Theory on laser sputtering by high-density valence-electron excitation of semiconductor surfaces

382

Surface

Theory on laser sputtering of semiconductor surfaces Hitoshi Institute Received

by high-density

Science 248 (1991) 382-410 North-Holland

valence-electron

excitation

Sumi of Materials Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan 6 August

1990; accepted

for publication

11 December

1990

A theory on sputtering by sub-bandgap lasers is developed on the presumption by Itoh and Nakayama that rupture of a surface bond by two-hole localization leads finally to ejection of an atom from the surface due to Coulombic repulsion between the two holes and lattice relaxations. The lasers excite electrons in the valence band to unoccupied intra-gap surface states, giving rise to free holes in the valence band and surface negative charges which attract holes and let them move two-dimensionally along the surface. The theory is applied to calculate the ejection efficiency per laser pulse of Ga neutrals from the intrinsic Ga sites and that of Ga cations from defect sites of Ga adatoms on the GaP(l10) surface. It is predicted that both of them should increase exponentially with the laser intensity when it exceeds a certain critical value, although the former should increase only quadratically and the latter only linearly when the laser intensity does not exceed this value. The former is much smaller than unity, meaning that the sputtering yield of Ga neutrals does not fade with the repetition of laser pulses, while the latter is not much smaller than unity, meaning that the sputtering yield of Ga cations fades rapidly with repetition, in agreement with experiment. Abrupt superlinear rise of the sputtering yield observed can be reproduced in the region of exponential increase of the efficiencies.

1. Introduction Recently much attention has been aroused by laser sputtering of semiconductors and insulators including various III-V and II-VI compounds and oxides [l]. Intensities of laser pulses used for the sputtering are so low that the crystal surface has not yet been brought to melt. In this sputtering, therefore, energies of laser-excited electrons in the surface are utilized for ejecting atoms directly, not through heating the lattice. This sputtering reveals itself with following two striking features: First, the number of atoms ejected by laser irradiation rises superlinearly with the laser intensity when it exceeds an apparent threshold value above which the sputtering becomes detectable [2-51. Second, the sputtering is observed for laser energies below the band gap as well as above it [6,7], and it becomes most efficient for sub-bandgap lasers with a yield an order of magnitude larger than that for band-to-band ones [8,9]. We can suppose that the sputtering occurs through a process similar to laser- or electron-stimulated desorption of an adsorbate chemisorbed on a substrate. Historically, the desorption has been described in terms of a model proposed independently by Menzel and Gomer [lo] and by Redhead [ll] in 1964. In their model called the MGR model, desorption of an adsorbate occurs through rupture of the adsorbatesubstrate bond triggered by excitation of electrons in the bonding state of the bond to an antibonding one [12]. This model does not specify a concrete mechanism for excitation of electrons in the bonding state to the antibonding one. In 1978, Knotek and Feibelman observed that desorption of an adsorbate ionically bonded to the substrate arises when energies of incident electrons or photons exceed a threshold value corresponding to a core-excitation energy of an atom participating in the ionic bond [13,14]. This fact enabled them to specify a mechanism of antibonding-state formation, called the Knotek-Feibelman 0039-6028/91/$03.50

0 1991 - Elsevier Science Publishers

B.V. (North-Holland)

H. Sumi / Theory on sputtering by sub-bandgap lasers

383

mechanism, that Auger decay of a deep core hole produced by incident electrons or photons gives rise to simultaneous creation of two (or more) shallow holes in the adsorbate, and then it is Coulombically repelled by the substrate, being brought to desorption [LS]. A similar mechanism has also been applied to understanding desorption of an adsorbate covalently bonded to the substrate in which Auger decay of a deep core hole gives rise to simultaneous creation of two holes in the adsorbate-substrate bond and the bond is ruptured. The adsorbate is desorbed due to Coulombic repulsion between the two holes residing on the both sides of the ruptured bond [16]. The laser sputtering of compound semiconductors which will be discussed in the present work, on the other hand, is induced by lasers with energies of the order of the band gap. These energies are much smaller than the core-excitation energies giving the threshold in the ~otek-Feibelman mechanism. In order for the laser sputtering to occur, rupture of a covalent bond on the surface must at least take place, with two holes becoming localized at the bond by some mechanism. In this laser sputtering, however, holes initially produced by laser irradiation are delocalized in the free states in the valence band, with simultaneous production of free electrons in the conduction band in the case of band-to-band excitations, or, of localized electrons in intra-gap surface states in the case of sub-bandgap excitations. Itoh and Nakayama [17,X3] considered that two of the free holes produced in the valence band get to localize at a single bond on the surface by forming Anderson’s negative-U state [19] stabilized by strong electron-phonon interaction on the surface. They considered that it leads to rupture of the bond and the rupture is followed by the ejection of an atom since the two localized holes on the both sides of the ruptured bond repel each other due to Coulombic repulsion between them. An aim of the present work is to calculate the rate of two-hole localization at a bond on the surface by extending the previous treatment [20] on a problem of single-electron localization in the phonon field at a deep-level defect in semiconductors. The rate is regarded as giving the rate of ejection of atoms from the surface, in accordance with the presumption by Itoh and Nakayama 117,181. The present work, therefore, does not aim to justify the presumption directly, but to clarify whether the rate of laser sputtering given by it is consistent with the observations or not. An atom on the surface is fixed to it by three bonds, for example, in III-V compounds. In order for the atom to eject after rupture of a bond by localization of two holes, therefore, rupture of remaining two bonds must follow automatically, which requires an energy of about 2-3 eV estimated from the width of the valence band. The energy is supplied by the Coulombic repulsion of the two holes separating during the ejection and by lattice relaxations around the ruptured bonds, the former being estimated later in the text at about 1 eV and the latter being able to exceed easily l-2 eV. Therefore, their presumption itself seems plausible, but its unambiguous justification would require elaborate theories on actual lattice relaxations around ruptured bonds and it is outside the scope of the present work. In order for the two holes to get localized by means of negative-U-state formation at a single bond, they must first joint each other, and it is accomplished by quantum-mechanical tunneling against the Coulombic repulsion between them. When a large number of electrons and holes are produced by laser excitations, Coulombic repulsion between a specific pair of holes becomes weakened due to screening by other carriers. Itoh and Nakayama 117,181argued that the observed superlinear rise of the sputtering yield with the laser intensity could be brought about by a rapid increase in the rate of quantum-mechanical tunneling against the Coulombic repulsive potential which could become narrower and narrower with increasing carrier concentration by increased screening. In this respect, however, the present work considers another mechanism than they do: When the laser intensity is increased, the number of free holes produced in the valence band is increased. Then, their distribution, which is regarded as thermalized with the lattice, changes from the Boltzmann distribution to the degenerate Fermi one. In this degenerate distribution, the Fermi energy of holes is higher than the bottom of the free-hole band, and the average kinetic energy of free holes increases with the laser intensity in proportion to an increase in the Fermi energy. The increase in the average kinetic energy of free holes leads to a decrease in the average distance

384

H. Sumi / Theory on sputtering by sub-bandgap lasers

of quantum-mechanical tunneling against the Coulombic repulsion between the two holes. Therefore, it brings about an increase in the rate of two-hole localization, without the help of increased screening. The tunneling distance is usually so small, at most 10 A, that screening by other carriers is not effective at such a short distance in usual cases, except when holes are produced by lasers in such a large number that their average mutual distance decreases to a value much smaller than 10 A. This case is realized when holes are piled up at least to a few tenth of the total width of the valence band. Therefore, enhancement in the rate of the quantum-mechanical tunneling brought about by the decrease in the average tunneling distance should take place through the above-mentioned mechanism long before it should take place through the increased screening of Coulombic repulsion by the free carriers produced. It will be shown that the enhancement in the rate mentioned above manifests itself as an exponential increase of the sputtering yield with the Fermi energy when the distribution of laser-excited free holes can be regarded as degenerate. When the surface is irradiated by sub-bandgap lasers, electrons in the valence band are excited in intra-gap surface states and are fixed along the surface as surface negative charges. These surface charges give rise to an upward bending of the valence band toward the surface. Then, it should occur that free holes left in the valence band move two-dimensionally, being confined in the bent part of the valence band, so long as the Fermi energy is not very large. In this situation, the Fermi energy becomes directly proportional to the total number of free holes moving in the bent valence band and hence to the intensity of the laser producing them. Then, it can be predicted that the sputtering yield should rise up exponentially with the increase in the laser intensity when it exceeds a certain critical value where the Boltzmann distribution of holes produced in the valence band begins to change to the degenerate Fermi one. Below this critical intensity, the sputtering yield should increase quadratically with the laser intensity. The plan of the present work is as follows. In section 2, we develop a theoretical scheme for calculating the rate constant of two-hole localization at a bond on the surface. In section 3, the scheme is applied to calculate the current of Ga neutrals ejected from the (110) surface of GaP, for which the most extensive data has been accumulated by Itoh and his co-workers. In section 4, application is made to calculate the current of Ga cations ejected from defect sites of Ga adatoms on the (110) surface of Gap. The ejection of Ga cations occurs much earlier with increasing the laser intensity than that of Ga neutrals from the intrinsic Ga sites [9]. Both Ga neutral and cation ejections are discussed from a unifying viewpoint in the last section together with further details.

2. Rate constant for two-hole localization at a bond Two-hole localization can be realized either simultaneously or successively. A free hole can localize with a loss in its kinetic energy of the order of the half-width of the valence band. Then, simultaneous localization of two free holes suffers twice as much loss in the sum of their kinetic energies. In the semiclassical approximation for the electron-phonon interaction, the thermal activation barrier for a transition from free to localized states is proportional to the square of the energy loss suffered by the localization (with the proportionality constant given by the inverse of four times the energy of lattice relaxation associated with the localization) [20]. Using this fact we can easily show that the process of simultaneous two-hole localization has an activation energy larger than that of successive localization. Therefore, let us investigate only the latter. In the latter, the second hole makes a transition from a free state to a localized state at a bond where the first hole has already been localized. This second step is rate-limiting with a small rate. Before this step, the first localized hole should have been thermalized well with the lattice by inducing lattice distortions around it. Further localization of the second hole at the bond makes it ruptured. We consider, therefore, that localized states for the first hole can be found at various sites, intrinsic or extrinsic, on the surface as metastable relaxed ones with a thermalized population. Such localized-hole states can be made in various

H. Sumi / Theory on sputtering by sub-bandgap lasers

385

situations depending on atomic arrangements on the surface under consideration. Concrete examples of them can be found in sections 3 and 4 on the (110) surface of Gap. As described above, our problem of treating successive two-hole localization can be reduced to a problem of treating single-hole localization only in the second step. Then, the problem becomes similar to the one, treated in ref. ]20], of the capture of free carries by deep-level defects in se~conductors. An essential difference between the two problems is that a transition of the second hole from a free state in the valence band to a localized state at the first localized hole must be performed by quantum-mechanical tunneling against the Coulombic repulsion between the two holes. Therefore, the present problem requires an essential extension of the treatment in ref. [20], as developed below. Let us write by A$ the total number of bonds holding the first localized hole in thermal equilibrium. Moreover, let us write by I? the rate constant for a transition of the second hole from a free state to a localized state at one of these bonds. Then, the total rate for formation of the two-hole localized states on the surface is given by i=N,R.

(2.1)

This J is regarded as equal to the current of sputtered atoms, as mentioned in the previous section. Both Nd and R increase with increasing the intensity of the laser with concomitant increase in the total number of holes produced in the valence band. Let us start from a situation that the first hole has already been Localized on the surface with a certain amount of lattice distortions around it. Let us write by Q the coordinate of an additional lattice distortion induced when the second hole is also localized at the bond at the first localized hole. The energy of the lattice distortion can be written as v,(Q)

=

tQ”v

(2.2)

by adequately adjusting the scale of Q. The energy of hole-phonon interaction induced by the second-hole localization can be written as -(2S)“‘Q by introducing a coupling energy S. This S represents the lattice-relaxation energy associated with the second hole localization since )Q* - (2S)“‘Q = [Q (2S)‘j2]* - S. When the second hole remains free in the valence band, the total energy of the system composed of the second hole and the lattice distortion Q can be written as Ef + Vo(Q) where Ef represents the kinetic energy in a free-hole state specified by a suffix j. We consider that Er constitutes a continuum extending from zero with changing j, with the density-of-state function written as p(E)

= c+-Ef). f

(2.3)

When the second hole is also localized at the first localized hole, the total energy of the system is changed to f Q” - (2S)“‘Q supplemented by the loss in kinetic energy for hole l~a~zation at a bond, written as E,, and also by the Coulombic repulsive energy between the two holes localized in a single bond, written as U. Then, it is given by Vb(Q)=:[Q-(2S)1’2]2-S+E,+U.

(2.4)

The energy Eh should be of the order of half the total bandwidth for the free motion of a hole. Multiple curves of Ef + V,(Q) for various js above V,(Q) together with V,(Q) are schematically shown in fig. 1 as a function of Q. The formation of two-hole localized state at a bond on the surface and the following ejection of an atom from the surface can be described with reference to fig. 1 as follows: Let us first consider the second hole in a certain free state f on the adiabatic potential E, + V,(Q) where Er corresponds to the initial kinetic energy in the free state. At some point of Q, say Qf, this adiabatic potential crosses the adiabatic potential V,(Q) for the hole localized to the bond at the first localized hole. When the lattice fluctuates thermally from Q = 0 up to Q = Qf, the second hole with kinetic energy Ef can

H. Sumi / Theory on sputtering by sub-bandgap lasers

386

E$U-S

Fig. 1. Adiabatic potential V,(Q) for the second hole staying in the lowest free state in addition to the first localized hole, and that V,(Q) for the second hole staying in the localized state at the first localized hole, as a function of lattice distortion Q associated with the two-hole localization. Above V,(Q), there exist multiple adiabatic potentials for the second hole staying in one of the free states in the valence band.

\

make a transition to the localized state. After the transition the lattice relaxes now along the adiabatic potential V,(Q). During the relaxation, we suppose that the lattice distortion Q exceeds a critical value for ejection of an atom from the surface. The rate constant R for formation of the two-hole localized state is given by the thermal average in Ef of the rate of the transition to the potential V,(Q). To calculate R we must know the matrix element for the transition mentioned above of a hole from a free state f to the localized state at the first localized hole. Let us write it by Mfb and introduce a spectral function of the matrix elements for various fs as

g(E)= CIM/h12@-E,). The average occupancy function at temperature

(2.5)

at the free-hole state f with energy T and energy E = Ef,

F(E)=l/{exp[(E-E,)/k,T]

E, is determined

+I},

by the Fermi

distribution

(2.6)

where E, represents the Fermi energy. The distribution in the magnitude of the lattice distortion Q can be taken as the classical one proportional to exp[ - Vo(Q)/k,T] at temperature T since the laser sputtering has been observed at room temperature. When laser excitations cause elevation of the temperature of the free-hole system, T in eq. (2.6) should be replaced by the effective free-hole temperature, larger than the lattice temperature T. The matrix element M,h is determined by quantum-mechanical tunneling of a free hole with kinetic energy E, to a localized state at the first localized hole against the Coulombic repulsion between them. Since it is small, it can reasonably be treated with the second-order perturbation theory in calculating the rate constant R. Then, based on Fermi’s golden rule we can calculate R by 1Mfh ) ?I[ E, + V,(Q) - V,(Q)], representing energy coincidence between the two adiabatic potentials Es + V,(Q) and V,(Q), averaged with the initial occupancy F( Ef) at the free-hole state f over the thermal distribution of the magnitude of Q proportional to exp[ - V,(Q)/k,T], as

R=2(2~/A)CSdQIM/hl*B[E,+

V,(Q)-

&(Q)]F(Ef) exp[-K/',(Q)/kJ]

f

//dQ

exp[ - &tQ)/k~l.

(2.7)

387

H. Sumi / Theory on sputtering by sub-bandgap lasers

The first factor 2 originates in spins. Su~ation over f in (2.7) can be rewritten into integration over E with the spectral function g(E) of eq. (2.5), and integration in Q there can be performed by using eq. (2.2) for V,(Q) and V,(Q) - V,(Q) = Eb + U - (2S)“*Q, resulting in R = [4n/( h?Sk,T)]“*/dEg(

(2.7a)

E)F( E) exp[ - ( Eh + U- E)“/(4Sk,T)].

Let N, represent the total number of free holes produced in the valence band as a result of laser irradiation on the surface. Let us assume that Nh is much larger than the total number of the first localized holes on the surface waiting for an arrival of the second hole. Then, the Fermi energy E, in eq. (2.6) is related to Nh by

N,=~CF(E/)=~J~E~(E)/{~~P[(E-E,)/~,T~+I>,

P-8)

f

where we have used eqs. (2.3) and (2.6). Let us write by Ed the energy of the relaxed metastable state for the first localized holes on the surface. Then, exp[(& - EF)/kBT] m 1 must be satisfied under the assumption mentioned above. Energy Ed is related to the total number Nd of the bonds holding the first localized holes in thermal equilibrium, as Nd=N,F(Ed)“N,expl-(Ed-EF)/kST],

where NO represents the total number In the limit of small laser intensity, that E, becomes much lower than the and the distribution function F(E) of function

for

exp[(E,-E,)fk,T]

~1,

(2.9)

of bonds capable of holding the first localized hole on the surface. the total number of holes produced in the valence band is so small band bottom (at energy zero) of free holes with exp( E,/k,T) +z 1, free holes can well be approximated by the Boltzmann distribution (2.10)

B(E)=exp[-(E-E,)/k,T].

Then, in this limit, N, of eq. (2.8) can be approximated Z=2xexp(

-Ef/k,T)

= Z/dEp(E)

by Zexp(E,/k,T),

where (2.11)

exp(-E/kJ)

f

represents the effective number of the free-hole states in the valence band at temperature of exp(E,/k,T) s 1 for the low hole density limit is equivalent to Nh/Z = exp( E,/k,T)

+z 1,

with

F(E)

T. The condition

= ( Nh/Z) exp( - E/R,T),

in the limit of small laser intensity. Also eq. (2.9) tends to Nd = N,( Nh/Z) exp( - E,/k,T) Introducing these equations into R of eq. (2.7a) and then into J of eq. (2.1), we get J = ~~(~~/Z)*C

exp( -E,/k,T),

in the limit of small laser intensity,

(2.12)

in this limit. (2.13)

with C= f4~/(AZSkgT)]*‘*IdEg(

E) exp[ -E/k,T-

(E,+

U-

A!?)~/(~S~,T)].

(2.14)

In this limit, therefore, the current J of atoms ejected from the surface is proportional to the square of the laser intensity which is proportional to the total number Nh of free holes produced by the laser in the valence band. This feature is directly related to the origin of the laser sputtering in the Itoh and Nakayama model [17,18] where it occurs when two laser-produced free holes come to localize at a single bond on the surface. For general intensities of the laser, eq. (2.7a) for R can be simplified by taking into account the energy

H. Sumi / Theory on sputtering by sub-bandgap lasers

388

dependence of g(E): Let us consider a situation that a free hole with kinetic energy E( > 0) is going to be localized to a bond at the first localized hole. The two holes are electrostatically interacting through the Coulombic repulsive potential C(l)

= e*/+r>,

(2.15)

where c and r represent respectively the static dielectric constant at the surface and the distance between the two holes. After the free hole localizes at the bond, the distance between the two holes shrinks to a value a of the order of the bond length. Then, the Coulombic repulsive energy in the two-hole localized state U introduced in eq. (2.4) is U= .&+~3).

(2.16)

When E c U, it is not allowed in the classical beyond the classical turning point at a distance r,(E)

mechanics

that the free hole joins

the first localized

hcie

(2.17)

= e2/(eE),

which is larger than u. in order for the free hole to join the first localized hole beyond Y,(E), the free hole must tunnel through the Coulombic repulsive potential C(r) by a distance rc( E) - a. The situation is illustrated in fig. 2. The spectral function g(E) defined by eq. (2.5) represents the mean square average of the matrix element Mfh of the tunneling mentioned above at E = E,. The matrix element depends on the tunneling distance exponentially. Since the present tunneling distance r,(E) - a = e2( CJ- E )/( cEU> decreases with increasing E for E K U. g(E) should increase very rapidly with increasing E towards U. In fact, in the appendix it will be shown that g(E)

= N-‘h(E)p(E)/‘(

with N representing

p(E)‘+

the total number

(2.18)

~~~(~)~tE~/~~2~~ of bonds

on the surface,

and

0, h(E)

=

tan -‘[(U/E

exp ( - 4( A/E)“* 1,

- 1)“2]

+ 4( A/U)“‘(l

- E/U)“2)

for

E I 0,

for

Oi

for

E 2 U,

E-c U,

(2.19)

kineticenergy

-+__

0

a

r,(E)

--.-

r

Fig. 2. Coulombic repulsive potential between two holes, one of which is localized at the origin while the other staying in a free state is approaching with a kinetic energy E. The classical turning point of the free hole is r,(E), while it is going to localize at a distance u from the first localized hole by quantum-mechanical tunneling against the potential.

389

H. Sumi / ‘Theory on sputfering by sub-bandgup lasers

and p(E)=N-‘9’j[b(E’)p(E’)/(E-E’)] where 9

dE’,

(2.20)

represents taking the principal part of the integral, and the const~t

A is given by

A = e4Pn,/(2e2A2),

(2.21)

with the use of the effective mass m,, of free holes. Note here that A measures the strength of the Coulomb field on free holes, corresponding to the height of the acceptor level bound by a unit charge e in the effective-mass approximation if E takes its bulk value of the static dielectric constant in the crystal. At the surface, e is smaller than it, and A is much larger than the acceptor height. It will be shown in section 3 that A is estimated at about 0.25 eV while U is at about 1.0 eV. When 0 < E << U, using the expansion of +y*)-’ + $(l +y2)-’ -t 0[(1 +Y*)-~]} for y > 1, we get tan-‘y = $r -y{(l b(E)

+ S(A/U)“‘[l-

= exp{ -27r(A/E)“*

E/(6U)]

},

for

0 < E K U,

(2.39a)

and see that g(E) given by eq. (2.18) increases very rapidly in proportion to exp[ - 2n( A/E)‘/*]. In eq. (2.7a) for R, the Fermi distribution function E(E) for free holes with kinetic energy E decreases exponentially with increasing E as exp( -E/k,T) when E exceeds the Fermi energy E,. The E increase of g( E j mentioned above is much more rapid than the E decrease of F(E) for E > E, since A % k,T at room temperature, so long as E, is much lower than U. In eq. (2.7a), therefore, the E integration for R is determined dominantly by a contribution from a region of exp[( E - E,)/k,T] S- 1 but E K U. In this E region we can approximate F(E) in eq. (2.7a) by the Boltzmann distribution function of eq. (2.10), and eq. (2.7a) tends to R = exp( E,/k,T)

C,

for

E, much lower than U,

(2.22)

where C is given by the same expression as eq. (2.14). Further using eq. (2.9) for Nd, we can get an approximate formula for the current f of eq. (2.1) of atoms ejected from the surface as J = No exp(2EF/k,T-

E,/k,T)C,

in the general case.

(2.23)

In the limit of small laser intensity where eq. (2.12) is satisfied, J of eq. (2.23) tends to eq. (2.13) getting proportional to the square of the total number N, of holes produced by the laser in the valence band. In the general case, J is proportional to the square of exp( E,/k,T). Therefore, J increases exponentially with the Fermi energy EF when it increases beyond the band bottom of free holes at energy zero. This situation takes place when free holes are produced in such great numbers in the valence band as to be describable by the degenerate Fermi distribution, in other words, when Nh increases beyond the effective number Z of free-hole states in the valence band at temperature T with increasing the laser intensity. When the surface is irradiated by sub-bandgap lasers, electrons are excited in the intra-gap surface states localized on the surface, in concomitant production of free holes in the valence band, as mentioned before. In this case, surface charges by these electrons give rise to an upward bending of the valence band towards the surface, and free holes with low energies move two-dimensionally, being confined in the bent part of the valence band under the influence of the surface electrons, as also mentioned before. Then, the density-of-state function p(E) for free holes in the valence band defined by eq. (2.3) takes a two-dimensional form independent of energy E at least in the low energy region above the band bottom (at energy zero) of free holes, and it can be written as P(E)

= N/W,

for E 2 0,

(2.24)

H. Sumi / Theory on sputtering by sub-bandgap lasers

390

where N represents the total number of bonds on the surface as before and W is an energy scale of the order of the total bandwidth for the two-dimensional motion of a hole in the valence band. When eq. (2.24) is satisfied, Z determined by p(E) in eq. (2.11) tends to Z = (N/W)2k,T,

(2.25)

and N, in eq. (2.8) tends to N, = (N/ W)2k,T exp( E,/k,T) This equation

= exp( Nh/Z)

enables

J = [exp( N,,/Z)

Then,

we obtain

with eq. (2.25)

- 1.

us to express - l] 2J,,

ln[l + exp(E,/k,T)].

(2.26)

J of eq. (2.23) in terms of the reduced

in the general

hole density

NJZ

as

case,

(2.27)

with J, = N,C exp( - Ed/kBT).

(2.28)

A detailed condition for eq. (2.27) is given later by investigating eq. (2.7a) for R in more detail. In the limit of large laser intensity for the degenerate hole distribution in eq. (2.27), the current atoms ejected from the surface increases exponentially with N, as J = exp(2NJZ)

J,,

in the limit of large laser intensity

for exp( N,,/Z)

> 1,

that is, J increases exponentially with the laser intensity proportional to N,. Now, at the boundary between the two limits of small or large laser intensity, as J= J,,

at

exp(N,/Z)

J of

(2.29) J, of eq. (2.28) gives J

= 2.

(2.30)

An approximate formula for J, is obtained below, together with the detailed condition for eq. (2.27). Since b(E) = 1 for E 2 U in eq. (2.19) and p(E) = N/W for 0 I E I W in eq. (2.24) with W considerably larger than U, we can approximate p(E) of eq. (2.20) as a constant of the order l/W, with which we write p(E)’ as l/(cW’) by using a coefficient c of order unity, although c can be a little smaller than unity. Since b(E) < 1 for E -=KU in eq. (2.19), the second term in the denominator of eq. (2.18) is much smaller than its first term, and eq. (2.18) tends to g(E)

(2.31)

=cWb(E).

Using that b(E), from which g(E) is obtained in eq. (2.31) can be approximated by eq. (2.19a), we can show that the exponent of the integrand in eq. (2.14) for C has a sharp peak with changing E at E, = (2rk,TS/A)2’3A”3,

(2.32)

A = 2S - E,, - CJ+ ;( A,‘U)“2SkgT,‘U.

(2.33)

with

E, represents an average kinetic energy of a free hole with which it tunnels most dominantly to a bond at the first localized hole against the Coulombic repulsion between them. As will be shown later, E, can be estimated at about 0.2 eV, and the condition for eq. (2.19a) is well satisfied since U can be estimated at about 1.0 eV. Eq. (2.22) from which we obtained eq. (2.23) and then eq. (2.27) was obtained by approximating the Fermi distribution function F(E) in eq. (2.7a) by the Boltzmann one in the energy region around E,. Therefore, eq. (2.27) can be justified when exp[( E, - E,)/k,T] B 1, which can be rewritten with eq. (2.26) as exp( N,,/Z)

- 1 -=K exp( E,/k,T).

Since E, ze k,T

at room temperature,

(2.34) eq. (2.27) can be justified

over a wide range of NJZ.

H. Sumi / Theory on sputtering by sub-bandgap lasers

391

The exponent of the integrand in E of eq. (2.14) can be approximated by the second-order expansion in E - E, for estimating C, from which we obtain the magnitude of J, in eq. (2.27) as J, = :W,

(2.35)

exp( -E&J),

with

L- = ~(W/~)(8r/~)[(2~k,TS)2A/A5]1’6 X exp( -X/k,T-

3a[~tA/(2?rkJS)]“~

+ 8(~f/U)“~),

(2.36)

and X=

( Eb + U)2,‘(4S).

(2.37)

Note here that X represents the energy of the crossing point between the adiabatic potentials V,(Q) and V,,(Q) seen from the bottom of V,(Q) in fig. 1.

3. Ejection of Ga neutrals from the GaP(llO)

surface

--of the GaP(llO) and (111) surfaces gives rise to the ejection of Ga neutrals with a constant yield over thousands of times of repetition of laser shots [8,9]. It has been inferred from the constancy of the yield that the Ga neutrals are ejected from the intrinsic Ga sites on the surface, not from the defect sites of Ga adatoms adsorbed on the surface. In order to calculate explicitly the ejection rate of Ga neutrals with the theory developed in the previous section, we must know the structure of real surfaces, especially, their reconstruction in some detail. It has been considered [21,22] that reconstruction on the GaP(llO) surface in the vacuum does not destruct the original lattice period of the surface whose structure is well known [23-251. On the --- bulk crystal, similarly to that on the GaAs(ll0) (111) surface, on the other hand, reconstruction changes the original lattice period of the bulk crystal into the (2 x 2) superperiod. Even for GaAs, however, atomic arrangement in the (2 X 2) reconstructed structure of the surface has not fully been clarified yet [26-281. Therefore, calculation of the ejection rate of Ga neutrals will be performed only for the GaP(llO) surface. On the GaP(llO) surface, each atom has only three bonds with neighboring atoms. If the original atomic arrangement of the bulk crystal were retained also on the surface, each atom would retain an electronic structure of sp3 hybridization suitable for fourfold coordination. One of the sp3-hybrid orbitals projecting out from the surface would be occupied by only one electron, forming an unpaired dangling bond. Since energy of the dangling bond on Ga is higher than that on P, an electron in the former tends to move to the latter with the former completely emptied while the latter is completely occupied. Associated with this change in the electronic structure, reconstruction in atomic arrangement takes place in such a way [21-241 that Ga atoms on the surface sink a bit into the bulk, forming sp2-hybrid-like bonds suitable for planar threefold coordination with neighboring three P atoms together with a completely unoccupied nonbonding p-like orbital. On the other hand, P atoms protrude a bit from the surface, forming p3-like bonds suitable for cubic threefold coordination with neighboring three Ga atoms together with a completely occupied nonbonding s-like orbital. Such an anti-phase vertical shift of Ga and P atoms on the surface is called the surface buckling, and its presence has in fact been confirmed by the scanning tunneling microscopy for GaAs [25]. On the GaP(llO) surface, the completely unoccupied nonbonding p-like orbital on each Ga atom constitutes an unoccupied intra-gap surface state. Its presence has in fact been confirmed through inverse photoemission studies at a location about 0.3 eV below the conduction band minimum within the forbidden gap of about 2.3 eV [29]. The completely occupied nonbonding s-like orbital on each P atom, on It has been observed that pulsed laser excitation

H. Sumi / Theory on sputtering by sub-bondgap lasers

392

the other hand, constitutes an occupied surface state buried in the valence band. Its presence has been confirmed through photoemission studies [30] at a location about 0.8 eV below the valence band maximum, in agreement with theoretical calculations [31] which locate its dominant dispersionless part at about 0.7-0.8 eV below the valence band ma~mum, and also with angle-resolved photoe~ssion studies 1321. The laser sputtering experiment for the ejection of Ga neutrals from the GaP(llO) surface [9] was performed by a laser with an energy of about 2.1 eV which is about 0.2 eV below the bandgap. The laser can excite electrons in the valence band only to the unoccupied intra-gap surface states, leaving free holes there. We have no possibility of exciton production since the indirect exciton with a binding energy as small as 20 meV even in a bulk crystal [33] is located too high to be excited by the laser used, besides that the absorption coefficient due to excitons is very low in GaP with an indirect bandgap structure. Strong excitation with sub-b~dgap lasers, therefore, gives rise to many free holes in the valence band together with surface negative charges due to electrons localized in the intra-gap surface states on Ga atoms. Free holes produced in the valence band are very rapidly thermalized within the band, and some of them become thermally populated also in the surface states on P atoms buried in the valence band. Their population is very small, however, since the energy of the surface states is much lower than the valence band maximum compared with the thermal energy k,T at room temperature, although it is certain that they are present. Let us consider the case where these localized holes that are thermally populated in the surface states on P atoms form the first localized holes in the general theory of laser sputtering presented in section 2 when it is applied to the ejection of Ga neutrals from the GaP(110) surface. Then, the energy of these surface states corresponds to Ed appearing in section 2, and Nd given by eq. (2.9) corresponds to the total number of surface bonds extending from the P atoms holding a localized hole in the surface state since N, tends to the total number N of bonds on the surface in this case. Note here that Ed should be a little smaller than the value of 0.7-0.8 eV obtained as the binding energy of these states in the photoemission experiments or calculations, since 0.7-0.8 eV contains only the electronic part of the total energy Ed which is lowered by an energy of local lattice relaxations occurring after localization of a hole in the state. The left half of fig. 3 illustrates localization of a hole in the surface s-like state on a P atom. Following the general theory of laser sputtering presented in section 2, we consider next that another hole gets to localize at one of the two surface bonds extending from the P atom holding the first localized hole, by means of quantum-mechanical tunneling against the Coulombic repulsion between the two holes. When the second hole localization takes place at the second bond from the bottom in fig. 3, for example, the bond is considered to rupture automatically with the two holes separated from each other on both sides of the ruptured bond as illustrated in the right half of fig. 3: In fact, if the bond remained connected by a single electron even after the localization of the second hole there, intra-ato~c Coulomb energies would be raised very much, as explained below. A covalent bond connecting P and Ga atoms in GaP has a polarity of about three to one, meaning that about three fourth of the two electrons in the bond is localized

___.

additional

‘---

Fig. 3. Localization of a hole at a surface state on a P atom on the GaP(ll0) surface in the left half, and rupture of a bond from the P atom triggered by additional localization of one more hole to the bond in the right half. The solid lines represent surface bonds while the dashed ones represent bonds toward underneath atoms.

H. Sumi / Theory on sputtering by sub-bandgap lasers

393

on the P side while about one fourth is localized on the Ga side [22]. Then, if the bond remained connected in the right half of fig. 3, the Ga core with + 3 electronic charges would be surrounded approximately by 1: electrons, leaving 1: positive charges at the Ga site, while the P core with + 5 electronic charges would be surrounded approximately by 4: electrons, leaving $ positive charge at the P site. This unbalance in the charge distribution between the Ga and the P sites would raise very much the intra-atomic Coulomb energies at the both sites. In the situation shown in the right half of fig. 3, on the other hand, the Ga and the P atoms on both sides of the ruptured bond share two unpaired electrons one by one. In this case, about one positive charge is found both at the Ga and the P sites of the ruptured bond, and the intra-atomic Coulomb energies are greatly reduced. Although the interatomic Coulomb energy is raised thereby, it stays much smaller than the intra-atomic ones raised otherwise. After rupture of a bond due to the two-hole localization illustrated in the right half of fig. 3, the bond becomes opened furthermore because of Coulombic repulsion between two + 1 electronic charges localized on the both sides of the ruptured bond. Moreover, local lattice relaxations should take place triggered by a change in the charge distribution around the ruptured bond. In the terminology of section 2, these motions in atomic and electronic configurations around the ruptured bond correspond just to slipping down along the adiabatic potential V,(Q) in fig. 1 after a transition of a hole from a free state to the localized state at the first localized hole. Then, we suppose, in accordance with Itoh and Nakayama [17,18], that during the relaxation along V,,(Q) in fig. 3 a crossover takes place to another adiabatic potential on which the Ga core at the end of the ruptured bond is flung out together with three electrons, one from the unpaired orbit and other two from the two bonds to neighboring P atoms. This gives rise to the ejection of a Ga neutral and leaves three ruptured bonds around which electronic and lattice relaxations compensating the ejection take place furthermore as mentioned in section 1. The R formulated in eqs. (2.7) or (2.7a) gives the rate constant with which the state in the left half of fig. 3 changes into that in the right half. Various parameters in R can be estimated in reference to fig. 3: The energy Ed describes how high the state in the left half of fig. 3 is relative to the state where the first localized hole at the P atom is freed to the lowest free state in the valence band. It is smaller than the energy (of about 0.7 - 0.8 eV) of the filled s-like surface state on a P atom by an energy of lattice relaxations due to hole localization at the surface state. Taking the lattice-relaxation energy of about 0.2-0.3 eV, we suppose Ed = 0.5 eV later. The energy of Coulombic repulsion between the two holes in the situation in the right half of fig. 3 determines U of eq. (2.16) which is composed of a and 6. Let us estimate a at about 2.36 A, the P-Ga distance in the bulk crystal at room temperature [34], and the static dielectric constant L at the surface at about 6 corresponding to the middle between its bulk value = 11.0 at room temperature [35] and its value = 1 in vacuum. Then we get U = 1.0 eV. In addition to U, another energy loss is required to construct the state in the right half of fig. 3 due to the localization of a free hole at a bond followed by a concomitant change in the electronic structure at the bond shown in the right half of fig. 3. This energy loss was written as E,,, and we estimate it as Eb = 1 eV, since it should have a magnitude of about half the total width W( = 2 eV) for the two-dimensional motion of a hole in the valence band whose width is about 3 eV for the heavy-hole branch as determined experimentally [32] and theoretically [31,36]. After formation of the state in the right half of fig. 3, lattice relaxations around the ruptured bond take place including further opening of the bond due to Coulombic repulsion between the two (+ 1) charges on the both sides of the ruptured bond. The energy of the relaxation was denoted by S, by which the state in the right half of fig. 3 is lowered after relaxation. Let us assume S = 2.5 eV, for which Ed + U + Eh - S goes down to nearly zero and hence the two-hole localized state becomes relaxed nearly as low as the two-hole free state by virtue of the strong electron-phonon interaction on the surface. In order to estimate ET of eq. (2.32) we must know the value of A of eq. (2.21) measuring the effect of Coulombic repulsion of the first localized hole to a free hole on the surface. The heavy-hole mass m,, determined by cyclotron resonances [37] is about 0.67 times the bare electron mass m,, in GaP, being a little heavier than the mass (= 0.63 mb) obtained by band-structure calculations [36] due to a polaron

H. Sumi / Theory on sputtering by sub-bandgap lasers

394

effect. This value of m,, together room temperature E -,-= 0.16 eV,

with r I: 6.0 enables

(or, E,/k,T=

us to get A = 0.25 eV, then, at kgT = 26 meV for

6.3),

(3.1)

as the average kinetic energy of a free hole on quantum-mech~ical tunneling to a bond at the first localized hole against the Coulombic repulsion between them. It in fact satisfies E,. GC U together with exp( E,/k,T) z+ 1. Therefore, we see that the validity condition of eq. (2.34) for the approximate formula (2.27) of the current J of Ga neutrals ejected from the surface can be satisfied over a wide range of the reduced concentration N,/Z of holes produced by the laser in the valence band. Let us remember in eq. (2.27) that J, given by eq. (2.28) gives the critical magnitude of J at the critical laser intensity of eq. (2.30) where its laser-intensity dependence changes from the square one in eq. (2.13) to the exponential one in eq. (2.29). Since there are half as many Ga atoms as bonds (with the total number N) on the GaP (110) surface, J, per single Ga site is given by J,/$N =j, exp( -E,/k,T) with N,, = N where j, is determined by eqs. (2.28) and (2.14) with (2.31) as j,-744x105

s-1,

(3.2)

[or, 6.02 X lo5 s-’ by the approximate eq. (2.36)], for W= 2 eV and c = 0.6. The duration of a laser pulse used in the experiments [8,9] is T = 20 ns and the lifetime of a hole produced in the valence band is usually much smaller than this. In this situation, rJ,/$N gives the efficiency for the ejection of Ga neutrals from the surface by a single shot of the laser pulse at the critical laser intensity, with TJ~ giving the total number of Ga neutrals ejected during 7. Using eq. (3.2), we can estimate 7J,/&N given by TJ,/~N

= ~j, exp( -E,/k,T)

= 0.66 X lo-“,

for

r = 20 ns,

(3.3)

at room temperature when Ed is taken as 0.5 eV as mentioned before. This magnitude of the efficiency at the critical laser intensity is, however, much smaller than the efficiency observed in the experiments [8,9] since it has been estimated to be in the range of the order 10p5. This is, nevertheless, quite consistent with the observations, for example in fig. 2 of ref. [9], that an increase in the laser intensity by only about 40% gives rise to a marked increase in the sputtering yield by about 30 times [8,9]: At the critical laser intensity of eq. (2.30), the sputtering yield proportional to J of eq. (2.27) a [exp( Nh/Z) - 11’ increases only about 3 times under the same condition. In order to understand the observations, we must assume on the basis of the approximate eq. (2.27) that the magnitude of N,/Z in the experiments should lie in the range of 4-7. Moreover, in this range of N,/Z, eq. (3.3) with (2.27) gives the efficiency rJ/ fN a quite reasonable magnitude of the order of 10m5 consistent with the experiments. More detailed discussion on this point is given in section 5. From the theoretical point of view, however, the magnitude of eq. (3.3) is sensible very much to a small change in the value of Ed and X determined by Eh and S in eq. (2.37), since it is proportional to exp[ - ( Ed + X)/k,T] as derived from eq. (2.36) for j,. As the total number N, of holes in the valence band increases very much with the laser intensity, the condition eq. (2.34) becomes violated. In this case, we cannot apply eq. (2.27) for J. Instead, we must numerically calculate eq. (2.7a) for R by using eq. (2.6) for F(E) and eq. (2.31) for g(E) with eq. (2.19) for b(E). From this R, we can obtain J = N,R of eq. (2.1) with eq. (2.9) for Nd. Since we have already calculated the critical magnitude of the current J, = iNjc exp( -E;,/k&‘) with N, = N in eq. (3.2) or that of the efficiency of eq. (3.3), it is now sufficient to calculate only j=

J/J,

= 2 exp( E,/k,T)R/j,

= 2[exp( Nh/Z)

- l] R/j,,

(3.4)

independent of E,, in order to see the dependence of J on N,/Z related to E, intensity is proportional to Nh/Z. An explicit calculation was performed for lJ=l.OeV,

A=0.25eV,

E,,=l.OeV,

and

S=2.5eV

at

k,T=26meV.

by eq. (2.26). The laser

(3.5)

H. Sumi / Theory on sputteringby sub-bandgaplasers

395

10”

I--

($ _1,‘,/ I’ ,’

(en-l)’ 2oc ~

lo(

(

4

2

6

n

Fig. 4. Linear and logarithmic plots of the n (= N,,/Z) dependence of the current J of Ga neutrals ejected from the intrinsic Ga sites on the GaP (110) surface, where J is scaled by the critical magnitude J, of its value at the critical laser intensity for exp( n) = 2.

The calculated J is shown as a function of n = Nh/Z in fig. 4 for n I 7.2 both in the linear scale shown on the left ordinate and in the log~th~c scale shown on the right ordinate. Both n2 and [exp(n) - l]* are also plotted for comparison. We see therein that J rises up exponentially, deviating from n* at about n = 1, obeying approximately [exp(n) - 112 as derived in eq. (2.27) and that the approximate eq. (2.27) is well reliable.

4. Ejection of Ga cations from the GaP(110) surface It has been observed that pulsed-laser excitation of the GaP(110) surface gives rise initially to ejection of Ga cations although it fades rapidly and disappears as the pulsed excitation is repeated over about ten times [9]. The initial ejection of Ga cations rises up superlinearly with the laser intensity, having an apparent threshold intensity above which the ejection becomes detectable. The ejection of Ga neutrals treated in the previous section, on the other hand, becomes detectable at a laser intensity several times larger than that for Ga cation ejection. Moreover, ejection of Ga neutrals does not fade at least over thousands of times of repetition of laser shots. From these experimental findings, it has been inferred that Ga cations are ejected from defect sites of Ga adatoms adsorbed on the surface, --not from the intrinsic Ga sites from which Ga neutrals are ejected. It has also been observed for the GaP(l11) surface that ejection of Ga neutrals has a component rapidly fading with increasing number of repetition of the laser shot over about ten times as well as a component unfading with it. It seems that Ga neutrals are ejected also from --defect sites as well as from the intrinsic Ga sites on this surface. Atomic arrangement on the GaP(lll) surface is, however, not well known in comparison with that on the GaP(ll0) surface, as mentioned also in the previous section. Therefore, we concentrate our investigation only to the GaP(llO) surface hereafter also in this section. We can expect that a Ga adatom on the GaP(ll0) surface is coordinated as shown in the left half of fig. 5, where it is doubly bonded to two neighboring surface P atoms by two of the three p (or, sp2-hybrid-like) orbitals of Ga together with two nonbonded electrons in its s orbital (or, in the remaining one of the sp2-hybrid-like orbitals). This state can be constructed by the addition of a Ga cation, composed of a Ga core and two valence electrons, to the surface. Instead, if the Ga adatom gets one more valence electron,

396

H. Sumi / Theory on sputtering by sub-bandgap

lasers

hole localization

Fig. 5. Ga adatom on the GaP(llO) surface in the left half, and rupture of a bond from the Ga adatom triggered by localization of a hole to the bond in the right half. The solid lines represent surface bonds while the dashed ones represent bonds toward underneath atoms.

being constructed by the addition of a Ga neutral, the electron must occupy the remaining nonbonded p orbital resonating with the conduction band, and it will diffuse away from the Ga adatom into the bulk. Either one of the two s electrons of the Ga adatom cannot diffuse away because of strong intra-atomic Coulomb attraction to them. Since the state in the left half of fig. 5 can be constructed by the addition of a Ga cation to the surface, the Ga adatom is seen from surroundings as carrying + 1 charges. In other words, we can consider that a hole has already been localized at the Ga adatom on the surface. Then, following the theory mentioned in section 2, we consider a situation that a hole in a free state in the valence band comes to localize at either one of the two bonds from the Ga adatom. This hole must do so by quantum-mechanical tunneling against Coulombic repulsion from the first localized hole on the Ga adatom. Localization of the second hole at one of the two bonds from the Ga adatom leaves a single electron at the bond. As illustrated in the right half of fig. 5, however, the electron should automatically come to localize on the P side of the bond, rupturing the bond, since the nonbonding s-like surface state on the P atom is lower enough in energy than the s or p states on the Ga adatom. After localization of the second hole mentioned above, the P atom on one side of the ruptured bond has only one electron in the s-like surface state with a hole localizing there, while the Ga adatom on the other side of the rupture bond can be regarded as a Ga cation singly bonded to the surface P atom. These two atoms repel each other by Coulombic repulsion between them, inducing lattice relaxations around the ruptured bond. These motions in electronic and atomic configurations around the ruptured bond correspond to splipping down along the adiabatic potential V,(Q) in fig. 1. During the relaxation the state in the right half of fig. 5 is supposed to switch over to a state where a Ga cation is completely detached from the P atom to which it was singly bonded, as illustrated also in fig. 1. This is in accordance with the presumption by Itoh and Nakayama [17,18] as mentioned in section 1. According to the theory mentioned above, an essential difference between the ejection of Ga neutrals from the intrinsic Ga sites on the surface and that of Ga cations from the defect sites of Ga adatoms is the following: The first localized holes playing the role of nucleation cores for laser sputtering are found only with a very small thermal population in a metastable surface state on each intrinsic P atom in the former, while they are found always on every Ga adatom in the latter although their total number is much smaller than the total number of intrinsic P or Ga atoms on the surface. Let us represent the total number of bonds extending from all the Ga adatoms on the surface as Na. Since H of eq. (2.7a) gives the rate constant for localization of the second hole to one of these bonds in the present situation, the total rate for rupture of the bonds from ail the Ga adatoms on the surface is given by J, = N, R instead of J of eq. (2.1) with the same R. This J, is regarded as the current of Ga cations ejected from the surface, as before.

H. Sumi / Theory on sputtering by sub-bandgap lasers

397

Since the bond extending from the Ga adatom to a surface P atom is weaker than the intrinsic surface bond, both the energy loss for localization of a free hole at a bond and the energy of lattice relaxations after its rupture should be considerably smaller for the former than for the latter, for which they were written respectively as Eb and S. Therefore, Eb and S in the expression of R in eq. (2.7a) must be replaced by considerably smaller ones, written as Eba and S, respectively. Then, the average value of the kinetic energy of a hole with which it makes a transition most dominantly to a bond from a Ga adatom is given by eq. (2.32) in which Eb and S, are replaced respectively by Eba and S,. It is written as E,. Now, R can be approximated by eq. (2.22) under the condition of exp[( E,, - EF)/kBT] 2 1 with exp( E,/k,T) approximated by eq. (2.26). Then, J, given by N,R tends to J, = [exp(N,/Z)

- 11 JacT for

exp( NJZ)

- 1 <( exp( E,,/k,T),

(4.1)

with

J, = flv, A,,,

(4.2)

where jaj,, is given by eq. (2.36) with E;, and S therein replaced respectively by Eba and S,. When the laser intensity proportional to N, is very small, eq. (4.1) tends to J, = (N&Z)

J,, ,

for

N,/Z a 1.

(4.3)

That is, the current of Ga cations ejected from the defect sites of Ga adatoms increases linearly with the laser intensity so long as it is very small. This property arises from the microscopic mechanism of Ga cation ejection that it requires the localization of only one free hole at a bond to be ruptured. It is in contrast with Ga neutral ejection from the intrinsic Ga sites which increases quadratically with the laser intensity so long as it is very small, as shown in eq. (2.13), since it requires the localization of two free holes at a bond to be ruptured. When the laser intensity is very large, on the other hand, it tends to J, = exp( Nh/Z) Jac,

for

exp( NJZ)

B 1.

(4.4)

That is, the sputtering current increases exponentially with the laser intensity also here. At the critical laser intensity between the two regimes mentioned above, eq. (4.1) gives J, = J,, at exp( N,,/Z) = 2. We see that J,, of eq. (4.2) gives the ma~itude of the current of ejected Ga cations at the critical laser intensity above which the current increases exponentially. In order to estimate Jac of eq. (4.2). let us take, for example, Eba = 0.5 eV and S, = 1.5 eV, considerably smaller than Eh = 1.0 eV and S = 2.5 eV, respectively, taken in eq. (3.5) for the intrinsic surface bond, while we take the same values for U and A as in eq. (3.5). Since Eba + U - S, = 0, we have assumed that the localized hole state at a bond from a Ga adatom in the right half of fig. 5 becomes relaxed as low as the lowest free state in the valence band. At k,T= 26 meV at room temperature, eq. (2.32) with Eh and S replaced by Eba and S, gives E Ta

=

0.18 eV,

(or, E,/k,T

= 7.1).

(4.5)

It in fact satisfies E,, K U together with exp( E,/k,T) >> 1. Therefore, we see that the validity condition in eq. (4.1) for the approximate formula of the current J, of Ga cations ejected from the surface can be satisfied over a wide range of the reduced concentration N,/Z of holes produced by the laser in the valence band. The critical magnitude of the current of ejected Ga cations per single defect site of a Ga adatom is J,,/ iNa which equals 2C of eq. (2.14) calculated by eq. (2.31) with Eha and Sa used instead of Eb and S, given by jac = J,/$N,

= 4.77 x IO6 s-l,

(4.6)

[or, 4.59 X lo6 s-i by the approximate eq. (2.36)] for W = 2 eV and c = 0.6 as before. It is important to note here that observed actually in the experiments [8,9] is not the steady current of Ga cations ejected from the surface, but, the total number of them ejected by a single shot of a laser pulse,

398

H. Sumi / Theory on sputtering by sub-bandgap lasers

or, a quantify proportional to it, which depends on the duration time r of the laser pulse. The latter is related to the efficiency for ejection of Ga cations from the defect sites of Ga adatoms by a single laser shot. Taking into account the duration time T of a laser pulse used in the experiments [8,9], we obtain rj,

= 0.0954,

for

7 = 20 ns.

(4.7)

This means that the efficiency is about 0.1 at the critical laser intensity, that is, the sputtering yield of Ga cations by a single laser shot decreases at a rate of about 0.1 every time the laser pulse is repeated. This situation is close to the observation shown in fig. 1 of ref. [9] that the sputtering yield decreases at a rate of about 0.5 every time the laser pulse is repeated with a fluence of 0.35 J/cm2. This observation can be understood as a situation at a laser intensity a little larger than the critical one for exp( Nh/Z) = 2. These situations are in contrast with those for ejection of Ga neutrals from the intrinsic Ga sites where the critical magnitude of the efficiency in eq. (3.3) is much smaller than unity. The efficiency is very small on the order of 1O-5 even at N,/Z = 4-7 where Ga neutral ejection can be detected in the experiments, as noted in the previous section. This difference between the two situations for ejection of Ga cations and neutrals originates mathematically in the factor exp( - E,/k,T)( -K 1) which appears only in eq. (3.3) for the efficiency of Ga neutral ejection, but does not in eq. (4.7) for that of Ga cation ejection. It originates physically in a difference in the mechanism of laser sputtering between them that the first localized hole playing the role of a nucleation core for laser sputtering can always be found on every Ga adatom, while its probability of finding at any intrinsic Ga site on the surface is much smaller than unity, being limited by the thermal population factor exp( - E,/k,T) of the first-localized-hole state with energy Ed. Since rjaC of eq. (4.7) is not much smaller than unity, the efficiency for Ga cation ejection should be described, not by rJ,,/ f N,, but by a more accurate equation, F,=l-exp(-rJ,/:N,)=l--exp{-rj,,[exp(Nh/Z)--l]},

(4.8)

where the second relation is applicable under the condition in eq. (4.1). As mentioned before, it has been observed [9] that the sputtering yield of Ga cations by a single laser shot decreases at a rate of about 0.5 between two successive shots every time the laser pulse is repeated with a fluence of 0.35 J/cm2. This means that Fa equals just 0.5 at this fluence. Combining this fact and the expression of F, of eq. (4.8) with eq. (4.7) we can correlate the magnitude of N,/Z to the fluence of a laser pulse with duration time 20 ns, as N,/Z = 2.1 at a laser fluence of 0.35 J/cm’. Since N, represents the total number of holes produced in the valence band by the laser, it is determined principally by the laser intensity, not by the fluence, becoming proportional to the former. However it becomes proportional to the laser fluence, too, so long as the duration time of the laser pulse is fixed. Therefore, this relation can easily be extrapolated to other laser intensities or fluences by proportionality. Many interesting results can be derived from this extrapolation, as will be discussed in the next section. As the total number N, of holes in the valence band increases very much with the laser intensity, the condition in eq. (4.1) becomes violated. In this case, we must numerically calculate eq. (2.7a) for R by using eq. (2.31) for g(E) with eq. (2.19) for b(E). This R enables us to obtain J, = N,R. Since we have already calculated in eq. (4.7) the critical magnitude of the current Ja, = +N, j,, at the critical laser intensity, it is now sufficient to calculate only j, = J,/J,, = 2R/ja, in order to see the dependence of J, on N,/Z. We take for the calculation: U = 1.0 eV, A = 0.25 eV, E,, = 0.5 eV,

and

S, = 1.5 eV

at

k,T=

26 meV.

(4.9)

The calculated J! is shown as a function of n = N,/Z in fig. 6 for n I 7.2 both in the linear scale shown on the left ordinate and in the logarithmic scale shown on the right ordinate. Both n and [exp(n> - l] are also plotted for comparison. We see therein that J, rises up exponentially, deviating from n at about

H. Sumi / Theory on sputtering by sub-bandgap lasers

0

Fig. 6. Linear and logarithmic plots of the n (3 N,,/Z) dependence of the current 3, of Ga cations ejected from the defect sites of Ga adatoms on the GaP(ll0) surface, where J, is scaled by the critical magnitude .I,, of its value at the critical laser intensity for exp( n) = 2.

6

4

2 n

n = 1, obeying appro~mately is well reliable.

399

[exp(n) - l] as derived in eq. (4.1), and hence that the approximate

eq. (4.1)

5. Discussion The main issue of the present theory is to give the current of atoms ejected from the surface as a function of the density of holes moving two-dimensionally along the surface in the valence band. The calculation is based on the assumption of the steady state, and hence the current does not include such a quantity as the duration time of the laser pulse which is used to produce holes along the surface in the valence band. The density of holes should be proportional to the intensity of the laser irradiating the surface. The hole-density dependence of the current can be obtained by setting the five parameters given in eq. (3.5) or (4.9), as shown by 7 (= J/J,) in fig. 4 or by j;, ( = J/f,,) in fig. 6. Observed actually in the experiments [8,9] is, however, the sputtering yield of atoms ejected by a single laser shot with a duration time r = 20 ns. The sputtering yield is related with the efficiency for ejection of atoms by a single laser shot, which is given by F, of eq. (4.8) for ejection of Ga cations from the defect sites of Ga adatoms on the surface. The sputtering yield by the pth shot for p = 1,2,3,. . . is given by Y, = (1 - I$)‘-‘F,.

(5.1)

When the efficiency is much smaller than unity as in the case of ejection of Ga neutrals from the intrinsic Ga sites, the sputtering yield of eq. (5.1) does not depend on the repetition time of the laser shot, and has a magnitude nearly equal to the efficiency itself. The efficiency for Ga neutral ejection is given by the rate constant multiplied by T, as F = rJ/tN

= T( J,/$N)

J/J,

= TjCJexp( - E,/k,T),

(5.2)

where J represents the current of Ga neutrals ejected from the intrinsic Ga sites with the total number ;N on the surface, and we have used eqs. (3.3) and (3.4) in obtaining the third equality. Since F-=x 1, the sputtering yield of Ga neutral ejection observed should equal F and it can directly be converted to the rate constant F/T = j,Jexp( - E,/k,T) proportional to $ shown in fig. 4. For ejection of Ga cations, on the

H. Sum’ / Theory on sputtering by sub-bandgup lasers

400

other hand, the efficiency is not much smaller than unity and hence it must be given by eq. (4.8). Since the second equality in eq. (4.8) is justified only under the condition in eq. (4.1), in general cases it must be given by F,= 1 -exp[-7(Jac/fN,)J,/J,,]

= 1 -exp(-r&x),

(5.3)

where we have used eq. (4.6) and .?a = J,/J,, in obtaining the second equality. Since F, is not much smaller than unity, the sputtering yield for Ga cations of eq. (5.1) decreases with the repetition number p of the laser pulse, in agreement with observations [8,9]. In this case, the rate of decrease in the sputtering yield between successive two shots is given by 1 - Fa, and does not depend on p. Therefore, the expe~ment~ data on the p decrease in the sputtering yield should be analyzed so as to derive its decreasing rate equal to 1 - F, from which we can derive by eq. (5.3) a rate constant j,,J, proportional to .???shown in fig. 6. In order to calculate the efficiencies F and Fa of eqs. (5.2) and (5.3), we must first obtain j, exp( - E,/k,T) and j,, therein. They could be calculated by setting W, c and Ed additionally to the parameters in eqs. (3.5) and (4.9), where W represents the total bandwidth for the two-dimensional motion of a hole in the valence band, c the coefficient appearing in eq. (2.31) while E, is the energy of the first-localized-hole state for Ga neutral ejection. When W = 2.0 eV, c = 0.6,

and

Ed = 0.5 eV,

(5.4)

are adopted as before,_ j, exp( - Ed/k&T) is given by eq. (3.2), while j,, is given by eq. (4.6). Combination of these values with J shown in fig. 4 and .?! shown in fig. 6 gives the efficiencies F and F, of eqs. (5.2) and (5.3) by a laser pulse with duration time r = 20 ns. They are shown in fig. 7, where F is so small compared with F, that F is shown in 10e5 units. As mentioned in the previous section, the efficiency Fa of Ga cation ejection from the defect sites of Ga adatoms was observed to reach about 0.5 at the laser fluence of 0.35 J/cm2. In fig. 7, F, = 0.5 is realized at n = 2.136. Although n (3 Nh/Z) is determined principally by the laser intensity, not by the fluence, becoming proportional to the intensity, the fluence is proportional to the intensity so long as the duration time r of the laser pulse is fixed. In the experimental data [8,9] analyzed below, r had been fixed at 20 ns.

Fig. 7. n (- Nh/Z) dependence of the efficiency F, for ejection of Ga cations from the defect sites of Ga adatoms on the GaP (110) surface and the efficiency F (in 10m5 units) for ejection of Ga neutrals from the intrinsic Ga sites by a laser pulse with duration time of 20 ns.

17. Sumi / Theory on sputtering by sub-bmdgap

lasers

401

(In this respect, see the note added in proof, too.) Therefore, under this condition, n is proportional to the laser fluence, and n value at F, = 0.5 obtained above enables us to get a calibration between them n (= N&Z)

= 6.10 x (1aser fluence

in units of J/cm*),

(5.5)

for a laser pulse with a duration of 20 ns. When the laser fluence is 0.42 J/cm’, for example, it corresponds to n = 2.56 in eq. (5.5), and the left ordinate of fig. 7 gives F, = 0.67. It means that the sputtering yield by a laser shot with fluence of 0.42 J/cm* should decrease at a rate of about 0.33 (= 3) every time a laser shot is repeated, in good agreement with the observation shown in fig. 1 of ref. [9]. The relation shown in eq. (5.5) can be applied also to Ga neutral ejection from the intrinsic Ga sites for which the efficiency F is shown on the right ordinate in fig. 7. When the laser fluence is 0.75 J/cm2, for example, it corresponds to n = 4.58 in eq. (5.5), and fig. 7 gives F = 4.89 X 10e7. When the laser fluence is increased 40% to 1.05 J/cm*, it corresponds to n = 6.51, and fig. 7 gives F = 1.40 X 10e5. Then, the sputtering yield z: F increases by about 29 times when the laser fluence is increased 40% from 0.75 to 1.05 J/cm’, just in agreement with the observations shown in fig. 2 of ref. 191. Since F is of the order of 10-6-10P5, the sputtering yield should not fade over thousands of times of repetition of the laser pulse irradiation, also in agreement with the observations and consistent with the estimation on the magnitude of the sputtering efficiency in refs. [8] and 191. From the theoretical point of view, however, the magnitude of F given by eq. (5.2) is very sensitive to a small change in Ed which was chosen as in eq. (5.4) for the F in fig. 7. Ejection of Ga neutrals from the surface begins to be detected at the laser fluence of 0.75 J/cm2 in ref. [9] and this magnitude was called the threshold laser fluence for superlinear rise of Ga neutral ejection there. It is located at an n of about 4.6 as mentioned above. Fig. 4 shows, however, that the exponential increase in the sputtering efficiency with the laser fluence first manifests itself not at an n of about 4.6, but at an n of about 2, which corresponds to a laser fluence of 0.3 J/cm2 in eq. (5.5). This shows therefore that the threshold laser fluence obtained experimentally [8,9], above which the sputtering yield was observed to rise up superlinearly, has no physical meaning since it is determined artificially by the lowest detection limit in the experiments. In fact, neither linear nor quadratic dependence of the sputtering yield on the Iaser fluence has been observed below the threshold laser fluence. Simply no data has been measured below it. The situation mentioned above can also be applied to ejection of Ga cations from the defect sites of Ga adatom, but, with an interesting conclusion in this case: We see in fig. 1 of ref. [9] a series of sputtering yields of Ga cations obtained at the 1st to the 7th laser shot with a fluence of 0.35 J/cm2. The sputtering yield obtained at the 7th shot is about l/25 that obtained at the 1st shot. The latter corresponds to the total number of ejected Ga cations equal to half the total number jN, of Ga adatoms on the surface since the efficiency Ed is about 0.5 at a laser fluence of 0.35 J/cm’ as mentioned before. Then, the former corresponds to that equal to 0.02 X $V,. This shows that the lowest limit for detection of Ga cations is at least lower than this number. Then, the lowest detection limit in the efficiency is lower than 0.02. At an efficiency of 0.02f -=z l), ejected Ga cations can be detected with the same yield at least over about ten laser shots as apparent in eq. (5.1). We see in fig. 7 or by eq. (4.8) with eq. (4.7) that Fa = 0.02 corresponds to n - 0.192, and by eq. (5.5) that this value of n can be produced by a laser with a fluence of about 0.03 J/cm’. We see in fig. 6 that at n = 0.192, the current J, of ejected Ga cations begins just to deviate from the linear dependence on n. We expect, therefore, that when we can derive J, or a quantity proportional to it by an analysis shown below eqs. (5.2) and (5.3), we can detect a switch in the dependence of J, on the laser intensity from the linear one to the exponential one by laser shots with a fluence around 0.03 J/cm’ with a duration time r of 20 ns.. Almost all the essential features in the laser sputtering observed so far have thus been reproduced quite well by the present calculation when parameters are set as shown in eqs. (3.5), (4.9) and (5.4). We do not claim, however, that this set of parameters is unique: The critical magnitude J,, of the current of Ga

H. Sumi / Theory on sputtering by sub-bandgap

402

lasers

cations in eq. (4.1) is sensitive to a change in both Eba and S, through the X of eq. (3.37), while the relative ratio between J,, and the critical magnitude .I, of the current of Ga neutrals in eq. (2.27) is sensitive to a change in Ed as well as Eh and S as seen in eq. (3.35). Therefore, the set of parameters shown in eqs. (3.5), (4.9) and (5.4) is only one of many which can reproduce the experimental observations. The present theory calculated the rate of rupture of a bond at the first localized hole triggered by additional localization of the second hole, under the presumption that the rupture of the bond leads finally to the ejection of a Ga neutral or cation, in accordance with Itoh and Nakayama [1’%,18]. The current of Ga neutrals or cations determined in this way has reproduced quite well the experimental observations. This seems to strengthen the presumption mentioned above, although a concrete mechanism correlating the rupture of the bond to the ejection of a Ga neutral or cation remains concealed in a black box. As a step toward uncovering the black box, it must first be checked by an analysis shown below eqs. (5.2) and (5.3) that the current of Ga neutrals should become proportions to the square of the laser intensity or fluence, while that of Ga cations to the laser intensity or fluence itself, when n becomes small compared with unity at the laser fluence determined by eq. (5.5). The present analysis showed that the laser intensity where superlinear rise of the sputtering yield of Ga neutrals has been observed corresponds to the region of N,/Z 2: 4.5-6.5. In this region, the Fermi energy E, of holes produced in the valence band amounts to about (4.5-6S)k,T, as derived from eq. (2.26). Since the total bandwidth W for the two-dimensional motion of a hole was estimated at about 2 eV, the Fermi energy estimated above means that holes are piled up to about l/17 to l/12 of the total bandwidth. Even in this case of relatively low level of hole a~umulation, it may be possible that the electronic temperature in the hole system becomes higher than the lattice one of the bulk crystal. If this is the case, the temperature appearing in the Fermi distribution function F(E) of eq. (2.6) for holes must be replaced by an effective hole temperature T,,, different from the lattice temperature T, in calculating the rate constant R of eq. (2.7a). The Th must be used also for the temperature appearing in eqs. (2.25) and (2.26). Even in this case, we can expect that the current J or J, shows an Nh/Z dependence similar to that shown in fig. 4 or in fig. 6, since this is mainly determined by the sharp rise with E of b(E) shown in eq. (2.19a) which determines g(E) appearing in eq. (2.7a) for R through eq. (2.31). Let us investigate the extent of accumulation of surface charges on the surface due to electrons localized in intra-gap surface states at Ga sites. The total number N, of the surface localized electrons is the same as N, of holes produced by sub-bandgap lasers in the valence band. From the definition of n = NJ.2 and 2 of eq. (2.25), we obtain N,/:N = (4k,T/W)n, (5.6) where $N equals the total number of Ga sites on the surface and W represents the total bandwidth for the two-dimensional motion of a hole in the valence band, being estimated at about 2 eV. When n = 1, for example, eq. (5.6) shows that a localized electron is found on the average at every twentieth Ga site on the surface at room temperature since 4k,T/ W = l/20. Surface negative charges due to these electrons with such a high density bend the valence band upward strongly toward the surface. Then, holes confined in the bent part of the valence band can move only two-dimensionally in a very thin layer along the surface. This justifies posteriorly our initial assumption on the two-dimensionality of hole motion in the valence band at n as low as about unity. Let us estimate the average tunneling distance of the second hole for a transition to the first localized hole through the Coulombic repulsive potential. It can be related to the average kinetic energy of the hole derived from the E integration in eq. (2.7a) for R as JEh(E)F’(E)

exp[ -(E,+

U-

E)2/(4Sk,T)]

dE

Er

(5.7) /b(E)F(E)

exp[-(E,,+

I/--E)*/(4Sk,T)]

dE

*

H. Sumi / Theory on sputtering by sub-bandgap lasers

403

Since e’/(r~) gives the classical turning point of the hole with energy E on tunneling to a distance a from the first localized hole, the average tunneling distance is given by a D, of D, = e2/@)

-a.

(5.8)

The average distance for tunneling to a bond from a Ga adatom can be obtained similarly, and it is written On the other hand, the average distance D between the two holes can be determined by as D,. NhD2 = N;12, where D represents the area per a single bond on the surface. Using eq. (5.6) together with N, = N,, there, we obtain D as a function of n = N&Z, as D = [(Q/n)

W/(2kBT)]1’2.

(5.9)

Using the lattice constant of 5.45 A [34], we find ft = (5.45)2/8 A2 on the GaP(llO) surface. Fig. 8 shows D,, I), and D for WY 2 eV and k,T= 26 meV in units of a as a function of n. Since a is about the Ga-P bond length = 2.36 A, we see in fig. 8 that both D, and D,, are in fact in a range of 10-7 A for n I 7.2 as noted in section 1. We note moreover that they never exceed D. This means that screening by other holes or electrons is not effective in such a short distance as D, and DTa, justifying posteriorly the initial assumption for the unscreened from [eq. (2.15)] of the Coulombic repulsive potential. As mentioned before, superlinear rise of the sputtering yield of Ga neutrals has been observed also from --the GaP(lll) surface [8]. Atomic arrangement in the (2 x 2) reconstruction on this surface has not been clarified yet, even for more familiar GaAs in III-V compounds [X-28]. It is probable, however, that most P atoms on the top of the surface are coordinated to underneath Ga atoms by three bonds, since the original (unreconstructed) structure of this surface has this coordination of bonds at every P atom. Then, similarly to the situation on the (110) surface, a P atom should construct the three bonds by p orbitals, and have two nonbonded electron in the remaining s orbital which is located deeper in energy than the top of the valence band composed of s-p hybridization. Therefore, the P atom on the surface should have a completely occupied surface state buried in the valence band. We consider that a hole localized in the surface state on a P atom plays the role of a nucleation core for the ejection of a Ga neutral on the same surface. Then, the equations derived in the present theory can be applied also to the laser sputtering from --the GaP(111) surface without essential modifications. However, it has been observed by photoe~ssion --studies 1381 that the GaP(lll) surface has a filled surface state at an intra-gap position 0.8 eV above the top of the valence band. We consider that there should exist another filled surface state below the top of

Fig. 8. n (E N,,fZ) dependence of the average mutual distance D between holes in the valence band, and the average tunneling distance D, and D,, of a hole for a transition against Coulombic repulsion to a bond from the first tocahzed hole in the former and from a Ga adatom in the latter. They are scaled by the final tunneling position n from the center of the Coulomb potential, a being of the order of the bond length.

404

H. Sumi / Theory on sputtering by sub-bandgap lasers

the valence band, too, based on the physical reasoning mentioned above, and that the filled intra-gap surface state, if it exists, does not play any---role in the laser sputtering under consideration. In the laser sputtering from the GaP(lll) surface, it has been observed [8] that the ejection of Ga neutrals has two components, one rapidly fading with the repetition of a laser pulse over about ten times, and another not fading with it at a laser fluence as low as 0.24 J/cm*. The former is similar to Ga cation ejection from the GaP (110) surface, while the latter is similar to Ga neutral ejection from it. For the (110) surface, however, the latter have never been observed at such a low laser fluence. This means that ---the critical magnitude, J, in eq. (2.27), of the current of Ga neutral ejection is not so small for the (111) surface as for the (110) one. Since J, is given by eq. (3.35), we consider that the relaxed --- position Ed of a hole localized in the surface state buried in the valence band is a little smaller in the (111) surface than in the (110) one where Ed was estimated to be about 0.5 eV as in eq. (5.4).

Note added in proof Very recently Hattori et al. [39] and Nakai et al. [40] observed that the yield of Ga neutral ejection from the GaP(llO) surface decreases to about a half after 8000 repetitions of a laser pulse with a fluence of 1.0 J/cm’. The pulse duration is the same as used in their previous experiments [8,9], although it is expressed as 10 or 28 ns in their new papers. Their observations enable us to estimate the efficiency F per single shot from the relation (1-F)8000 = 0.5, as F = 9 x 10w5. This value is larger than the F = 8 X 10e6 calculated in fig. 7 for n = 6.1 corresponding to the fluence value in eq. (5.5). One can adjust the calculated F to the observed one easily, for example, by lowering Ed in eq. (5.2) a bit from 0.5 eV used in fig. 7 to 0.44 eV. In the above papers [39,40], moreover, changing their previous interpretation, they suggest the possibility that Ga neutrals are ejected from defect sites. This possibility should be checked by microscopic observations. If this is true, N, in eq. (2.9) must be regarded as the total number of bonds extending from such defects on the surface in the general theory presented in section 2.

Acknowledgements The author would like to thank Professors N. Itoh and Y. Nakai and Mr. K. Hattori of Nagoya University for showing their data prior to publication and for helpful discussions. His thank is also due to Dr. M. Georgiev of Bulgarian Academy of Sciences for drawing his attention to the present problem with valuable discussions.

Appendix. Matrix element for two-hole localization against Coulombic repulsion In order to see explicitly rapid energy, E, dependence of g(E) defined by eq. (2.5) we must calculate the matrix element M,h for the tunneling of a free hole with kinetic energy E to a bond at the first localized hole against the Coulombic repulsion between them. Let us start from a situation that a hole has already been localized on the surface, being metastabilized by lattice distortions. As in section 2, this situation is taken as the reference frame where the origin is defined in both energy and additional lattice distortions. We consider a free hole besides the first localized one. When the free hole also gets localized at the first localized hole, a lattice distortion is further induced, which is described by a coordinate Q. Before localization of the free hole, energy associated with the lattice distortion Q is written as V,(Q) = :Q’ of eq. (2.2). The lattice distortion Q gives rise to an interaction potential for the free hole. Let us write it as - b(r) Q where r represents the coordinate of the free hole while the linear dependence of the interaction

405

H. Sumi / Theory on sputtering by sub-bandgup lasers

on Q takes into account that it arises only when the lattice is additionally distorted on the reference frame mentioned above. The free hole interacts electrostatically with the first localized hole via the Coulombic repulsive potential C(r) of eq. (2.15). The kinetic energy operator of the free hole is written as K. Then, the Hamiltonian for its interaction with the lattice distortion Q is: H=K+C(r)-b(r)Q+

v,(Q).

(A-1)

The hole-phonon interaction potential - b( r)Q should give rise to a bound state of a hole when Q is large enough, since a hole can additionally be localized at the first localized hole against the Coulombic repulsion between them by virtue of the strong hole-phonon interaction. When the bound state becomes deep enough at a large Q, the binding energy should become approximately linear in Q as (2S)“2Q in terms of the notation in section 2, and the adiabatic potential associated with the bound state tends to V,(Q) = Eb + U - (ZS)‘/*Q + V,(Q) of eq. (2.4). where Eb, U and S represent respectively the energy loss due to hole localization in the bound state at a single bond, the Coulombic repulsive energy in the bound state due to two-hole localization in a single bond, and the energy of additional lattice relaxations due to additional hole localization at the bond. The bond is ruptured as a result of the two-hole localization. Let 1b) represent the hole bound state obtained when Q is large enough, for example at Q = Qb. Remaining free-hole states unbound by the lattice distortion are written as 1f)s, which satisfy orthogonality condition (b 1f) = 0 for any f. Then, for a large enough Q = Qs

K+W+-bWQ,=

CE~IJ)(fl+[E,+~-(2S)“2Q,] f

IbXbl

>

64.2)

should be satisfied, with (blKlb)=E,,

(bIC(r)lb)=U,

and

(blb(r)lb)=(2S)“2.

G4.3)

When Q is not so large as Qb, we cannot diagonalize Ei by I b) and I f)‘s, and a term of mixing between them arises. In the static-coupling scheme of ref. [20], this term gives the matrix element of a transition between these two kinds of states, &$,,, introduced in eq. (2.5). Since - b(r)Q = - b(r)Q, + b(~)(Q* - Q), we can rewrite K + C(r) - b(r)Q as:

K+C(r)-b(r)Q=

CEflf)(fl+[E,+U-(2S)1’2Q]

IWO4

f

+[b(r)-(2S)“21b)(bl](Q,-Q>, where we used eq. (A-2) at Qb and then -(2S)‘/*Qb I b)(b I = -(2S)‘/2Q 1b)(b 1- (2S)‘/2 Q). Note here that (b 1[b(r) - (2S)“2 / b)( b I] I b) vanishes due to eq. (A.3), while

(A.4 1b)(b

l(Qh -

(fl[b(r)-(2S)“*Ib)(bI]

Ib)=(flbtr)lb)=O(N-“2),

(A.51

(fI[b(+

lr>=(flb(r)lf)=O(N-‘),

b4.6)

and

@#“IWl]

where N represents the total number of bonds on the surface, since b(r) is nonvanishing only in the nei~borhood of the bond where the free hole is going to localize, while I f) is a free state extended all over the surface. The matrix element of eq. (AS) m~tiplied by Qs - Q in eq. (A.4) can induce a transition of a hole in a free state ) f) to the localized state I b) when Q deviates from Q,,, while that of eq. (A.6) induces modification of the free states by mixing between them when Q deviates from Qb. Since the latter is very small and is not primarily important in determining the rate of a transition of a free hole to the localized state, we can reasonably neglect the f-f mixing terms of eq. (A.6). The matrix element ~~~

H. Sumi / Theory on sputtering by sub-bandgap lasers

406

introduced in eq. (2.5) for a transition between terms of eq. (A.5) multiplied by Qh - Q as &,,=

QKf

(Q,-

Neglecting

the f-f

lb(r) mixing

H=

v,(Q)]

to the f-b

mixing

(A.71

terms of eq. (A.6) means

Z:(~dfXbl++WG(f

ey. (A.4) with eq. (A.@ into eq. (A.l),

T(Ef+

states corresponds

lb).

[b(r)-(25)“‘lb)(bI](Q,-Q,~ Then, substituting

a free to the localized

IfXf

I + v,(Q)

IbXbl

I)=H’.

we can rewrite

(A4

H of eq. (A.1) into the form

+H’,

(A.91

with V,(Q) defined by eq. (2.4) and C, 1f)(f j i 1b)(b 1 = 1. The meaning of eq. (A.9) is apparent: V,(Q) represents the adiabatic potential for the two-hole localized state drawn in fig. 1, while each of E, + V,(Q) for various f’s represents one of the multiple adiabatic potentials in fig. 1 for a situation that the second hole remains free in the valence band around the first localized hole, and H’ represents an interaction inducing a transition between the two kinds of states. Note here that H’ becomes effective only in the Q region where V,(Q) crosses with [El + V,(Q)]s in fig. 1. The electronic part, H - Vo(Q), of the Ha~lton~an H of eq. (A.9) is written as He(Q) + H’, where

Ho(Q) = CE, IfXf

I+&(Q)lbXbl

(A.10)

t

f

with ‘-- (2s)“‘Q. Since H is originally H(Q)

= H,(Q)

written + H’=

(AX)

as eq. (A.l), K-t

C(r)

its electronic

part

H,(Q)

+ H’ can also be rewritten

- b(r)Q.

(A.12)

Since the diagonal matrix element of H’ of eq. (A.8) at the state ) b) vanishes although with any one of I f )‘s, the spectral function g(E) of eq. (2.5) can be expressed as

H’ connects

g(E)=(blH’G[E-H,tQ)]H’lb). In order to calculate

the right-hand

as

I b)

(A.13) side of eq. (A.13), we use an operator

identity

Of

[z-H(Q)]-'=[z-H,,(Q)]-'+[z-Hi]-'H'[z-H(Q)]-',

(A.14a)

obtained from the first equality in eq. (A-12) for a complex variable z. Taking the expectation both sides of eq. (A.14) at the state I b) diagonal~ng Ho(Q) of eq. (A.lO), we obtain (bI[z-H(Q)]-‘H’Ib)= Multiplying

[z-Eh(Q)](bI[z-H(Q)l--‘Ib)-1.

H’ from the right on the both sides of eq. (A.l4a),

[z-H(Q)]-%‘=

[z-H,(Q)]-‘H’+

value of the

(A.15) we obtain

[z-Ho(Q)]-lH’[z-H(Q)]-lH’.

another

form of it (~.i6)

407

H. Sumi / Theory on sputtering by sub-bandgap lasers

Taking the expectation value of, the both sides of this equation at the state I b) and using that (b I ff ’ I b) vanishes for H’ given by eq. (A.8), we obtain

[z-Eb(Q>](bI[z-H(Q)l-‘rSr’Ib)

(b~H’[z-ll(Q>]-lH’Ib)=

= [z-Eb(Q)12(bi[z-H(Q)I-11b)-

[=-&(Q,l~

(A-17)

where the second equality was obtained by introducing eq. (AX) into the first equality. Finally, we can find one more form of the identity (A.14) by multiplying H’ from the left on the both sides of eq. (A.16), as H’[z-H(Q)]-‘H’=H’fz-~~(Q)]-1H’+H’~z-~~(Q)]-*H~[z-~(e)]-‘H’.

(A.I~)

Since H’ given by eq. (A.8) connects the state 1b) only with any one of the free states 1f) while it does not connect between any free states, the expectation value of the second term on the right-hand side of eq. (A.18) can be decomposed as (b/H’[z-Ho(Q)]-‘H’[z-H(Q)]-‘H’jb) (A.19)

=(bIH’[z-H,,(Q)]-‘H’Ib)(bI[z-H(Q)]-’H’lb).

Taking the expectation obtain

value of the both sides of eq. (A.18) at the state 1b) and using eq. (A.19), we

(~~H’[Z-~~(Q)]-~H’~~)=(~~W’[~-H(Q)]-’H’~~)/[~+(~~[~-II(Q)]-~H’~~)]

=z-E*(Q)-(blfz-H(Q)]-‘lb)-‘,

(A.20)

where the second equality was obtained by using eqs. (AX) and (A.17). The spectral function g(E) defined by eq. (A.13) is given by the imaginary part of the left-hand side of eq. (A.20) for z = E + ia with 6 representing a positive infinitesimal number. Then, the second equality of eq. (A.20) gives g(E)

= --71-l Im(b/H’[E-H,(Q)+is]-lH’Ib) =rr-l

Im(bI[E-H(Q)+ia]-‘lb)-‘.

(A.21)

Since EI(Q) in the second equality of eq. (A.21) can be expressed as the right-most side of eq. (A.12), g(E) given by the second equality does not include explicitly the value Qb of the lattice coordinate Q around which the Hamiltonian H(Q) was diagonalized as eq. (A.2). This feature is very important for justifying the static-coupling scheme for calculating the multiphonon nonradiative transition rates developed in ref. 1201. Now, H(Q) in the second equality of eq. (A.21) is given by the right-most side of eq. (A.12). A component of H(Q) expressed as - b(r)Q in eq. (A.12) represents an attractive potential for the second hole produced by the lattice distortion Q around a surface bond at the first localized hole. When Q is large enough, it gives rise to a bound state for the second hole, that is, a two-hole localized state at the single bond. The bound state has been written as I b) for the second hole in the limit of very large Q = Qb_ This means that I b) has an amplitude do~n~tly in the same space region as b(r) has. Since (b ] b(r) lb) = (25)‘/2 m . eq. (A.3), we can introduce an approximation b(r) = (2s)“’

1b)(b

(A-22)

1 ,

in calculating the right-hand side of the second equality of eq. (A.21). Since eq. (A-12) ensures [z-H(Q)]

-t = [z - K-

C(r)] -’ - [z-K-

C(r)] -‘b(r)Q[z

- H(Q)]

-I,

(A-23)

H. Sumi / Theory on sputtering by sub-bandgap

408

for any complex

variable

z, the approximation

lasers

(A.22) allows us to obtain

(~~~L-~(Q)]-~~~)-‘~(~~[z-K-C(~)]-’~~)-’+(~S)”~Q.

(A.24)

Let us remember here that K represents the kinetic energy operator for the free motion of a hole along the surface while C(r) represents the Coulombic repulsive potential of eq. (2.15) exerted by the first localized hole. Since C(I) is centrifugally decaying from the location of the first localized hole at r = 0, the eigenvalue of the Hamiltonian K + C(r) can be classified by the absolute magnitude of the two-dimensional wave vector k, with eigenvalue E;, = (t~k)~/(2rn,) where m,., represents the hole mass in its free motion described by K. Its wave function is wave-like outside the classical turning point rc(Ek) given by eq. (2.17) for C(r,) = Ek, while it decays exponentially inside 1;1(Ek) toward r = 0. The situation is schematically described in fig. 2. When the decay rate of the wave function at r inside the classical turning point t.,( Ek) is written as a(r), the wavef~ction is given by ek(r)

= N-‘j2

exp

r,(G)

i/

I

e(x)

dx

I,

(A .25)

with\k,(r)=N-‘I” at r=rc(Ek), where N represents the total number of bonds C(r) does not change very rapidly, (Y(T) can approximately be given by ix(r) = {(2mi,)[C(r)

on the surface.

So long as

(A.26)

-E,]}“2,‘A,

since [~~(~)~2/(2~~) = C(r) - Ek(> 0). When C(r) is given by eq. (2.13, integration in x in eq. (A.23 with eq. (A.26) can be performed analytically by changing the integration variable from x to y = [C(x) Ek l”2, resulting in $(r)

= N-l” for

exp( - 2( A/E,)“’

C(r)

tan-’

[C(r)/E,-1]“2+2[A/C(r)]1’2[1-E,/C(r)]”2),

> Ek,

(A.25a)

with A = ~4~~/(2~2~~).

(A.27)

Let us consider, as in section 2, that after the free hole gets localized at the bond at the first localized hole, the distance of the two holes in the two-hole localized state shrinks to a value a of the order of a bond length. Let us write as b( Ek)/N the absolute square of the amplitude of the wave function of a free hole with momentum k at the coordinate value a. It is given by 1\k,(a) I 2 when Ek < U, while by N-i when Ek > U, where U, defined by C(a) in eq. (2.16), represents the Coulombic repulsive energy between the two holes in the two-hole localized state. Therefore, eq. (A.25a) gives b(E) as 0, h(E)

=

exp( -4(

A/E)“’

tan -‘[(U/E-1)1’2]

i 1,

+4(A/U)“2(1-E/U)“2),

for

E 10,

for

O
for

E 2 U. (~.28)

Let us consider here that I b) represents a strongly localized state of a hole bound by the field of a attractive potential overlong the strong lattice distortion which produces at r = a a strong short-ranged Coulombic repulsion from the first localized hole, and hence 1b) should have a dominant amp~tude at the location r = a from the first localized hole. When a free-hole state with wave function (A.23 or (A.25a) is expressed as a ket vector I k), therefore, (b I k) can be approximated by the amplitude !Pk( r) at r = a of the state I k). Then, according to the definition of b( Ek)/N mentioned above, we obtain

lWW*=

I?r5;.(a) l”=b(Ed/‘N.

(A.29)

H. Sumi / Theory on sputtering by sub-bandgap lasers

Since 1k) diagonalizes the Hamiltonian can be expressed as

4139

K + C(r), the first term on the right-hand side of eq. (A.241 (A .30)

In section 2, the free-hole states were specified by suffix f with energy Ef instead of momentum k in the present section. The density-of-state function p(E) for these states introduced in eq. (2.3) is nothing but that for state 1k)s with energy E,+. Then, using eq. (2.3), we can approximate the right-hand side of eq. (A.30) by using eq, (A.29), as (h]I~-K-C(r)]-‘j6)~N-‘f[b(~~~p(E’)/(z-E’)]

dE’.

(A.311

When the complex variable z in eq. {AX) is replaced by E + ia with a positive infinitesimal number 6, the quantity on the right-hand side of eq, (A.31) can be split into its real part given by the principal part of the integral and its imaginary part, as (bl[E-K-

C(r) +i6]-‘lb)

=p(E)

-i&(E)P(E)/N,

(A.32)

with (A.33) where 9 represents taking the principal part of the integral. The relation (A.24) ensures that the spectral function g(E) given by the second equation of (A.21) is determined by the imaginary part of the inverse of the left-hand side of eq. (A.32), as, with b(E) of eq. (A.28),

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