Theory of internal photoemission in sandwich structures

THEORY OF INTERNAL PHOTOEMISSION IN SANDWICH STRUCTURES

J.

KADLEC

Max-Planck-Instjtut für Physik und Astrophysik, 8 München 40, Föhringer Ring 6, Germany

(~

NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters) 26, No. 2 (1976> 69-98. NORTH-HOl.LAND PUItLIsHINC; COMPANY

THEORY OF INTERNAL PHOTOEM1SS1ON IN SANDWICH STRUCTURES J. KADLEC Max-Planck-Institut fur Physik und Astrophj’~ik,8 Munchen 40, Fohringer Ring ó, Germany Received February 1976

Abstract: The internal photoemission in sandwich Structures is treated as a multi-step process. The excitation functions for electrons in the electrodes are obtained as a result of a rigorous optical analysis of the light propagation and absorption. The e1ectron~--electron and ehectron—phonon interactions are described in terms of energy-dependent mean free paths. The calculation of the quantum yield and photocurrent includes the electrons which escape without scattering, after one scattering, and after two scattering processes. In calculating the probability for electrons to escape over the inner potential barrier within the structure account has been taken of both their being scattered within the barrier region and their quantum-mechanical character. The present calculations can conveniently be used for theoretical investigations of the photoemission in its dependence on various parameters of the structure. The formulae also retain their validity for the photoemission of holes when the quantities due to electrons are correspondingly replaced. An adaptation of the theory to the vacuum photoentission from thick as well as very thin samples is possible without difficulties.

Contents: I. Introduction 2. Optical analysis and excitation function 3. Electron transport in electrodes 3. 1. Transmittive photoemission 3.2. Reflective photoemission 4. Escape of electrons

71 73 77 8() (7 91

5. Quantum yield 6. Conclusion References Appendix A Appendix B

Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS> 26. No. 2(1976) 69-98. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders niust be accompanied by check. Single issue price Dtl. 12.50. postage included.

92 93 95 9o 07

J. Kadlec, Theory of internal photoemission in sandwich structures

71

1. Introduction The internal photoemission is frequently used to investigate the electronic properties of metal—insulator—metal (MIM) or metal—insulator—semiconductor (MIS) sandwich structures [1, 21. The most powerful tool of this method is probably the examination of the spectral dependence of the photocurrent flowing between the electrodes of the structure. Such measurements have also been performed with varying the other parameters of the structure, e.g. the insulator thickness, applied voltage, or thickness of the electrodes. The analysis of experimental results is almost exclusively based on the simple Fowler relation [1—41 YF(w) (hw 0)2, (YF.(w) is the quantum yield per absorbed photon, hw is the photon energy, and 0 is the photoelectric work function), obtained originally for the photoemission from metal into vacuum. Although in many cases this relation is very useful to describe the main features of the photoresponse above the threshold, it fails in explaining some detailed observations on sandwich structures, especially if the electrodes and/or the insulator are very thin [2, 4]. In the last years some attempts have been made to modify the Fowler relation in order to understand the particular observations. For example, Rouzeyre [5] has introduced some of the multiple reflections of the light in the sandwich into the calculation, Berglund and Powell [61 and Blossey [71 gave the treatment of the electron scattering in the barrier. Schuerme-yer et al. [81 have applied the Monte Carlo method for calculation of the photocurrent. However, the internal photoemission in sandwich structures has never been described in all its aspects by a unified complex treatment. It is the purpose of this paper to outline such a theory which would be able to account for a variety of important effects influencing the photocurrent in sandwiches. At the same time in striving for a general treatment we also realize the disadvantage of such an approach: it cannot be used directly to analyze the experimental results. The complicated expressions can be effectively dealt with only by a computer. On the other hand, various functional dependences of parameters can be accounted for without involving strong simplifications. The question of the influence of a certain variable (e.g. electrode thickness) upon the photocurrent can be answered quite accurately. This is the way in which our theory can promote the understanding of internal photoemission in sandwich structures. Moreover, the calculation of electron scattering in the electrodes is believed to be of interest also in the theory of ultraviolet photoemission spectroscopy of metals and of secondary electron emission. Some problems treated in this work arise only in the case of a thin-film sandwich structure and have no analogy in the vacuum photoemission from a semi-infinite solid. One of them is the presence of multiple reflections of light on the interfaces within the structure and their influence on the excitation of electrons. The exact analysis (within classical optics) of the propagation and absorption of the light wave is performed in section 2. Another special feature of sandwich structures is that the currents entering the barrier region from the two electrodes are in opposite directions; the net photocurrent is then calculated (in terms of quantum yields) as a difference of these currents (section 5). Transport of electrons through the electrode to the interface involving onestep and two-step scattering is calculated in section 3. The scattering probabilities are expressed in terms of energy-dependent mean free paths for electron—electron scattering 1e (inelastic processes) and for electron_phonon* scattering l~,(elastic or nearly elastic processes). The total mean free path 1 may be written as 1 = ielp/(ie + —

*By electron—phonon scattering we denote all lattice-related (defect, impurity, and phonon) scattering processes.

72

J. Kadlec, Theory of internal photoernisxjon in sandwich structures

E

FERMI LEVEL

EFO

/

~~/// -

/ E ~‘/i~

Fig. I. Diagram of the energy scale at an interface e.g. metal--insulator): the energy T is measured from the Fermi level.

The energies are measured [91 with respect to the Fermi level ~

=

0) of the electrode under

consideration (see fig. 1). Using the parabolic-band approximation* for the excited electrons with energy E ~ 0m’ the magnitude of electron hnomentum is then p = [21fl(E + E0)I 1/2 and its 0m is the maxirnuni barrier height, in the x-directed component 1/2 where effective electron mass,p~ E =the[2m(E~ energy+inE0)] excess of the Fermi level EF and E~the part of energy above EF associated with p~. Then the kinetic energy E~of an electron (within the electrode) associated with p~is E~+ E 0. The value E0 is given by the energy difference between the surface potential barrier (or inner potential barrier [I 0]) J7~and the barrier height (or photoelectric work function) 0: E0 = V0 [91 In the special case when the free-electron approximation remains valid down to the bottom of the conduction band, the energy F0 is equal to the energy E~which ——

~.

is the difference between the Fermi level and the bottom of the conduction band. Concerning the escape of electrons from the electrode, we doubt the often expressed conviction that the quantum-mechanical character of electrons can be neglected in calculating the probability of escape over the barrier. The smearing of the quantum-mechanical coefficient 0m transmission (in comparison to the classical [11--1 3] over the energy in the vicinity of the barrier maximum zero-or-one probability) can lead to a change of the functional dependence of the quantum yield on photon energy ~iw for 11w ~ Section 4 shows an approximate way to evaluate the escape probability which accounts for both the quantum-mechanical character of electrons and the possibility of their scattering in the barrier region. We should mention that, although all the considerations are made for the emission of electrons into the conduction band of the insulator, the calculations in this paper are also directly applicable to the emission of holes into its valence band. As long as the parabolic-band approximation is valid t’or excited holes it is sufficient br the description of the hole emission to replace the quantities related to the electron emission (effective mass, energy F 0, density of filled states, mean free paths, potential barrier, and so on) by the corresponding quantities related to the hole emission. *f/or non-parabolic bands see the comments in section 6.

J. Kadlec, Theory of internal photoemission in sandwich structures

73

Table I Problems of the internal photoemission in (MIM) sandwich structures

L Quantum Energy Distribution of Electrons at the M-I Interface

I

LTransport over/through the

___

Transport of Electrons in the Electrode

J

LExcitaton

Yield

Propagation and Absorption of the Light Wove

___

Scattering in Insulator

Function

Barrier

[ QM-Transmissivity L~Lthe Barrier

Inelastic

Scattering

LElastic

Scattering

Excitation of Electrons

A summary of the problems involved in the description of the internal photoemission in sandwich structures is presented in table I.

2. Optical analysis and excitation function A rigorous optical treatment of the absorption in a sandwich structure will be given in this section. The analysis is based mainly on the work of Pepper [141. Consider a plane-parallel sandwich structure (see fig. 2) consisting of three media M1—M2—M3 (e.g. metal—insulator—metal). The structure is in optical contact on the left-hand side with a nonabsorbing medium M0 and on the right-hand side with a medium M4 (e.g. vacuum and glass substrate, respectively). Assuming, for

simplicity, that all the media M1 are homogeneous and isotropic*, we can characterize them by a complex dielectric function e1(w) and a thickness d1, where w is the circular frequency of the light and the index! denotes the particular region. The dielectric function is related to the index of refraction n.(w) and the extinction coefficient k1(w) by e1(w) = [n1(w) + ik1(w)] 2 The plane separating M1.~1from M1 is referred to as the interface!. According to the usual experimental conditions the analysis will be restricted to normal incidence** of the light from M0 on the first interface, denoting this direction as the positively oriented x-axis with the origin (x 0) at the first interface. If necessary, the real and imaginary parts of a complex quantity are denoted by primes and double primes, respectively. *The treatment of uniaxially anisotropic media is indicated in Pepper’s paper [141. **The case of oblique incidence can be treated in a similar way distinguishing now between the two directions of polarization [141.

74

.1. Kadlec, Theory of internal photoemission in sandwich structures

/~

Mo

(vacuum)

-

(metal)

M~

~//

-

J~insulatorr/ -

~

/~3./~

(metal,

/

PA4

1

(substrate)

I~~sem~onducthr.?,j

1

2 /~

~ (~~////// 0

//

//

3/- ~ /.

____

d

1

~/

/4

/ /

//////

_______

d,.d2

Jig. 2. Scheme of a plane-parallel sandwich structure: the interface I x

//~~~) d,.d2.d3

=

x

0) is illuminated from the medium M0.

Assuming that absorption occurs via one-electron volume processes [9, 14]. the excitation function PexcjU. w, x) in the medium M1 defined as the number of electrons (per unit energy. unit length in x-direction, and incident photon of energy hw) excited to energy F at the position x is given by PeXcI(E.

w, x)

=

P01(E, w)’ i~1(w,x)

(1: ~ 0).

(1)

Here P01(E, w) is the normalized distribution function 191 for electrons excited to energy F, and the volumetric absorptance r~1(w,x) is defined as the ratio of the number of photons absorbed per unit volume at the position x to the number of photons incident per unit area of the irradiated interface 1. The distribution function P01(E, w) involves the density of initial and final states and describes the kind of optical excitations (direct or nondirect transitions). It has been treated in detail elsewhere [9, 10. 15—191. Our chief interest in this section will be devoted to the exact evaluation of the quantity r?1(w, x) which gives the spatial distribution of the excited electrons along the x-ax is. Following Pepper [14] the volumetric absorptance i~1(w,x) is given by the negative divergence of the Poynting vector S1 of the electromagnetic field in divided by the incident flux S~: divS(2) The Poynting vector is defined by

S1 ~Re{E7(p.

t)X H1(p, t)}.

(3)

where E1(p.

t) and H1(p, t) are the electric and magnetic fields, respectively, and the star superscript means a complex conjugate; p is the position vector, t is time. The electric and magnetic components of the electromagnetic field may he taken as plane waves;

E1(p,

t)

H1(p, t)

=

E,~exptik~p

iwt) + E,: exp(.1k1 p

iwt)

=

H exp(ik p

iwt) + H1 exp(ik7 p

iwt),

(4)

.1. Kadlec, Theory of internal photoemission in sandwich structures

75

where E1 and H1 are the complex amplitudes*, k1 is the propagation vector, and the superscript plus or minus denotes the quantities due to the wave propagating in the positive or negative x-direction, respectively. The amplitudes E1 and H1 of each wave are connected by Maxwell equations [14, 17]: k,XH1—we~e,E1,

(5)

k1XE1w~,H1,

where e,~,and ji,,, are the permittivity and permeability of the vacuum (the media are assumed to be nonmagnetic). Using the proper boundary conditions [14, 171 for normal incidence, it follows that the tangential components of the propagation vector k1 in all the media are zero. Let us denote the x-component of k by Then from the law of reflection ~.

=

—kJ~

(6)

e~,

=

is the vacuum wavelength of the incident light. In order to calculate the Poynting vector in M. it is convenient to express E~and E1 in terms of

where 7m,,,,

EJ=R,E~.

(7)

The complex coefficients T, and R1 can be found (see Appendix A) by solving the system of two linear equations —

—

._

/

_

(8)

0

9”~mconnecting

the amplitudes of the electric vector in the media

where matrix Mm_i the and square Mm is determined by 1

=

rn

.

—rm

[~

+

+

m

-

+

-

rn

m—1

rn—i Pm

exp[i(k~—Iç_~)p~] mP[~(An~~n_i)Prn] —rrnexp[i(k~ ~k~i)pm] CXP[1(k~~k~i)Prn1 (9)

The amplitude reflection coefficient rrn at the interface m is given by rrn = (em—i ~m)1(~m—l + (Fresnel reflection coefficient). The relation between E~and E~can be obtained from the condition that the wave in the final medium Mf propagates only in the positive x-direction, i.e. E~ 0: —

1E\ ~

—

)~f~f_i

—

—-

.

(10)

~

In particular, f = 4 for our sandwich structure (we assume that M

4 is semi-infinite in the positive x-direction). With regard to the usual experimental situation we specialize the above results to a *The complex amplitude in M1 is defined by the time-averaged field at the position p

f—i).

(0, 0, 0) (not by the field at the interface

J, Kadlec, Theory of internal phoroemission in sandwich structures

76

structure where only the media M~and M3 can absorb light (e.g. MIM or MIS). Consequently, we are interested in S1 and S3. First we calculate the complex coefficients 1. and R1 ( = 1, 3) using eqs. (7, 8, 9, 10). We obtain

T1

=

r1+l R [1 + r3r4 exp(2i~3d3)+ r2 exp( 2i~2d2){r3+ r4 exp(2i~3d3)} I r1 + I R exp(2i~1d1)[r2 ~1 + r3r4 exp( 2i~3d3)} + exp(2i~2d2){r3 + r4 exp(2i~3d3)} I,

=

T3

r1 + I R (r2 + l)(r3+ l)exp[i(~1

R3

—i— (r2 -F’

-----

~3)d1]

exp[i(~2

---

~3)d2).

(11)

r1 + 1

l)(r3+ l)r4exp[i(~1 + ~3)d1lexp[i(.~2 + ~3)d2]exp(2i~3d3),

where

R

{ 1 +r1r2 exp(2i~1d1)} ~ 1 +r3r4 exp( 2i~3d3)F + exp(2i~2d2){r2+r1 exp(2i~1d1)} {r~+r~ exp(2i~3d3)}.

=

Using the definition (3) and eqs. (4, 5, 6. 7) we obtain for the negative divergence of the Poynting vector in the medium M1 the expression 2 lE~l —divS 2 exp(---a 2 exp(+a 1= a~[ 1T11 1x) + 1R11 1x) + 2Re{(T7R1) exp(—2i~x)}j, ~——

where the absorption coefficient a1 is defined asa1 the incident photon flux is

2

S~ = lE~l ° 2~

=

2~’.and ~

=

Re~1L

~‘

(12)

lm{~.}.Because

(~is real).

(13)

the volumetric absorptance is given by ~ 1(w, x)

=

a.

[j7’.J2

exp(—ct1x) + IR1P exp(+a1x) + 2Re{(T7R1)exp(—.2i~x)}1

(14)

and the excitation function by Pexcj(E,

w, x)

=

P01(E, w)a1

[~~

2

exp( —a1x) + 1R1

2

exp(+a1x) + 2Re{(T7R,) exp(— 2i~x)}1. (15)

The three terms in the expression can physically be interpreted as the contributions due to the electromagnetic wave propagating in the positive x-direction, in the negative x-direction, and the

interference field between these two waves, respectively. Let us denote these three terms of the excitation function successively by superscripts 1 (positively directed term), 2 (negatively directed term), and 3 (interference term).

.1. Kadlec, Theory of internal photoemission in sandwich structures

77

Then Pexcj(E, w, x)

=

(16)

~ P~CI(E,w, x).

We introduce further the formal absorption coefficients ct

2~’,

=

and the formal operators* j3J(~.)~ lT1l2Re~ j3(w)~ IR 2Re{ 1I 2Re~(77R,)

~

—cs~~—2~’,

~

(17)

I.3k(w):

(def.: ,5)(w)X (def.: i3,2(w)X

IT, 2Re{X}), 1l IR 2Re{X}), 1I

(18)

(def.: ,31(w)X = 2Re{(T$R 1)X}).

Then the k-term of the excitation function is P~~,1(E, w,x)

A1(E, w)~3~(w)exp(—a~x) (k

1,2,3)

(19)

and the excitation function Pexc,j(E, w, x)

=

A1(E, w)

ki

~(w)

exp(—a~x),

(20)

where we have denoted A1(E, w)

F01(E, w)-~—cs,1.

(21)

The excitation function in the form (20) will be used for the calculation of the electron transport in the next section.

3. Electron transport in electrodes In this section the transport of electrons excited at the position x in the medium M, of thickness d1 to the interface will be calculated. The complicated transport problem is approached by describing the most important interactions between the excited electrons and the medium during their motion to the interface. It is assumed (see fig. 3) that an electron can (i) reach the interface without scattering, or (ii) suffer one elastic (electron—phonon: e—p) or one inelastic (electron— electron: e—e) scattering before reaching the interface, or (iii) suffer two scattering processes in one of the four possible combinations e—p/e—p, e—p/e—e, e—e/e—p, e—e/e—e before reaching the interface. The scattering processes of higher orders (three-step and more) are neglected because *The definition of~/(w)and ~/(u) including the operator of the real part Re{ is not obvious at the moment, but it simplifies the later formalism.

78

J. Kadlec, Theory of internal photoemission in sandwich structures

TE

RE dNpeo

-

~

//

‘

_____

e-p dN

0

~IIIii e-e

::::~: dN.eo

•

I

0

x

___________~

x

x’

d1

__________,__._L.

d1 +d2

x’

I

x

5’

—

X

d,+d2+ d3

Fig. 3. Scheme of the scattering processes witlun the electrode,

the probability for contributing an electron with sufficient energy after suffering such many-step scattering is very small in most cases of practical interest. Likewise, with regard to the energy re-

gion of interest (not too far above the barrier maximum) the possibility of scattering by plasmons will not be considered. The calculation of items (i), (ii), and (iii) represents an extension of the results of Berglund and Spicer [18] first to the finite medium thickness, and second to the two-step scattering processes (e.g.: e—p/e.---p). Furthermore, the calculation includes not only those electrons ,

which escape from the illuminated surface of a (semi-infinite) solid (henceforth reflective photoemission, RE), but also the electrons escaping from the interface on the opposite side to the illuminated surface of a (relatively thin) emitting film (l1enceforth transmittive photoemission, TE).

Contrary to the common practice [9, 181 we do not immediately evaluate the energy distribution of the escaped electrons, but rather first the differential energy distribution dN1(E, z, w) of electrons reaching the interface (before escaping) which depends on the electron energy E and the photon energy hw, as well as on the direction (relative to the surface) of electron momentum p. The distribution d1V1(E, z, w) is defined as the number of electrons excited per unit energy by the incident photon hw which arrive at the interface with energy E and the x-component of p being ~ The variable z in the distribution is equal to the cosine of the angle ~9between p and the .v-axis. The reason for this treatment is to preserve the explicit dependence of the distribution on the energy E~associated with the electron motion in the x-direction, where E~= + E0 = p~/2in. Similarly to previous work [9, 18], the following simplifying assumptions will be made: 1) The optically excited electrons are distributed isotropically in direction. 2) The probability of scattering processes may be described in terms of energy-dependent mean free paths. 3) Both elastic and inelastic scattering may be considered as isotropic. 4) The surface scattering is neglected. As a supplement to the discussion [9, 181 of some of these assumptions we present the following remarks. Concerning 3), this assumption is the only one which allows a relatively simple analysis of the scattering. In the case of the inelastic scattering in the energy range under consideration, this assumption upon isotropy is based on the examination [20] of the solutions of the

J. Kadlec, Theory of internal photoemission in sandwich structures

79

Schroedinger equation for the scattering problem. The possible anisotropy of elastic events can roughly be taken into account by thinking of two scattering mechanism [211, an isotropic and a completely forward. Because the scattering is elastic (or nearly elastic), the forward component changes neither the energy nor the momentum. But such a process can at most influence the apparent cross section for the elastic scattering. Moreover, the photoemission experiment~on sandwiches are normally performed with (evaporated) polycrystalline or amorphous electrodes at temperatures equal or above 77°K.~Under such circumstances (especially at T = 300°K) the possible anisotropy of the e—p scattering is practically washed out [221. Isotropic elastic scattering has also been assumed in Ballantyne’s calculation [19] of the phonon energy loss. Concerning 4), it has been shown [231 that the kind of surface scattering (specular or diffuse) does not change the number of electrons which are able to escape. The presence of surface scattering could, in principle, increase the number of these electrons [23] but it is not likely because of the very small escape cone for the electrons with energy just above the barrier height. On the other hand, electrons with energies several electronvolts above the barrier have a short mean free path for e—e scattering and the surface scattering increases the probability for the loss of the electron by a collision with another electron to the energy below the barrier height. Therefore we probably do not introduce a significant error by neglecting the surface scattering. In distinction to the previous work [3, 5, 6, 8, 9, 181 we do not assume the classical (zero-or-one) escape probability which is a gross approximation not only to the case of a thin insulator barrier, but also for the vacuum photoemission at high external fields if the electron energy is not far enough from the barrier height. This question has been investigated in detail elsewhere [11]. Under the assumptions 1) through 4) we formulate the probability of reaching the interface and the scattering probabilities in the medium M.. The probability for an electron with energy E and momentum p to reach the interface from a distance ~ without scattering (see fig. 4a) is [9, 181 Psurf,/(E, z, x)

=

~exp(—~/l1z),

(22)

where l~= l,(E) is the total mean free path in M1 and z = cosi~. The probability for an electron with energy E’ and momentum p’ to be scattered isotropically by another electron (e—e scattering) after moving a distance r’ at an angle J3’ (see fig. 4b) and produce a secondary electron with energy E and momentum p is~’ exp(—r’/l) e1~scat,j(E’,E, r’, tY) 5iflJ3ISeeI(EI, E) (23) ,

where l = 11(E’) and ~ = lej(E’) are the total mean free path and the mean free path for e—e scattering of an electron with energy E’, respectively. Seej(E’, E) is the distribution function [9] for secondary electrons produced by e—e scattering a primary electron energy E’. 1 andofmomentum p’ to be with scattered isotropically The probability for an electron with energy E by a phonon (e—p scattering) after moving a distance r’ at an angle i3’ and produce a secondary electron with energy E and momentum p is analogously

eThe probability (23) is a generalization of expression (37) of ref. [9] which describes the e—e scattering probability in the case when no other scattering mechanism is allowed.

.1. Kadlec, Theory of internal photoernission in sandwich structures

5(1

I

M

1

I

~

-------

~_~1II

Fig. 4. Scheme of the geometry for the evaluation of the probability, a) of reaching the interface, and b) of scattering.

pPscati(E,

F. r,

- exp( ----r’/r) i~’) =

~sins~’S5~1(L’. L)

(24)

.

0/

where = l~1(E’)is the mean free path for c--p scattering. Sepj(E’ F) is the distribution function t’or secondary electrons produced by c--p scattering of a primary electron with energy F’. This distribution reduces to a 6-function at F’ in the case of perfectly elastic scattering. Corresponding to all possible electron interactions assumed in the excited medium M1, the coinplete differential energy distribution dN1(F, z, w) is given by .

diV1

-

=

diV01

+

dN~01+ dNeoj

+

dN~~01 + dIV~.01+ dN5~01+ dN~501

(25)

where the individual terms on the right are due to ehectrons which: --

reach the interface without scattering (dN01) scatter once before reaching the interface (dNp0j and diVe01)

scatter twice before reaching the interface (dN~~01, dIVpeoj~~epOj and dIVee0j). We now return to the case of a sandwich structure where only the media M1 and M3 absorb light and, consequently, can produce electrons. Our interest will be devoted to the electrons which are directed into the nonabsorbing medium M2. In other words, we are interested in the differential energy distribution dN1(E, z, w) due to the transmittive photoernission t’rom M1 through the interface 2, and dN3(E, z, w) due to the reflective photoemission from M3 through the interface 3. In the next parts of this section the terms of’ diV1 and diV3 will be calculated.

3. 1 Transmittioe photoemission from M1 through the interface 2;j .

=

/

a) no scattering: diV01 The distribution is given by the product of the excitation function (20) and the probability (22) to reach the surface integrated over all possible Positions ot’ excitation .v (fig. Sa):

diV01(E, z, w)f

dxP5~~1(F, W, X)Pç~,~~ 5(E,

:, v).

(26)

Performing the integration we obtain

~

z, w)=~A1(F,w)~ T~1(E,:,w),

(27)

J. Kadlec, Theory of internal photoemission in sandwich structures

0

d

x

1

x’

0

a)

x

d1

0 x” x

b)

81

x~

d1

c)

Fig. 5. Scheme of the geometry involved in the calculation for electrons which a) do not scatter, b) scatter once, c) scatter twice before reaching the interface.

where we defined

j3~f(w)T~1(E,z, w)

T~1(E,z, w)

(28)

and T~1(E,z, w)

=

k

11z

cx1l1z

—

(exp(_~_)

11z

1

—

exP(--a~di)} .

(29)

b) e—p scattering: dN~01 Consider an electron with energy E’ at the position x’. It is scattered by a phonon after moving a distance r’ at an angle i~ (see fig. Sb) with the probability ~F5~5~,1and reaches the surface with the probability ~Surf The distribution is then given by the product of the excitation function taken at energy E’ at the position x’ with these two probabilities integrated over all possible positions of excitation x’, positions of scattering x, and over 9’ all(xprimary energies E’ contributing* to = x’ + r’ cost9’) we have the final energy E. Introducing polar coordinates r’ and t ~.

d

11w

dN~01(E,z, w)

=

1

f

1~surf,1

X (~P~11~~ (E’, E,

r’,

n

(di--x’)/cos~9’

1r/2

dE’f dx’ Pexc 1(E’, w, x’)[/

di~’f

+

f

--x’/cos~3’

d~’f

(30)

(E, z, r’, s3’)),

~‘~

where the expression in the double-brackets is to be applied as an integral operator on the function behind it. After some calculations (see Appendix B) we obtain:

i

11w

dN~ 01(E,z, w)

=f

~

-~

(31)

dE’A1(E’, W)Sepi(E’,E)~1k=1 T~01(E’,E, z, w)

where we defined *Strictly speaking, the lower limit in the integral over E’ should read E also gain the phonon energy SEp. Since I~Ep‘~ E (e.g. ~

—

~Ep instead of E, because the electrons can, in principle,

~ 10-2 eV in metals) one can write E

—

E.

82

J. Kadlec, Theory of internal photoemission in sandwich structures

T~Ø1(E’,E, z, w)

ji~(w)T~01(E’,E, z, w)

(32)

and

)

T~01(E’,E, z,

+

—~—-~~

ct1(a111z

1)

—

The expressions

--

=

(lZ)2

+

t~p(~~1)Qhh1

exp(—a~d1)Q~61 + exp(-- --—)Q~5,1 11z

e(_

+~

a1

-

d1

Qi2,i

Ei(--) 1 11

+

exp t--(a~di+ L

--

(33)

11z

Q are defined later by eqs. (42), F1(~)is the exponential integral of the first

order (see Appendix B). c) e—escattering: diVeoi Using the same consideration as in the calculation of dN~01we obtain for diV501 the following result: d1V501(E, z, w)

f

=

dE’A1(E’, W)Seet(E’, F)

~—

2I~1T~1(E’.F,:,

(34)

w).

eI~~_

E

where

T~ 1(E’,E, z, w) and

(35)

T~Q1(E’,F, z, w),

is determined by eqs. (32, 33, 42).

d) c—pie—p scattering: dN~~0~ We proceed to the mathematically more complicated case of two-step scattering. An electron excited at the position x” to energy F” will be scattered by a phonon to energy F’ at’ter moving a distance r” at an angle t~” (see fig. 5c) with the probability pPscat,i(F”. F’, r”. i~”). It travels on at a random angle i3’ and will be scattered by a phonoti for the second time to energy F after moving a distance r’ with the probability pPscat,i(E’~E, r’, t9’). Introducing polar coordinates r”, i~” and r’, ~3’at the positions x” and x’, respectively (x = x’ + r’ cosi~’,x’ .v” + r” cos~”)andintegrating the product of the probabilities over all possible positions of excitation x”, positions of scattering x’ and x, and over all energies F” and E’ contributing to the final energy E, we have

z, w) =

,f

dE”f

ii

(d 1 —x”)/eos~”

iT/2

x~(f d~”/

dE’fdx”P

dr”+fd~”/

(F” w, x”)

—x”/ cosh”

dr’j(f

(d ~-x”

ir/2

—r” cos ,3” )/cos ,9’

d~’/

dr’

—(x”+r”cos~”)/cos~’

+

f d~’f

dr’ )~~(pPscati(E”~ F’, r”, ~

~p~scat,

~(E’, F, r’,

~‘) ~surf,

1(L’, z, r’,

a’)), (36)

83

J. Kadlec, Theory of internalphotoemission in sandwich structures

where the expression in the double-brackets is again an integral operator on the function behind it. The calculation of the multiple integral is too lengthy to be shown. Some essential steps are given in Appendix B. The result reads*: hw c1N~~OLW,z,

~

E”

o.~)f dEuf

dE’A1(E’, W)Sep1(E”,E’)Sepi(E’,E)

T~~01(E”, E’, E, z, w),

81p11p1 ,,l, ki ~

(37)

where we defined T~ 01(E”,E’, E, z, w)

(38)

j3~f(w)T~~01(E”, E’, E, z, w)

and T~~01(Eu, E’, E, z, w) 3 [exP(_a~di)Qi,i_exP =

—

—

11z

k 2 (a111z— l~z (a1) 1) (exP(_cs~di)Q~iexp -

11zd1

~~)]Q3

2,1—exp [_(a~i+

(11z) a111z— 1

——---~--

(39)

(_ ~-~-) Q

11z

(— ~-~-) 11z Q~,

d1 [exp(_a~di)+exp(_~__)] Q9,1+ [1 2

(l1z)

i—

exp

[- (a~di+~~i-)Ij 11z Q~+Q~ -

d1

+exP[_(ct1di+7_.)]]Qio,l)

d 1 1) exp(_ _)Qt2,iQ~7,i_exp_a~di)Qii,tQ~8,i

—

k k a1(a111z

—

(liz)2 E1(-~-)(ex~[_(a~di+

—

(ak)2

—

1 + Q4,i}

~±)]

(

Q12~l_Qiii}

~-~)] E1(~)—Q~7,1+exp [~_(a~cii+ ~~i.)] Q~8~1}

E1(~) [exp(_a~di)—_exp (_-

—X~1(E”,E’, E, z, w). The function X~1is given by

X~1(E”, E’, E,

z, w) f =

du X~1(E”,E’, E, z, w, u),

(40)

where x~1converges quickly to zero for higher u. For many cases of practical interest the function X~1is negligible. If necessary, X~1can be obtained by numerical integration of x~1,where *Concerning the limits of the integrals over energy, see the footnote given in connection with (30).

84

J. Kadlec, Theory of internal photoemission in sandwich structures k ~ XTI(L ,

=

,

E, L, z, w, U) = 3d (liz ) 1 k /l1z(l~u+1~’) \ / d1 \ /l1z(l~u+l~’)~1 exp [---(d1/l~+d1u/l~’)I I exp(--a1d1)L2i~-——i—exp i-iL2i—-—-—~ I I — a~l1z--1 L \l’~’(l1z—-l’~) / liz! \ l~(l1z+l~) Li

r

-

—

rexpi-—1a1d1+--—-I r i k d1 ~i L21—---/l~’(l1zu+l~ )~ /1’(l1zu—I —l~)\1 exp(—diu/l~) -——i---L2,----—-

+

-

L

L

11z /1

\

\ l’~(l~z+l’~’) 1

—

-

~l’~(l1z—l’~’) 1]

i’~

riexpi---ia1d1+--—iIL2i--—------———i----L21—-—------r I k d1 ~ (l1zu+l~’)~ /1~(l1zu—l~’)\1exp(—d1u/l~’) 1(1 ~ii -~

+

L

L

11z1J

\.

11zd1 + ----- ---—---—----———--—

(a~)2(a~liz_ 1) +

[exp

[_

r

r

\ l’1 1z+l’1) /

-----

\ l~’(l1z—1~) /i

rh exp(—a1d1)L2i k ~ l~u+l~’ I d1 —---—-—--— i —exp u—

~ j lçu+ly ~i exp[—(d1/l+d1u/I~’)1 iL2u ——-----~ ii —-—-—---—\ 11z / ~l~’( I +a~l)1i —

L

‘l~’(1 —a~l’~)I

L2(~’~_~~1)] ~çl’~(1 —--a’~l~’) 1’~

(a~di+~-)1 L2(~~c-~i ~ 11z

l’~(I +a’fl’)

~ k d1 ~ /l~(u+a~l~’) /l~(u—cs~1’)\1 + icxpi--ta1d1+-----iiL2t—------————p---L21—----—-——--—-i I exp(--d1u/l’1’) L L \ 11z IJ \l~(I +a’~/~) I \l~(1 --- a’fl~)!J JY

l1z(a~l1z+1 )d1 +-—--——~--~----—--

~

+

r

r

r

~ d1 ~1 exp[—(d1/l~+d1u/l~’)I

k

[exp(—a1d1)---exp~---i—~j

,,

d1 ~

rexp(-—d1u/l’1 [~

flY

-

The expressions +

=

In (i+

exp(—d1u/l’1’) ------~-----

L2 i~l+~-u

\1

/l’i

L2~y Ujjj

~) ~7L~J

L2(~,a)].

+

(41)

Q are defined:

f d~ d1 ~) Q~1-L~(-~

~xp

+

+ln (1— ~~~E1(~L 11z / L \ l~’ -

\

L2(

~2

j

~~—-——--—-—

L2~j~ uj+

[~+exp ~_(a’~di+ p-)]] [~/~u

~-‘~

——-

-

+--~

i’/

---In

---

~

~

l~: —1

—ln(l

d1

[

-~

+

\

-

d1 ~

+

±~ 11z / L

-

[L2(l

+ \ l,’~

l1zl,~+ 1;’)

~

1

d L2(~

+

(42)

_L)_E1(_L ~_L~ ~‘ ~i l~z/ 2

---L

/ l~ iljZ—1 I /l1z-_l;’~ / l~: /11Z—1’\ 21-------—--I+L21--—---- I+L2l—-----—-~---L2I----——-J+L2I------—I --L2( —) ----L2~-—--\11z+1’! \11z+1’ / \l~z—l’ \11z_1’! \11z_-l ‘ \11z 1-’ /~: -

—~+

---

Q21

+ln(l

-

1’ d~ d1 ~t)Q141~~’1(~ ~-)Qii,j_exp -

ln(l +~)[E~(~+~)---ln(I --L2(~/~,)

+L2~,) / /.z \

+

+

L2H,) l•z+f

---

d, [~(y

ln (1-

d/

~) [E~(~

+

6

-

11z(1

1

~~)1[L2(l+~,)-~L2(F(~~). -

iT

+ 1’)

---

2

-

~L2(~7)—

lz

+L2~-~) L2(~7-~) /lz+f’ -L2(T~-~) l-z l.z+1Y + iT -

-

~.

J. Kadlec, Theory ofinternal photoemission in sandwich structures

d l~ l;’(l,z+l;) +exP(~)[L2(~)_L2(1(1 ÷fI))]+exP _________

=

=

+exp (_-~)[L2(-~-)_L2 d t’ (l;1(lIz_l;))] 1(l1z—17) i-exp

i

d

~_

~)

\

r

/

85

~_L2(1~,u1~)1

[i~4-~ /

~ (l,z+1) -I

(42 cont.)

(— ) d~r [L2(.~) ~\ _L2(1~’1))1 ,,‘(11z—l) —I

l+a1, k1” ~

d.

[

d.

Q~1= +ln(l —a’l,)Q’81—ln (1 _~j;,)Q~o,j+exp~_(1i;’ j+j)1 +ln

(1+L I’ ~ i;’

L2( —

i;+i;’ 17(1 _aj~l,~))] ______

d. d.~ 1—a’~ E1(—1÷_1_)+L2( ‘“~L( 1+a1 ~iy)_L2(1_a~i;i) \ l—a~l, ~ 17 l+a~l~

/l+aklI,) ii l—a71

2

+L

2(

1 ) —L 1 _aklII ________ ________ k l—a~l7 2(1_a~)2(i_~i;)+L2(1_aulj)

if

________

=

k1’/ \ L2( d. +-_~L~) d. +L2(1~H-a1-;~‘;)— 11 )~1(—~

‘/

(

l•/

1—a.1/.

/

ill!

)

1 2 l+a~l7 _L2(Il+a~dlFI\ “‘ )+L

(

l+a~l,

~

=

[

+ln(l +a~l)Q~7,1—ln(1+ a~/ / ~Q~s,j+exp[_(_L kill! i; +.~)1 i;’ —~ L2( 1

+ln( ~ +L

_—,

)

I

l—a”l~ / ~,

+ .I~.’) 11l~’ 11+11? ‘ / / _L2(l,~(l+a~l)

l+a”f

—

______

L2(

~ //

2

1

\ if )~~~L2(1+aIdlH) l+a,’l, jj 6’

—exp (_ y)[L2(~) ~ll —L2 (l;1(1÷a~l;) 11 +a~l7))]_exP(_~)[L2~~1 d r (l;~_L2(k)], l;(l+a~’)

d1 l7(l—a~~’l)~ ___________ Q~,1= —exp ~_.~—)[L2~_.~_)_L2(, _a~))l~(ly)~2Uy)_ d ~i; ~ 17(1 a~l;) d _exp(_.~)

ii

Q 9,j

.

=

=

+.~_

+ ~-

[~1,

[exp (_

+1d~[exp(_

1

d. ~

d

d d

d —exp (~-~)L2(.~-)

~)Q22,1_E1(-~)]

~)Q2O,f_Ei(..~)]

d-

‘

~

d —

exp

id d.\ 1’ Qii~/Ei(,—~+--~-)+ln(l+_!_), i~ l,z l,z

Id1

1’

~—

Q121=E1(-~L_”

+

ln(1

~1~ l~z

___~,

J. Kadlec, Theory of internal photoemission in sandwich structures

86

Qi

3jEl(+~)+ln(l

a~dj) + hn( 1

+

a~1),

E~(~ + a~dj)+ ln(l

+

a~l7),

+

=

=

---~),

Qi4,jEi(~_~)+ln(l

+~,

=

F~(-~ a~j) + ln( I —

Q~81= F1(—~ au,) —

+ ln(l

(42cont.)

-

- -

a~l’).

Q2o,j=Ei(-~f_~)+ln(l ---~7)~

Q22,j=Ei(~_~)+ln(l

1)

where 1 is the index of the medium (j = 1, 3) and the superscript k corresponds to the three terms of the excitation function (20), k = 1, 2, 3. F1(~)is the exponential integral of the first order, L2(~T)is the dilogarithm, and ln(~)is the principal branch of the logarithmic function in the complex plane (cf. Appendix B). e) c--p/c—c, e—e/e-—p, and c—c/c—c scattering, resp.: dNpeoi~dNepoi~and dNeeoi~resp. In the same way as in the evaluation of diV~~01 we obtain

z, w)

~peoi(E,

~

~

X

=

f~”fdE, A 1(E”, W)Sepi(E”, E’)Seei(E’, F)

T:eot(E”, E’, E, z, w).

(43)

p1 el k—i

z, w)

~epoi(E,

x

81,,ll,

~

=

~ dE”f

dE’ A 1(E”, w) Seei(F”n F’) S5~1(E’.F)

T~01(E”,F’, E, z, w),

(44)

el p1 k—i

z, w)

~eeoi(E,

x

1

81e11e1

k1 ~

=

f

dE”fdE’

A

1(E”,

T~01(E”,F’, F, z, w).

W)Seei(E”,

F’)

Seei(E’, F)

(45)

J.

Kadlec, Theory ofinternal photoemission in sandwich structures

87

The calculation shows that T~’eoi(E”,E’, E, z, o.) = T~’~O1(E”, E’, E, z, w)

=

T:eoi(E”, E’, E, z, ci.,)

T~~O1(EU, E’, E, z,

~)

(46) where T~~01 is determined by eqs. (38—42). 3.2. Reflective photoemission from M3 through the interface;f

=

3

The terms of the differential energy distribution dN3 are calculated below analogously to the evaluation of the terms in dN1. a) no

scattering: dN03 d1+d2+d3

dN03(E, z, w)

J’

=

dXPexc3(E, w, X)PSUrf 3(E, z, x)

(47)

d1 +d2

because the possible positions of excitation are between d1 ÷d2 and d1 integration we have dN03(E, z, w) ~A3(E,

w)

k=i

+

d2

+

d3. Performing the

(48)

R~3(E,z, w),

where we defined R~3(E,z, w)

(49)

Lb~(w)R~3(E, z, w)

and z, w)

=

exp[—a~(d1 +d2)]

13z a313z + 1

—

exp[_(a~d3+ E~2~)]) 13z .

(50)

b) e—p scattering: dN~03 We write a parallel expression to (30): hw

dN~03(E,z, ~

=

.f

d1+d2+d3

dX’Pexc,3(E’, w, x’)

dE’f

(51)

E

xE[f

?r/2

(x’—d1 —d2)/cosi~’

dt~’f

The calculation results

dr’

ar

+ fdi~’f

(x’—d1 —d2—d3)/cosi3’

drl]1(pPscat,3(EI, E,

r’, 1~’)Psurt3(E,z, r’, a’)).

88

J. Kadlec, Theory of internal photoemission in sandwich structures 3

diV~03(E,z, w)

f

=

dE’A3(E’, W)Sep3(E’1 F) ~

~

R~03(E’,F, z, w)

(52)

E

where we defined

~(w)R~03(E’, F, z,

R~03(E’,F, z, w)

(53)

w)

and R~03(E’,F, z, w)

+ ~

=

k

exp[—a~(d5 + d2)l[+

+

a3(a3l3Z

+

exP[_(a~d3 +

I)

(l~z)2

—)] Q~6,3 13z

[_(a~d3

(Qii,3

--

~

a3

13

+

~)]Q123}

(54)

—)]

exp(---a~d3)+ exp(_~ 13z

c) c—c scattering.’ diVeo3

1

FIW

diVeo3(E, z, w)

f

=

dE’ A3(F’, W)See3(E’, F)

~

~-

E

R~03(E’,E, z, w)

(55)

e3k_t

where R~03(F’,F, z, w)

R~03(E’,F, z, w)

(56)

and R~03is determined by eqs. (53, 54, 42).

d) e---p/e-—p scattering: dNppo3 The parallel expression to (36) is: E”

11w

dN~~03(E, z, w)

=

f E

n/2

f

dx” Pexc3(E”~w,

E

dr”+fd~”/

is

dr”)

(x” --d 1 -—d2 —d3 —r”cos,3”)/cose’

dr’+fd~’f

d~’f

X (pPscat,3(E”.

(x” d1 —-d2—-d3)/cosh”

ir

(x”—d 1--d2 —-r”cos,9”)/cos~3’

ir/2

(f

d1+d2+d3

dE’

(x”—d1 --d2)/cosit”

x~fj’ d~f

x

d.P]IIf

F’, r’’, t~”)pPscat3(E’, E, r’,

t~’)Ps~~f 3(E,

dr’}~ z, r’, ~3’)).

(57)

The calculation results

diV~~03(E, z, w) =

ff

w)S5~3(E”,E’)Sep3(E’, E) 8l’~3l’~3 k=1 R~~03(L, F’, F, z, w),

(58)

J.

Kadlec, Theory of internalphotoemission in sandwich structures

89

where we defined (E” E’, E, z, w)

~k

~~(w)R~~03(E”, E’, E, z, w)

(59)

ppO3

and R”ppO3 (E” E’, E, z, w)

[÷a313z+ 1 (Qi,3 3 _______

x +

exp[—a~(d1 + d2)]

=

~3~

132 _____________ (a~)2(a~l3z+

1)

—

exp

[— (a~d3

d3

+

—

13z

exp (— ~-~-)Q33 + exp(—a~d3 )Q4, 13z /

d3 ti

(— exp[_(a~d3

/

~32’J L~+ Q~,3+ exp(—a~d3)Q~ 3

([i

-~-‘~-~-~

+

~-~-~lQ9,3+ [exp(_a~d3)

exp [_(a~d3+ 13z 1JJ

____________ 2

d3\ l~I

exPI,,____JQ~,3)

exP(_~)]Qio,3) 13z

d

(13z)

+

+

I

3

______

\l

k

a~(4l3z+ 1) (Qii.3~.3 exp [_(a~ d3 + —) 132 JI 2E (13z) 1(~) / d3 )Q12,3 exP(_a~d3)Qii3) — exp~—----1 13z —

—

13z

Id3

(_ [1

+ (~4)2 E1I,,—~—) 13 +

—

~-)

exp [_ (a~d3÷132 £~-)]]E1(.~) exp (_ 13z Q~3 + exp(—a~d3)Q” 13 —

X~3(E”,E’, E, z, w)].

(60)

The expressions Q are again defined by (42). The function X~3is given by

X~3(E”,

E’,

z, w)

E,

=

f du 43(E”, E’, E, z, w, u),

(61)

where x~3(E”,E’, E, z, w, u)

3d

=

(13z) 3 C [L2(l3z(l’31~+~) \

= —

a313z+ 1 d3)

+[exp( +

+

[exp

132 —

(

~-~-“)lL Il?.z(l’3u+1~’))1

l~(l3z—l~) )—exp[_(4d3+

i3ziJ

1(l 1 (l’3 3zu+1 3)

~

~—(d3/l~+d3u/l~)1

i’

l~’(l3z+l~)-I

l~(13z+l~) ) — exp(_a~d3)L2(l~(l~,z—l~’) 3 3z_l~)y] -I exp(—d3u/1~)

_______

_______

_________

I,’)

d3 132

\

13zd3

_________

(l~(l32u+l~)) exp( a~d 13(132+13)

C[L(

(a~)2(a~l3z+l)

—

—

)

l~u+l’~

—

l~(1+a~t3) ______

exp

Il’3(l32u_l’~))1exp(—d3u/13,) 3)L2~~-~~l(1 1’) i l’

[ (a~ —

l’3u+l~)1exp[—(d3/l~+d3u/l~)J d3+__)IL2 13Z ~ (l~1(l_a~l~) i d3

\1

90

J. Kadlec, Theory ofinternal photoemission in sandwich structures

+

[ exp

+

(

—

[exp(

d3 (f3I(u_a~lI3)\ . Il~’(u+a~l~))] exp(—d3u//’3) — i3z) L2 l’~(l—a313), k” /I exP(_ct~d3)L2(,13( I l+a~l~) 13 ‘I 1) 1 /( l’3(u+a~l’3’)\’1 exp(—d3u/l’3 d 3’ L2 (t3(u_a3l3)~ J exp(_a~d3)L2(—,, i3z) l’~(l—4l~)/ \13( —--—

——

+

+

~)i

13z(a313z’l)d3 ([I

—

[exp(_-a~d3)

exp[_(a~d3+~~2-)]] L~l’3+d3u//’3’)] -‘I L2(1+—u 13 l3z 13 \ l’~

1 exp(— d3)] [exP(—_d3u/1 /l’~ 3) L2~—,-13 u

—

l3zd~[exp(a~d3)

+

comments

)

)

‘‘I exp(—d3u/l’~’) +

3

/13

L2~_~u)]} 13

~) ~xp~id3u/l3)il~u)j.

~-~)1 132 L

exp(

J

The

-

—

(a~)2

—

-~

—

II

_~L~3uhhl’3)L2(~/3 13

+

L2~

(62)

presented in connection with the function X~.1apply similarly for X”R3’

e) c—p/e—c, e—e/e—p, and c—c/c—c scattering,

resp..’ dNpeo3,

dI’I~ep03, and

dNeeø3,

resp.

The expressions corresponding to (43), (44), and (45) arc: /1w’

dNpeo3(E, z, w)

=

f

E”

dE”f dE’A3(E”, w)S5~3(E”,E’)See3(E’, F)

E

E

3

x

81p31e3

—-~---,---

~ ki

11w

dI\Tepo 3(E, z, w)

=

f

(63)

E’, E, z, w),

peO3 (F”,

~

E”

dE’A3(E”, W)Se53(E”, E’)Sep3(E’, F)

dE”f

E

E

3

x

,,l, k1 ~ 81e3’p3

~k epO3 (E”,

E’, E, z,

11w

dIVeeo 3(E,

z, w)

=

,f

E”

dE”f dE’A3(E”, W)See3(E”, E’)See3(E’, E)

E

x

(64)

w),

E

3

,,l , k1 81e31e3

~ ~‘eeO3

(F”, E’, F, z, w).

(65)

The calculation shows that Rk peO3

(F” E’, z, w)

(E” E’, E, z, w)

= ~k

=

Rk

epO3

where R~~03 is determined by eqs. (59—62, 42).

(E”, E’, F, z, w)

eeO3’

R~~ 03(E”, E’, F, z, w), (66)

J. Kadlec, Theory of internal photoemission in sandwich structures

91

4. Escape of electrons Consider a potential barrier (e.g. of an MIM structure) of the type indicated in fig. 6 with the maximum height ~m at the distance Xm from one interface. We look for the probability that an electron coming to the barrier from this interface with energy E and the momentum direction determined by z (i.e. with a particular energy E~)will surmount the barrier. Two effects will be accounted for: scattering of electrons in the barrier region, and their quantum-mechanical character. The probability of arriving at the position Xm without scattering is given [6] by exp(—xm/Xz), where A is the (energy-dependent) mean free path of an electron for (electron—phonon) scattering in the barrier region. Assuming now that the propagation of electrons which do not scatter before reaching the position of the barrier maximum is described by the quantum-mechanical transmission coefficient Pqm(EX, V), the escape probability of an electron with energy F and momentum direction due to z is Pesc(E, E~,z, V)

=

Pqm(Ex, V)

exp(—xm/Az),

(67)

where V is the external voltage applied to the structure the sign of V being that of the electrode M3. We note that Xm = Xm(V) and z = z(E, Er). In the above consideration two simplifying assumptions have been introduced. First, it was supposed, that all the electrons reaching Xm without scattering are accelerated by the built-in field to the opposite electrode and so contribute to the current, and all the electrons which scatter before xm are returned to the emitting interface and so do not contribute. In other words, the possibility of backscattering over the barrier maximum and/or of overcoming the maximum after a previous scattering before Xm have been neglected. Such an approach is based on calculations 3escthe changing [6, 24] showing that these processes can be described by a correction factor to ‘ slowly with xm/A compared to exp(—xm/Az). Moreover, this factor is for not too large xm/A in the

T~N M 1

M3

M2

M3

~(O)

metat=1

‘insulator ~metal

I

~

~

insulator

~Xm(V)

a) Fig. 6. Energy diagram of the

qV

b)

potential barrier in a MIM sandwich

structure: a) no voltage, b) voltage V applied.

92

J. Kadlec, Theory of internalphotoemission in sandwich structures

order of unity. The second simplification regards the separation of the escape probability into the part describing the scattering and the part due to the quantum-mechanical character of an electron. These two parts are actually not independent of each other. For a more correct approach it would be necessary to evaluate the transmission coefficient into an intermediate scattering state within the barrier region. Instead, we have identified the transmission coefficient for the electrons which do not scatter before Xm with the coefficient Pqm(Ex, V) calculated for A Evaluating Pqm(Ex, V), usual methods [12, 13, 25--281 of quantum mechanics can be used. The transmission coefficient Pqm(Ex, V) is a function increasing from zero to one with increasing energy E~.The functional dependence on F~is influenced by the shape and height of the barrier and, consequently, also by the voltage V applied to the structure. A forthcoming paper [11] is devoted to the evaluation of Pqm(Ex, V) for some barriers of practical interest and its influence -÷

on the photoemissive quantum yield.

We see that the escape probability (67) corresponds to that of Berglund and Powell [6] for electrons with E~>t~m when we assume a classical (zero-or-one) transmission coefficient. On the other hand, the escape probability is given by Pqm(EX, V) in absence of scattering (A -* oc),

We specialize our result to the sandwich structure in fig. 2 with a nonabsorbing medium M2 of thickness d2. Then the escape probability for electrons from M1 which have the distribution dN1(E, z, w) is given directly by (67), where Xm is the distance between the interface 2 and the position of the barrier maximum ~m• For electrons from M3 with the distribution dN3(E, z, w) we replace Xm by d2 Xm in the exponential function, and E~by E~ q V in the transmission coefficient (because of the shift of Fermi level in M3 by an amount q V, q is elementary tharge) and obtain: —

—

—

Pesc(E,Ex,Z,

V)_~Pq~(E~_qV,

d2—x’ ~ m).

(68)

V)exp(__

5. Quantum yield from M1 and M3: net quantum yield The number of electrons with energy between F, F + dE and the direction of momentum corresponding to values between z, z + dz which flow from the medium M1 into M3 is given by (69)

th1~1(1~’,z, W)Pesc(F, E~,z, V)dEdz

(note that E~= E~(E,z)). We obtain the energy distribution iV1(E, w) per unit energy and inci-

dent photon of electrons escaped from M1 by integrating (69) over all momentum directions contributing to F (i.e. over an escape cone in a general sense). However, because of the explicit dependence on E~in Pesc(E, E~,z, V) it is more convenient to split the composite variable z into the dependence on E and E~and to integrate instead of over z over all E~~ F01 + E contributing to a particular energy E. The constant E01 is equal to the value F0 (see Introduction) taken in M1. We substitute z = [E~/(F + E01)I 1/2 into (69). Then 2 N1(E, w) =~(E+E01)”

f

E01-fE ~x

~1/2d~(E

F [~~]

1/2~W)Pesc(E~

F Ex,[E~]

1/2 -

E 01—qV

(70)

93

I. Kadlec, Theory of internal photoemission in sandwich structures

and the quantum yield Y1(w) per incident photon (71)

Yi(w)fdENi(E. w). Correspondingly, for electrons escaping from M3 into M1 we obtain 12 dN

N3(E, w) = ~ (E ÷Eo3)~hf2f E

dE~~

1 3 (E, [F +E03

~)~esc

[~

(E, E~,

+E

1 1/2 V)(72) 03

03+qV

and w).

Y()JdEN(E

(73)

The net quantum yield Y(w) per incident photon flowing between the electrodes is then Y(w) = Y1(w)

—

Y3(w).

(74)

The lower limits F01 q V and E03 + q V in the integrals in (70) and (72), respectively, originate from the fact that the electrons which have E~below the Fermi level of the opposite electrode (cf. fig. 6) will find there practically no available states (T = 0°K), and therefore their transition probability is identically zero. Our consideration makes no attempt to involve the tunnel currents which flow between the electron states situated energetically below the Fermi level of the negatively biased electrode. These currents are usually eliminated (e.g. by lock-in amplifier method) in photoemission experiments. In other words, eqs. (71) and (73) describe the “true” photocurrents which are due to photoexcitation. Furthermore, because the contribution to the photocurrent of electrons which are excited to energies far below the barrier maximum is small, the exact value of the lower limit in the integrals is not of relevance. The problems discussed in this paragraph vanish, of course, if V = 0. —

6. Conclusion Following the problems connected with the internal photoemission in sandwich structures and reviewed in table I we have successively evaluated the excitation function (20), the differential energy distribution dN1(E, z, ~) due to the transmittive photoemission from the top electrode M1 (it equals the sum of the expressions (27), (31), (34), (37), (43), (44), and (45)), and the differential energy distribution dN3(E, z, w) due to the reflective photoemission from the base electrode M3 (it equals the sum of the expressions (48), (52), (55), (58), (63), (64), and (65)). Then accounting for both the scattering of electrons in the barrier region and their quantum-mechanical character, the escape probabilities (67) and (68) from M1 and M3, respectively, have been given. The net quantum yield Y(o.,) (74) is calculated as a difference of the quantum yields Y1(w) (71) and Y3(w) (73) originating from M1 and M3, respectively. The net photocurrent which can be measured in an external circuit can be obtained by multiplying the net quantum yield Y(w) by the elementary charge and by the photon flux incident per unit time.

94

J. Kadlec, Theory of internal photoemission

in

sandwich Structures

Although the general expressions for calculation of the quantuni yield are rather complicated, there can be found some conditions leading to considerable simplifications ..So, for example, if the mean free path l~for c—p scattering in the electrode for all electron energies under consideration is greater than the mean free path ‘e for c c scattering, then the contributions due to the twice-scattered electrons can be neglected. In the Introduction we have for simplicity assumed a parabolic-hand approximation for excited electrons. Then it was possible to separate simply the energy E~which is connected with the motion in x-direction. We will briefly discuss whether our calculation can be used in the present form when the above assumption is not valid. It can clearly be seen that the formulations of sections 2 and 3 (optical analysis and electron transport in electrodes, respectively) are independent of the energy-momentum relation. However, exploiting the formulae of sections 4 and 5 (escape of electrons and calculation of the quantum yield) it is necessary to separate F2. The sufficient condition to do this is that the partial derivative aF/ap~is independent* of the tangential corn ponents (p1,, p~)ofp. This condition is. of course, fulfilled not only for the parabolic energy-momentum relation. A more serious problem could, however, arise from the use of the generalized “escape cone”concept (i.e. the introduction of the variable = pr/p and its split into F and E~). The original treatment [181tacitly assumes that the energy F of an electron is unambiguously determined by the magnitude of its momentum, i.e. F = E(p). but F does not depend on the electron direction. This is, for example, the case for free electrons. A simple consideration shows that the sufficient condition to introduce the generalized “escape cone”-concept is the rotational symmet~yof energy-momentum relation with respect to the x-axis. in other words, the energy F~connected with the motion in the tangential direction (toSo, tilestrictly surface)speaking, may depend only on the 2)1~2 the utilization ofmagnitude the “escape of the tangential momentum p~ = (p~, + P cone”-concept means a restriction to special forms of the energy-momentum relation. On the other hand, if the range ~F of excited energies above the barrier maximum (~F hw ~m) is not too large with respect to FB + F (E~is the difference between the Fermi level and the bottom -of the conduction band), the mean contribution to the photocurrent comes from the electrons reaching the interface with a small angle t9 (see fig. 4a). But then the actual etiergy-momentum relation at these energies can be approached by a relation of required properties, e.g. by a fitted -

---

parabolic-band approximation. Moreover, in many cases of practical interest (metals) the actual Fermi surface [291 is not far from a sphere. So we believe that tile U5C of the “escape cone”concept presents no serious restriction on tile above calculation of the photocurrent, especially not for photon energies in the vicinity of the barrier height (hw So far we did not worry about time temperature at which the analysis is valid. And indeed, it was not necessary because all the temperature-dependent quantities ~P 01,S~1,S5~1,1~, l~1, ~ ~ etc.) enter the resulting expressions in such a form that any kind of temperature dependence can be accounted for. In this sense the presented formulae can be used without regard to the temper-

ature. We have found Ill] that the application of the quantum-mechanical transmission coefficient in the escape probabilities (67) and (68) is necessary in the investigation of the quantum yield in the vicinity of the threshold (hw ~m~’ especially when the barrier is very thin. The usefulness *Tlie author thanks Ku. Gundlach for this mathematically exact formulation.

J.

Kadlec, Theory of internal photoemission in sandwich structures

95

of the optical analysis in section 2 has been confirmed by the numerical evaluation [30] of the excitation function (20) which shows some interesting features of the spatial distribution of excited electrons.

Acknowledgements The author is grateful to K.H. Gundlach for many stimulating discussions and to W. Steinmann for some interesting hints at related papers. Thanks are due to Miss H. Tettenborn for computer programming and to Mrs. U. Lindner and Mrs. Ch. Degendorfer for their help during the preparation of the manuscript.

References [11 R. Williams,

Semiconductors and Semimetals, Vol.6, eds. R.K. Willardson and A.C. Beer (Academic Press, New York-London, 1970) p. 97 and references therein. [21J. Kadlec and K.H. Gundlach, Phys. Stat. Sol. (a) 37 (1976) and references therein. [31 R.H. Fowler, Phys. Rev. 38 (1931) 45. [4] G. Lewicki, J. Maserjian and C.A. Mead, J. App!. Phys. 43 (1972) 1764. [51M. Rouzeyre, Phys. Stat. Sol. 24 (1967) 399. 161 C.N. Berglund and R.J. Powell, J. App!. Phys. 42 (1971) 573. [7] D.F. Blossey, Phys. Rev. B9 (1974) 5183. [8] F.L. Schuermeyer, Ch.R. Young and J.M. Blasingame, J. App!. Phys. 39 (1968) 1791. 191 D.E. Eastman, Metals, Vol. 6, ed. R.F. Bunshah (John Wiley and Sons, Inc., 1972) p. 411. [10] H. Thomas, Proc. mt. Symp. on Basic Problems in Thin-Film Physics at Clausthal-Gottingen, 1965 (Vandenhoeck and Ruprecht, 1966) p. 307. [111 J. Kadlec and K.H. Gundlach, to be published. [12] S.C. Miller and R.H. Good, Phys. Rev. 91(1953)174. [13] S.G. Christov, Phys. Stat. Sol. 42 (1970) 583. [14] S.V. Pepper, J. Opt. Soc. Am. 60 (1970) 805. [15] J.J. Quinn, Phys. Rev. 126 (1962) 1453. [16] E.O. Kane, Phys. Rev. 127 (1962) 131. [17] F. Stern, SoL St. Phys., Vol. 15, eds. F. Seitz and D. Turnbull (Academic Press, New York-London, 1963) p. 299. [18] C.N. Berglund and W.E. Spicer, Phys. Rev. .136 (1964) A1030. [19] J.M. Ballantyne, Phys. Rev. B6 (1972) 1436. [20] P.A. Wolff, Phys. Rev. 95 (1954) 56. [21] R. Stuart, F. Wooten and W.E. Spicer, Phys. Rev. 135 (1964) A495. [22] F. Kus and J.P. Carbotte, Sol. St. Comm. 15 (1974) 127. [23) R. Stuart and F. Wooten, Phys. Rev. 156 (1967) 364. [24] E.O. Kane, Phys. Rev. 147 (1966) 335. [251 E.C. Kemble, The Fundamental Principles of Quantum Mechanics (Mc Graw-Hill Book Company, Inc., New York-London, 1937) p. 109. [26] C.R. Crowell and S.M. Sze, J. App!. Phys. 37 (1966) 2683. [27] K.H. Gundlach, Sol. State El. 9 (1966) 949. [28] B.A. Politzer, J. App!. Phys. 37 (1966) 279. [29] e.g. J.M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1962) p. 109. [30] J. Kadlec and K.H. Gundlach, J. Appl. Phys. 47 (1976) 672. [31] O.S. Heavens, Reports on Progress in Physics, Vol. 23, ed. A.C. Stickland (The Physical Society, London, 1960) p. 1. [32] P.H. Berning, Physics of Thin Films, Vol. 1, ed. G. Hass (Academic Press, New York-London, 1963) p. 69. [33] J. Kadlec, Math. Comp. 30 (1976). [34] V. Kourganoff and 1W. Busbridge, Basic Methods in Transfer Problems (Clarendon Press, Oxford, 1952) p. 253. [35] M. Abramowitz and l.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1965) p. 228. [36] K. Mitchell, Phil. Mag. 40 (1949) 351.

96

J. Kadlec, Theory of internal photoemission in sandwich structures

Appendix A Consider a system [31, 32] of optical media M0, M1, M2, 1, 2, ... f(see fig. Al). Denoting by

...

Mf separated by the interfaces (Al)

E~(p)E~exp(ik~p) and E~(p)=E;exp(ilçp)

(A2)

the time-averaged electric wave propagating in Mm in the positive and negative x-direction, respectively, solution of the equations of propagation leads to the following relations between the waves in Mm and Mm_i: E~i(Pm)=

rm

~l

[E~(Pm)+TmL~(Pm)]

(A3)

1 [TmE~(Pm)+~(Pm)]

(rm is the amplitude reflection coefficient defined in the main text,

Pm ~S the position vector at the interface m). Substituting for E~(Pm)from (Al) and (A2), and writing in the matrix form we have m-

~m

m

(A4)

,

E;~ where ____

+

exp[i(k~—k~_i)pm] 1 rmexp[i(k~

—

rmexp[i(k~—k~_i)pm1 .

k~_i)pm]

exp[i(k~

—

(A5)

k;?,_i)PmI

For convenience, we multiply (A4) with the inverse matrix C)7~ and obtain

1 N0

2 N,

rn-i

3 N2

m

m~.1

N,,,,,

~ic~) S

f—i

Mm

N,.1

N,

i(~)

S

S

S

~ (~) P1 P2 P3

Pm.i

~m

Pm,i

~“i-i

~-

Fig. Al. Scheme of a plane-parallel system of optical media.

~

I.

E~

Kadlec, Theory ofinternal phoroemis~ionin sandwich

structures

97

E~ (A6)

(E:)=~m(E:I:)’

where ~FI~mis given by eq. (9) in the main text. Using successively the recurrence formula (A6) the amplitudes of the electric vector in the medium M1 can be written in terms of the amplitudes inMk (k
~/~j—i

~

~k+2~k+l

(A7)

.

Appendix B The evaluation of (30) and (36) can be done in the following way. Performing the elementary integration over r’, x’ and r’, r”, x”, we obtain expressions which must be further integrated over the angles ~‘ and i~’, i~”. The substitution cost~’= ±t(cosi~”= ±r)converts the integrals into the type J’dt ,~

Q(t) f(t) H~1(a1t+b1)

(a1* 0)

(Bl)

where Q(t) is a (generally complex) polynomial in t of order p (p ~ q) and J~t)is one of the functions exp(—’y/t), ln(mt + n), ln(mt + n) . exp(—7/t), or exponential integral E1(m/t ÷ n). The solution of integral(B1) for real integrands has been published recently [33]. A generalization of the results for complex integrands must be used in the present calculation and the real part of the resulting formulae is taken to express the differential energy distribution (cf. (18) and the corresponding footnote). For convenience, we recall the general definitions of some complex functions occurring in the resulting formulae of this paper. The exponential integral of the first order E5(~)is defined for complex ~by [34, 35]: ~‘

E1(flJ dy

exp(—y)

(iarg~l
(B2)

1’

If ~ = t r iO is real (t> 0) we have E1(—t

±iO) =

—Ei(t); lir,

where ~

Ei(t)

=

f

_OO

expy dy

(~~~oo < t<

00).

3’

The dilogarithm L~) is defined by [36]:

(B3)

98

J. Kadlec, Theory of internal photoemission in sandwich structures

L2(~)--fdy~-~ 3’~

(iarg(l —y)I <

(B4)

~).

0

The principal branch of the logarithmic function ln(~)is defined by ln(~)fdyi

(Iarg~I<~).

[351: (B5)

If ~ = t ~ iO is real (t> 0) we have ln(—t

±iO)

=

ln(t) ±lir.

(B6)