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Surface potential of a free electron metal in the RPA
SURFACE
SCIENCE ll(l968)
SURFACE POTENTIAL
419-429 o North-Holland
OF A FREE ELECTRON
Publishing
Co., Amsterdam
METAL IN THE RPA
R. W. DAVIES M.I.T. Lincoln Laboratory*, Lexington, Massachusetts 02173, U.S.A.
Received 18 January 1968 ; revised manuscript
received 15 April 1968
The effects of correlation on the surface potential of a dense electron gas are studied in the random phase approximation for an interacting system. The model considered is the same as that originally proposed by Juretschke, and subsequently studied by Loucks and Cutler using the Bohm-Pines decoupling scheme. In the high density regime, we suggest as a reasonable approximation for the Slater average surface potential, the algebraic average of Juretschke’s pure exchange potential and a Thomas-Fermi type screened exchange potential. The resulting theory yields substantial agreement with that of Loucks and Cutler as a function of effective screening parameter.
1. Introduction The theory of the surface potential in metals has a rather long history, dating back to Bardeen’s calculation 1) of the work function of a free electron metal. Bardeen considered a model consisting of a fixed uniform positivestep charge density, occupying half of space, plus a system of neutralizing mobile electrons. Carrying out a Hartree-Fock analysis, with a crude allowance for correlation corrections, Bardeen concluded that “the barrier at the surface is due largely to exchange and polarization forces rather than to ordinary electrostatic (or Hartree) forces”. In 1953, Juretschke2) re-examined the exchange potential for a simplified model in which a free electron gas is confined to half of space by an infinite potential barrier at x=0. By replacing the actual wavevector-dependent exchange potential by the more tractable Slater averages) exchange potential, Juretschke obtained a surface potential which exhibited some distinct oscillatory structure as a function of distance from the barrier. More recently, Loucks and Cutler4) have used the Bohm-Pines5) de* Operated with support from the U.S. Air Force. 419
420
R. W. DAVIES
coupling scheme to investigate the effects of correlation in Juretschke’s model. These authors calculated the Slater average of the exchange-correlation potential, and obtained surface potential curves quite similar to the one computed by Juretschke, but modified in depth and shape as a function of an effective screening parameter j?‘= kc/k,, where k, is the Fermi wavevector and k, is the plasma cut-off wavevector. In essence, the analysis of Loucks and Cutler leads to a surface potential which is the sum of a screened Slater average exchange potential, plus some position-independent terms due to long-range correlation. We shall discuss the results obtained by Loucks and Cutler in more detail in section 4. In view of the fact that the Bohm-Pines decoupling approach to interactions in an electron gas is knowns) to be somewhat lacking in rigor, even in the high density limit, it would seem, perhaps, worthwhile to investigate the effects of correlation on the surface potential using a different approach. In this paper we shall consider the problem of correlation in Juretschke’s surface model by employing an RPA analysis of the single-particle selfenergy. It is, of course, well known that the random phase approximation is intrinsically a high density approximation, and hence the relevance of our analysis to real metals is certainly questionable. On the other hand, one point which our treatment rather suggests is that, at least for the Slater average potential, the net correlation correction may not be as drastic as one might at first suppose. In particular, it has been suggested7) that the basic effect of correlation is to replace the exchange potential by a screened exchange potential. We believe that such a treatment would considerably overestimate the role of correlation. In this connection, we remark that, although the only position-dependent term in the analysis of Loucks and Cutler is a screened exchange term, the long-range correlation terms in their theory have the effect of considerably reducing the total effect of correlation. Because of this, we find substantial agreement with their results as a function of effective screening
parameter.
2. RPA self-energy analysis Quinn and Ferrells) have calculated the single-particle self-energy in the RPA for the unbounded free electron gas. The results of their calculation are rather pertinent to the problem we wish to consider, hence we devote this section to a review of some of their analysis. The real part of the exchange-correlation self-energy at zero temperature, and in the RPA, is given by E(k) = E(k) + Efb:“,.(k),
(1)
SURFACE
c
421
POTENTIAL
v(d[Re ’&k-q + is) -11 K(q, ek -
P &k>ek--qZEf
-e(ef-s,); C
(2)
vOIReX(q,ek-l~k_q+is)-l]’
4 E,>Ek-q~&i4
m
(3) --m
4
In the expressions above Ek=A2k2/2m, v(q)=4xe2/q2, O(E) is the unit step function, and K(q, w) is the RPA (or Lindhard) dielectric constant. The first term in eq. (2) is the usual exchange self-energy. Now for k’s close to k,, the phase space restrictions for the last two terms in eq. (2) are such that it is a reasonably good approximation to replace the dielectric constant by its static limit K(q, 0). If one further approximates the static dielectric constant by its long-wavelength limit, i.e. by
K(q,O)=l+$
(4)
4
one may evaluate
eq. (2) and find
2(P)=-($)(2-2/?~an-1~-tan-1p+]
+
(PZ-l~-~Zln~~-1~2+~2+28tan-‘2~_~ln _____ 2P (P + 1)2 + p’
B
with P= k/kf, and p= k,lk,. We comment that a simple screened exchange the first three terms in eq. (5). At this point we note from eq. (5) that
2P
calculation
3 (9
corresponds
to
,!?(P = 1) = E’“‘(P = 1) = - e2k,/rc, i.e. exactly at the Fermi surface, the entire correlation effect is contained in the line integral term E,!!::. (k,), which turns out to be negative. Furthermore, as shown by Quinn and Ferrell, the total correlation energy of the system per electron is determined completely by the value of the single-particle self-energy at the Fermi surface. Thus, the negative value of EiiFrT(kf)
422
R. W.DAVIES
accounts for the fact that the effect of correlation is to lower the total energy of the system. However, we shall find in the sections which follow that the effect of correlation on the Slater average surface potential (at least for sufficiently small j?) is to raise the potential, in agreement with the results of Loucks and Cutler. It is rather clear from the discussion above, however, that this is purely a consequence of averaging the surface potential over occupied states, and that, in all probability, correlation would lower the surface potential for an electron exactly at the Fermi surface. The point to be noted here is that the correlation correction to the singleparticle energy levels changes sign somewhere between the Fermi surface and the wave vector (k-0.8k,) corresponding to the average over occupied states. This implies a certain cancellation of the correlation effect from kvalues in the neighborhood of the Fermi surface, and this is the basis for our suggestion that the correlation correction should not be too drastic if one considers an average over occupied states. At the same time, the above observations make it clear that a Slater type average is inappropriate for discussing physical properties which depend only on those single-particle levels in the immediate vicinity of the Fermi surface. For such properties, a treatment of the exchange-correlation potential more along the lines of that suggested by Kohn and ShamQ) would be preferable. A problem of some physical interest, which should be considered carefully in the above connection, is that involving the effective surface barrier for Fowler-Nordheim tunneling. At low fields, only those electrons immediately below the Fermi surface are involved in tunneling. As the field is increased, however, the peak in the normally directed tunneling current moves to lower k-values, and the width of the distribution becomes significantly broadened. Thus, while the Slater type average probably has some relevance for the high field case, it is inappropriate for the low field situation. To consider the line integral term* as given by eq. (3), one can introduce the variables a 0 k z= ’ P= V’-2-9 2k, ’ A k,qlm kr’ and upon performing
the angular *
-m
integration CG
0
with K= K (2k,z, 4ie,zv). The principal
difficulty
in evaluating
E,!!:: (P) is
* The sign of eq. (54) in the paper of Quinn and Ferrell is in error (see their eq. (35) which is correct).
SURFACE
that use of the static dielectric
423
POTENTIAL
constant
is simply not justified.
The contri-
bution of eq. (6) is known* to be negative for sufficiently small rS values, and the small z approximation to the dielectric constant always leads to a negative value for this term. However, as we have obtained no really reliable estimate of this term either in the bulk or in the surface region, we shall refrain from any further discussion of this term, except to conjecture that it should tend to somewhat lower the surface potential curves in the regions of high density. Finally we consider the average of eq. (5) over occupied states. We point out that eq. (5) is less reliable for small P values, because, for k’s well below k,, the static dielectric constant approximation becomes questionable. On the other hand, we argue that the average over occupied states weighs most heavily the states of high wave vector, and thus we feel that this error should not be too severe. The average of eq. (5) over occupied states then yields i?,,,, = +E;:,, + +E;;ky’ , where, in units (3e2k,/2z), formula 7,
(7)
Ei:;, = - 1, and Eiz;e”’ is given by the well-known
1_!?-$fitan-‘2 ” 8+1(l+~~)ln(l++)). 6
(8)
Remark that eq. (7) is basically a consequence of the phase space restrictions in eq. (2), and that the same form obtains using the exact static dielectric constant. 3. Correlation effects in a surface region We consider the same model of a surface discussed by Juretschke and by Loucks and Cutler. We confine an electron gas to the half-space x>O by an infinite potential barrier at x=0. The unperturbed wave functions appropriate to this model are $k (y) = (i)’
sin k,x ei(“Yy+kzz),
(4, k,, k,) =
z(m,, 2m,, 2m,),
m,,m,=O,i-l,f2
,...,
and the $‘s are orthonormal
over a volume
* See, for example, eq. (57) of ref. 8.
V=L3.
m, = 0, 1, 2, . . . ,
(9)
424
R.W.DAVIES
Introducing
the matrix
elements
pkkg(q) =
d3r $,*(r) eiq”’ tik, (r) y
(10)
s one can write for the RPA correlation 1
G, (4 = pv
ccc v (4)
51~
k’
v (4’)
Pkk’
self-energy
(4)
Pk’k
(-
4’)
sk’
(‘1
-
tl’)
n6,’
(b>
.
w’
(11)
Here we are using the finite temperature
formalism,
S,,(z, - C,,)= [Zl - I$ - Ek’]-’
with rfi(21 + 1)
z1 =
)
__-~
+
/.I,
/3
li=klT. B Furthermore,
the polarization
propagator
T&,,(<~,) satisfies the equation
r$,, (51’) = TM*(51,) - 5 %q” (&> v (4”) r&W (51,) 3 where, in the RPA
c
Pkk’t-
f-
(Ek)- f - @k’)
4)
Pk’kbi)
y-_E+-(p~’ k’
k
(12)
(13)
I’
kk’
From eqs. (9), (10) one sees that rr and rr’ will be diagonal in the q,,, qz indices but not in qx. Upon performing* the E-sum in eq. (13) (the terms which do not involve Kronecker deltas of k: can be handled by contour integration), we find that, to within corrections of order l/L, the diagonal part of eq. (13) is the standard RPA proper polarization function
s d3k
f-c& __k >-“r(E ~ ___ke!! >. &k -
Ek+p
+
It then follows from eq. (12) that, to within corrections of order diagonal part of nb,,(ll,) is the usual RPA polarization function
Presumably, polarization
(14)
(1,
the physical significance of the non-diagonal propagator is that the effective dielectric constant
l/L, the
part of the is x-dependent
* In analyzing this expression, it is convenient to split the sine functions into exponentials, and to recombine terms in such a way that kz and k’, can be summed over both positive and negative values.
SURFACE
425
POTENTIAL
in the region of the surface. The contribution to the self-energy function from the non-diagonal part of the polarization propagator is not easy to assess, and in the equations which follow we shall simply ignore it. With the approximation discussed above, we obtain for the correlation self-energy
Eq. (16) differs from the usual bulk expression only with respect to the value of the matrix elements pkkt(q). We can, therefore, immediately write down the analog of eq. (2) for the surface case. At zero temperature we will obtain
E(k) = - ;
cc v(q)
Pkk’(q)Pk’k (-
1 K(q, &k- &k’+ is) X
4)
1
- 1 -f?(sr-ek)+
~kk’(q)~k’k~-q)
c 4v(q) 1 +isj-l
c k’
k’
.
(17)
Making use of eq. (IO), we can then write down the natural generalization of the exchange potential corresponding to eq. (17). It is given by
“(‘)
1 -.______ = - z(r) @k(r) cs k,
d3r’ $,* (4 $4 tr’)
$k’ (r> +k (“1
k’
x-
1 V
c4
v(q) e iT(*-*‘)
Re --___-
1
K(q, &k- ek’ + is)
1
_ 1
R. W. DAVIES
426
1 xV
c
v(q) e
iq+
-r’)
1
~~
K(q,
Ek - ck’ + is)
-1
1 (18) .
Invoking the same approximations for the dielectric constant as discussed in section 2, and taking proper account of the phase space restrictions on the U-sums in eq. (18) it is a straightforward matter to verify that the Slater average of pk(r) over occupied states is given by P(r) = *V’“.(r) + +Vsc.ex.(r),
(19)
where Vex(r) is the pure exchange potential calculated by Juretschke, and where V”“. ex.(r) is the analogous screened exchange potential, corresponding to a screening factor ePskrr’. Here fl= k,/k,, and Y, has the same meaning as in eq. (8) of Juretschke’s paper. It is also easy to check that the form of eq. (19) obtains using the exact static dielectric constant, the only effect being to replace the screening factor e-Bkrr’ by some more complicated function of r,. 4. Discussion Loucks and Cutler have given the following average exchange-correlation potential
expression
V(r) = V”“““‘(r) + V,,,, . Here, Vi,. is a position-independent to long-range
correlation
for the Slater
(20)
term which Loucks and Cutler attribute
effects, and is given by
(21) with /I’ = kc/k,, and k, the plasma cut-off wave vector. The screened exchange term in eq. (20) was computed by Loucks and Cutler using an IBM 7074 for both a sine integral
screening
factor,
F@‘k,r,)
and the exponential
= 1 - f Si(/?‘k,r,),
approximation F(j?‘kfr,)
The results obtained by these authors are reproduced in fig. 1.
to it, i.e. N e-P’krrl.
for the Slater average surface potential
SURFACE
427
POTENTIAL
.8
I 0
0.2
0.4
0.6
0.8
kfx Fig. 1. The complete surface potential, including long-range correlation, as calculated by Loucks and Cutler. The screened exchange contribution to the curves was calculated for the sine integral screening factor.
To obtain the quantity p(r) relevant to our present calculation, we need, in principle, only average the result obtained by Juretschke for the pure exchange potential, with the results obtained by Loucks and Cutler for the exponential screened exchange term. Unfortunately, the numerical results from the calculation of Loucks and Cutler are no longer available. Therefore, in order to obtain a qualitative comparison of the two theories, we have made a crude visual extrapolation of the exponential screened exchange contribution from fig. 3 of their paper. The resulting curves obtained for p(r) from this extrapolation are displayed in fig. 2. The extrapolation is particularly crude as far as the details of the oscillations in the surface potential are concerned, but it should serve to indicate the gross structure. Comparison of figs. 1 and 2 shows reasonable agreement as a function of effective screening parameter. One difference which can be noted is that the curves of Loucks and Cutler become progressively flatter than ours as the effective screening parameter is increased. In our present theory, which the contribution from the line-integral correlation term to the Slater average potential has been completely ignored, we can reach no conclusion as to whether this discrepancy is real. We remark, however, that GadzuklO) has recently obtained similar results using a rather different self-energy analysis. Finally, it should be pointed out that the effective screening parameters in the two theories are somewhat different of the functions electron density,
428
R.W.DAVIES
0
2.0
4.0
6.0
8.0
kfx Fig. 2.
The contribution
p(u) to the surface potential for various values of 8.
i.e. our p involves
Cutler involves
the Thomas-Fermi wave vector, while p’ of Loucks the plasma cut-off wave vector. More explicitly,
and
p = k,/k, = 0.815 r$,
(22)
p = kc/k, 2: 0.47 rf ,
(23)
density electron gas*. Thus one has j?- 1.74/J’ in the high density regime. However, Loucks and Cutler prefer to regard the screening parameter /3’ more empirically, and hence have chosen not to specify its dependence on the electron density. In real solids this, is, perhaps, not an unreasonable point of view.
for the high
Acknowledgments The author wishes to thank C. T. Kirk for initiating his interest in this problem. Several discussions with K. J. Harte were very useful. The author would also like to thank Dr. H. Juretschke for supplying the numerical * See ref. 6 where it is pointed out that the original Bohm-Pines
estimate of j?’ is too small.
SURFACE
POTENTIAL
429
results from his exchange calculation. Finally, the support and encouragement of D. 0. Smith is gratefully acknowledged. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
J. Bardeen, Phys. Rev. 49 (1936) 653. H. J. Juretschke, Phys. Rev. 92 (1953) 1140. J. C. Slater, Phys. Rev. 81 (1951) 385. T. L. Loucks and P. H. Cutler, J. Phys. Chem. Solids 25 (1964) 105. D. Bohm and D. Pines, Phys. Rev. 92 (1053) 609; D. Pines, Phys. Rev. 92 (1953) 629. K. Sawada, K. A. Brueckner, and N. Fukada, Phys. Rev. 108 (1957) 507. J. Robinson, F. Bassani, R. Knox and .I. Schrieffer, Phys. Rev. Letters 9 (1962) 215. J. J. Quinn and R. A. Ferrell, Phys. Rev. 112 (1958) 812. W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133. J. W. Gadzuk, Quarterly Progress Report No. 66, Solid-State and Molecular Theory Group M.I.T., Oct. 15, 1967.
SCIENCE ll(l968)
SURFACE POTENTIAL
419-429 o North-Holland
OF A FREE ELECTRON
Publishing
Co., Amsterdam
METAL IN THE RPA
R. W. DAVIES M.I.T. Lincoln Laboratory*, Lexington, Massachusetts 02173, U.S.A.
Received 18 January 1968 ; revised manuscript
received 15 April 1968
The effects of correlation on the surface potential of a dense electron gas are studied in the random phase approximation for an interacting system. The model considered is the same as that originally proposed by Juretschke, and subsequently studied by Loucks and Cutler using the Bohm-Pines decoupling scheme. In the high density regime, we suggest as a reasonable approximation for the Slater average surface potential, the algebraic average of Juretschke’s pure exchange potential and a Thomas-Fermi type screened exchange potential. The resulting theory yields substantial agreement with that of Loucks and Cutler as a function of effective screening parameter.
1. Introduction The theory of the surface potential in metals has a rather long history, dating back to Bardeen’s calculation 1) of the work function of a free electron metal. Bardeen considered a model consisting of a fixed uniform positivestep charge density, occupying half of space, plus a system of neutralizing mobile electrons. Carrying out a Hartree-Fock analysis, with a crude allowance for correlation corrections, Bardeen concluded that “the barrier at the surface is due largely to exchange and polarization forces rather than to ordinary electrostatic (or Hartree) forces”. In 1953, Juretschke2) re-examined the exchange potential for a simplified model in which a free electron gas is confined to half of space by an infinite potential barrier at x=0. By replacing the actual wavevector-dependent exchange potential by the more tractable Slater averages) exchange potential, Juretschke obtained a surface potential which exhibited some distinct oscillatory structure as a function of distance from the barrier. More recently, Loucks and Cutler4) have used the Bohm-Pines5) de* Operated with support from the U.S. Air Force. 419
420
R. W. DAVIES
coupling scheme to investigate the effects of correlation in Juretschke’s model. These authors calculated the Slater average of the exchange-correlation potential, and obtained surface potential curves quite similar to the one computed by Juretschke, but modified in depth and shape as a function of an effective screening parameter j?‘= kc/k,, where k, is the Fermi wavevector and k, is the plasma cut-off wavevector. In essence, the analysis of Loucks and Cutler leads to a surface potential which is the sum of a screened Slater average exchange potential, plus some position-independent terms due to long-range correlation. We shall discuss the results obtained by Loucks and Cutler in more detail in section 4. In view of the fact that the Bohm-Pines decoupling approach to interactions in an electron gas is knowns) to be somewhat lacking in rigor, even in the high density limit, it would seem, perhaps, worthwhile to investigate the effects of correlation on the surface potential using a different approach. In this paper we shall consider the problem of correlation in Juretschke’s surface model by employing an RPA analysis of the single-particle selfenergy. It is, of course, well known that the random phase approximation is intrinsically a high density approximation, and hence the relevance of our analysis to real metals is certainly questionable. On the other hand, one point which our treatment rather suggests is that, at least for the Slater average potential, the net correlation correction may not be as drastic as one might at first suppose. In particular, it has been suggested7) that the basic effect of correlation is to replace the exchange potential by a screened exchange potential. We believe that such a treatment would considerably overestimate the role of correlation. In this connection, we remark that, although the only position-dependent term in the analysis of Loucks and Cutler is a screened exchange term, the long-range correlation terms in their theory have the effect of considerably reducing the total effect of correlation. Because of this, we find substantial agreement with their results as a function of effective screening
parameter.
2. RPA self-energy analysis Quinn and Ferrells) have calculated the single-particle self-energy in the RPA for the unbounded free electron gas. The results of their calculation are rather pertinent to the problem we wish to consider, hence we devote this section to a review of some of their analysis. The real part of the exchange-correlation self-energy at zero temperature, and in the RPA, is given by E(k) = E(k) + Efb:“,.(k),
(1)
SURFACE
c
421
POTENTIAL
v(d[Re ’&k-q + is) -11 K(q, ek -
P &k>ek--qZEf
-e(ef-s,); C
(2)
vOIReX(q,ek-l~k_q+is)-l]’
4 E,>Ek-q~&i4
m
(3) --m
4
In the expressions above Ek=A2k2/2m, v(q)=4xe2/q2, O(E) is the unit step function, and K(q, w) is the RPA (or Lindhard) dielectric constant. The first term in eq. (2) is the usual exchange self-energy. Now for k’s close to k,, the phase space restrictions for the last two terms in eq. (2) are such that it is a reasonably good approximation to replace the dielectric constant by its static limit K(q, 0). If one further approximates the static dielectric constant by its long-wavelength limit, i.e. by
K(q,O)=l+$
(4)
4
one may evaluate
eq. (2) and find
2(P)=-($)(2-2/?~an-1~-tan-1p+]
+
(PZ-l~-~Zln~~-1~2+~2+28tan-‘2~_~ln _____ 2P (P + 1)2 + p’
B
with P= k/kf, and p= k,lk,. We comment that a simple screened exchange the first three terms in eq. (5). At this point we note from eq. (5) that
2P
calculation
3 (9
corresponds
to
,!?(P = 1) = E’“‘(P = 1) = - e2k,/rc, i.e. exactly at the Fermi surface, the entire correlation effect is contained in the line integral term E,!!::. (k,), which turns out to be negative. Furthermore, as shown by Quinn and Ferrell, the total correlation energy of the system per electron is determined completely by the value of the single-particle self-energy at the Fermi surface. Thus, the negative value of EiiFrT(kf)
422
R. W.DAVIES
accounts for the fact that the effect of correlation is to lower the total energy of the system. However, we shall find in the sections which follow that the effect of correlation on the Slater average surface potential (at least for sufficiently small j?) is to raise the potential, in agreement with the results of Loucks and Cutler. It is rather clear from the discussion above, however, that this is purely a consequence of averaging the surface potential over occupied states, and that, in all probability, correlation would lower the surface potential for an electron exactly at the Fermi surface. The point to be noted here is that the correlation correction to the singleparticle energy levels changes sign somewhere between the Fermi surface and the wave vector (k-0.8k,) corresponding to the average over occupied states. This implies a certain cancellation of the correlation effect from kvalues in the neighborhood of the Fermi surface, and this is the basis for our suggestion that the correlation correction should not be too drastic if one considers an average over occupied states. At the same time, the above observations make it clear that a Slater type average is inappropriate for discussing physical properties which depend only on those single-particle levels in the immediate vicinity of the Fermi surface. For such properties, a treatment of the exchange-correlation potential more along the lines of that suggested by Kohn and ShamQ) would be preferable. A problem of some physical interest, which should be considered carefully in the above connection, is that involving the effective surface barrier for Fowler-Nordheim tunneling. At low fields, only those electrons immediately below the Fermi surface are involved in tunneling. As the field is increased, however, the peak in the normally directed tunneling current moves to lower k-values, and the width of the distribution becomes significantly broadened. Thus, while the Slater type average probably has some relevance for the high field case, it is inappropriate for the low field situation. To consider the line integral term* as given by eq. (3), one can introduce the variables a 0 k z= ’ P= V’-2-9 2k, ’ A k,qlm kr’ and upon performing
the angular *
-m
integration CG
0
with K= K (2k,z, 4ie,zv). The principal
difficulty
in evaluating
E,!!:: (P) is
* The sign of eq. (54) in the paper of Quinn and Ferrell is in error (see their eq. (35) which is correct).
SURFACE
that use of the static dielectric
423
POTENTIAL
constant
is simply not justified.
The contri-
bution of eq. (6) is known* to be negative for sufficiently small rS values, and the small z approximation to the dielectric constant always leads to a negative value for this term. However, as we have obtained no really reliable estimate of this term either in the bulk or in the surface region, we shall refrain from any further discussion of this term, except to conjecture that it should tend to somewhat lower the surface potential curves in the regions of high density. Finally we consider the average of eq. (5) over occupied states. We point out that eq. (5) is less reliable for small P values, because, for k’s well below k,, the static dielectric constant approximation becomes questionable. On the other hand, we argue that the average over occupied states weighs most heavily the states of high wave vector, and thus we feel that this error should not be too severe. The average of eq. (5) over occupied states then yields i?,,,, = +E;:,, + +E;;ky’ , where, in units (3e2k,/2z), formula 7,
(7)
Ei:;, = - 1, and Eiz;e”’ is given by the well-known
1_!?-$fitan-‘2 ” 8+1(l+~~)ln(l++)). 6
(8)
Remark that eq. (7) is basically a consequence of the phase space restrictions in eq. (2), and that the same form obtains using the exact static dielectric constant. 3. Correlation effects in a surface region We consider the same model of a surface discussed by Juretschke and by Loucks and Cutler. We confine an electron gas to the half-space x>O by an infinite potential barrier at x=0. The unperturbed wave functions appropriate to this model are $k (y) = (i)’
sin k,x ei(“Yy+kzz),
(4, k,, k,) =
z(m,, 2m,, 2m,),
m,,m,=O,i-l,f2
,...,
and the $‘s are orthonormal
over a volume
* See, for example, eq. (57) of ref. 8.
V=L3.
m, = 0, 1, 2, . . . ,
(9)
424
R.W.DAVIES
Introducing
the matrix
elements
pkkg(q) =
d3r $,*(r) eiq”’ tik, (r) y
(10)
s one can write for the RPA correlation 1
G, (4 = pv
ccc v (4)
51~
k’
v (4’)
Pkk’
self-energy
(4)
Pk’k
(-
4’)
sk’
(‘1
-
tl’)
n6,’
(b>
.
w’
(11)
Here we are using the finite temperature
formalism,
S,,(z, - C,,)= [Zl - I$ - Ek’]-’
with rfi(21 + 1)
z1 =
)
__-~
+
/.I,
/3
li=klT. B Furthermore,
the polarization
propagator
T&,,(<~,) satisfies the equation
r$,, (51’) = TM*(51,) - 5 %q” (&> v (4”) r&W (51,) 3 where, in the RPA
c
Pkk’t-
f-
(Ek)- f - @k’)
4)
Pk’kbi)
y-_E+-(p~’ k’
k
(12)
(13)
I’
kk’
From eqs. (9), (10) one sees that rr and rr’ will be diagonal in the q,,, qz indices but not in qx. Upon performing* the E-sum in eq. (13) (the terms which do not involve Kronecker deltas of k: can be handled by contour integration), we find that, to within corrections of order l/L, the diagonal part of eq. (13) is the standard RPA proper polarization function
s d3k
f-c& __k >-“r(E ~ ___ke!! >. &k -
Ek+p
+
It then follows from eq. (12) that, to within corrections of order diagonal part of nb,,(ll,) is the usual RPA polarization function
Presumably, polarization
(14)
(1,
the physical significance of the non-diagonal propagator is that the effective dielectric constant
l/L, the
part of the is x-dependent
* In analyzing this expression, it is convenient to split the sine functions into exponentials, and to recombine terms in such a way that kz and k’, can be summed over both positive and negative values.
SURFACE
425
POTENTIAL
in the region of the surface. The contribution to the self-energy function from the non-diagonal part of the polarization propagator is not easy to assess, and in the equations which follow we shall simply ignore it. With the approximation discussed above, we obtain for the correlation self-energy
Eq. (16) differs from the usual bulk expression only with respect to the value of the matrix elements pkkt(q). We can, therefore, immediately write down the analog of eq. (2) for the surface case. At zero temperature we will obtain
E(k) = - ;
cc v(q)
Pkk’(q)Pk’k (-
1 K(q, &k- &k’+ is) X
4)
1
- 1 -f?(sr-ek)+
~kk’(q)~k’k~-q)
c 4v(q) 1 +isj-l
c k’
k’
.
(17)
Making use of eq. (IO), we can then write down the natural generalization of the exchange potential corresponding to eq. (17). It is given by
“(‘)
1 -.______ = - z(r) @k(r) cs k,
d3r’ $,* (4 $4 tr’)
$k’ (r> +k (“1
k’
x-
1 V
c4
v(q) e iT(*-*‘)
Re --___-
1
K(q, &k- ek’ + is)
1
_ 1
R. W. DAVIES
426
1 xV
c
v(q) e
iq+
-r’)
1
~~
K(q,
Ek - ck’ + is)
-1
1 (18) .
Invoking the same approximations for the dielectric constant as discussed in section 2, and taking proper account of the phase space restrictions on the U-sums in eq. (18) it is a straightforward matter to verify that the Slater average of pk(r) over occupied states is given by P(r) = *V’“.(r) + +Vsc.ex.(r),
(19)
where Vex(r) is the pure exchange potential calculated by Juretschke, and where V”“. ex.(r) is the analogous screened exchange potential, corresponding to a screening factor ePskrr’. Here fl= k,/k,, and Y, has the same meaning as in eq. (8) of Juretschke’s paper. It is also easy to check that the form of eq. (19) obtains using the exact static dielectric constant, the only effect being to replace the screening factor e-Bkrr’ by some more complicated function of r,. 4. Discussion Loucks and Cutler have given the following average exchange-correlation potential
expression
V(r) = V”“““‘(r) + V,,,, . Here, Vi,. is a position-independent to long-range
correlation
for the Slater
(20)
term which Loucks and Cutler attribute
effects, and is given by
(21) with /I’ = kc/k,, and k, the plasma cut-off wave vector. The screened exchange term in eq. (20) was computed by Loucks and Cutler using an IBM 7074 for both a sine integral
screening
factor,
F@‘k,r,)
and the exponential
= 1 - f Si(/?‘k,r,),
approximation F(j?‘kfr,)
The results obtained by these authors are reproduced in fig. 1.
to it, i.e. N e-P’krrl.
for the Slater average surface potential
SURFACE
427
POTENTIAL
.8
I 0
0.2
0.4
0.6
0.8
kfx Fig. 1. The complete surface potential, including long-range correlation, as calculated by Loucks and Cutler. The screened exchange contribution to the curves was calculated for the sine integral screening factor.
To obtain the quantity p(r) relevant to our present calculation, we need, in principle, only average the result obtained by Juretschke for the pure exchange potential, with the results obtained by Loucks and Cutler for the exponential screened exchange term. Unfortunately, the numerical results from the calculation of Loucks and Cutler are no longer available. Therefore, in order to obtain a qualitative comparison of the two theories, we have made a crude visual extrapolation of the exponential screened exchange contribution from fig. 3 of their paper. The resulting curves obtained for p(r) from this extrapolation are displayed in fig. 2. The extrapolation is particularly crude as far as the details of the oscillations in the surface potential are concerned, but it should serve to indicate the gross structure. Comparison of figs. 1 and 2 shows reasonable agreement as a function of effective screening parameter. One difference which can be noted is that the curves of Loucks and Cutler become progressively flatter than ours as the effective screening parameter is increased. In our present theory, which the contribution from the line-integral correlation term to the Slater average potential has been completely ignored, we can reach no conclusion as to whether this discrepancy is real. We remark, however, that GadzuklO) has recently obtained similar results using a rather different self-energy analysis. Finally, it should be pointed out that the effective screening parameters in the two theories are somewhat different of the functions electron density,
428
R.W.DAVIES
0
2.0
4.0
6.0
8.0
kfx Fig. 2.
The contribution
p(u) to the surface potential for various values of 8.
i.e. our p involves
Cutler involves
the Thomas-Fermi wave vector, while p’ of Loucks the plasma cut-off wave vector. More explicitly,
and
p = k,/k, = 0.815 r$,
(22)
p = kc/k, 2: 0.47 rf ,
(23)
density electron gas*. Thus one has j?- 1.74/J’ in the high density regime. However, Loucks and Cutler prefer to regard the screening parameter /3’ more empirically, and hence have chosen not to specify its dependence on the electron density. In real solids this, is, perhaps, not an unreasonable point of view.
for the high
Acknowledgments The author wishes to thank C. T. Kirk for initiating his interest in this problem. Several discussions with K. J. Harte were very useful. The author would also like to thank Dr. H. Juretschke for supplying the numerical * See ref. 6 where it is pointed out that the original Bohm-Pines
estimate of j?’ is too small.
SURFACE
POTENTIAL
429
results from his exchange calculation. Finally, the support and encouragement of D. 0. Smith is gratefully acknowledged. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
J. Bardeen, Phys. Rev. 49 (1936) 653. H. J. Juretschke, Phys. Rev. 92 (1953) 1140. J. C. Slater, Phys. Rev. 81 (1951) 385. T. L. Loucks and P. H. Cutler, J. Phys. Chem. Solids 25 (1964) 105. D. Bohm and D. Pines, Phys. Rev. 92 (1053) 609; D. Pines, Phys. Rev. 92 (1953) 629. K. Sawada, K. A. Brueckner, and N. Fukada, Phys. Rev. 108 (1957) 507. J. Robinson, F. Bassani, R. Knox and .I. Schrieffer, Phys. Rev. Letters 9 (1962) 215. J. J. Quinn and R. A. Ferrell, Phys. Rev. 112 (1958) 812. W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133. J. W. Gadzuk, Quarterly Progress Report No. 66, Solid-State and Molecular Theory Group M.I.T., Oct. 15, 1967.