structure of Matter Research.
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