The theory of laser annealing of disordered semiconductors

Volume 80A, number 1

PHYSICS LEUERS

10 November 1980

THE THEORY OF LASER ANNEALING OF DISORDERED SEMICONDUCTORS M.NOGA1

Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland Received 17 July 1980

A theoretical explanation of the disorder—order phase transition in pulsed laser annealing ofion implanted Si is given. The phase transition is related to the Bose condensation of electron—hole plasmons.

Most recent publications dealing with the laser annealing of disordered semiconductors have assumed that this process is a strictly thermal melting—recrystallization first order phase transition [1]. However, in refs. [2,31 the authors recounted many observations pointing out that this transformation was not a thermal melting phenomenon. The authors [4] gave many arguments for the importance of the electron—hole plasma produced by the laser to the disorder—order phase transition which is expected to be of second order, From the point of view of these arguments we propose a possible theoretical explanation of the phenomenon which is based on the second quantization of electron— hole plasmonic waves generated by laser light. The laser light is considered as a classical circularly polarized electromagnetic plane wave with wave vector q and frequency w given by the vector potential A ‘(x, t) = a { a exp [i(q.x ‘-

. + a * exp[—i(q.x

—

*

—

(1)

,

.

.

.

where £ and a are polarization vectors satisfying the

relations =

£*.q

—

,

-

~t)J} .

wt)]

The magnetic permeability of the layer is assumed to be equal to one. Its electric permittivity is denoted by er. If the laser light did not excite electrons from their localized into delocalized states of the amorphous semiconductor then the energy density of the laser field would be given by the formula 2w2’2 2 0 Er a i 7TC The delocalized electrons produced by the laser are considered as a self-interacting electron plasma of local electron density n(x, t) moving in a rigid (general. ly nonuniform) distribution of localized holes with local density N(x, t). Effects associated with thermally excited electrons and holes will be ignored. Thus our system under consideration consists of three subsys. tems, namely the electromagnetic field, localized holes and the electron plasma which are coupled to each other by electromagnetic interactions. For this system we write down the energy density 5C m the following form: W

(l/8ir)(e rE2 +B2)+~(Mu2/n 2 0)(N_n0) + ~ mnu2 + (h2/8mn 2 ~b~ (3) 0)(Vn) where m and Mare the effective masses ofthe electrons —

E.

£ =

~

0

,

£ •E

=

‘

The constant amplitude a is related to the energy flux passing through an amorphous layer as given by the

and holes, respectively, u is the constant ion sound velocity, n 0 denotes the uniform density of the elecPoynting vector trons and holes produced by,the laser and u = u (x, t) 0 0 2 is the local velocity of the electron plasma. In formuSo = (c/4ir) E X B = (a /2ir) ~x1. (2) La (3) the first term is the energy density of the dec. On leave of absence from the Depaitment of Theoretical tromagnetic field, the second term describes the enerPhysics, Comenius University, 81631 Bratislava, Czechoslovakia. gy associated with an inhomogeneous distribution of 91

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PHYSICS LETTERS

the localized holes, the third term is the kinetic energy of the electron plasma, the fourth term is the energy due to existing gradients in the electron plasma and the last term takes into account the exchange forces acting between delocalized electrons. The last term is evaluated in a first approximation by assuming a non-

E=

A-~A+VA,

Then the Maxwell equations (5) become equivalent to the following pair of equations:

(~rk2) a2A/at2

—

=

0,

—

—

(Er/C)

v~~ (9a)

,

= (41re/er)(n N). (9b) The variational principle gives us the field equation

(4)

the Maxwell equations VX B

2~A=

+(4ire/c)n[V~— (e/mc)A]

—

+ V’ (n u)

—

_c-1A.

We choose the radiation gauge V• A = 0.

The known equations of motion for the dynamical variables of the system are: the continuity equation

a (n

B = VX A

c~A,

—

~-~+(e/mc)A,

~ ire2(3n/ir)413

—

Vp

and the dot denotes time derivative. The lagrangian density (7) is invariant under the gauge transformations

degenerate electron gas and the Coulomb force acting between the electrons [5]. In this case the constant b = 1-re2h2/(mkBT), where e is the electron charge, kB is the Boltzmann constant and T is the temperature. By assuming an absolutely degenerate electron gas then the exchange forces lower the energy density ~JC by the amount [5] z~iC=

—

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—

for the localized holes

(Er/c) FJE/at =(4ire/c)nu,

VXE+c—laB/ato,

VB

=

0,

N—n

2)(mi9+ ep), 0 =(n0/Mu and that for the delocalized electrons

erVE=4ire(n—N),(5)

(10)

—(h2/4mn and the hydrodynamic equation for the electron plasma which is taken in the form 1VP,(6) m au/at + m(v V)o= eE + (e/c) u X B n~ whereP is an as yet unspecified pressure. We do not yet know the field equation for the localized holes. We cannot immediately apply the procedure of second quantization because we do not apriori know the canonically conjugate variables. That is why we look for a lagrangian density £ which would imply the energy density (3) and the field equations (4)—(6). A lagrangian with these properties is ‘

£

=

(1/81r)(erE2

—

—

—

(n

—

(Mu2/2n

2 0)(N

+

B2)

~

where

bn2 t~ =

—

—

— ~

iV)(mó~+ep)

The last equation reduces to eq. (6) by taking its gradient and utffizing the relation (8). The nonlinear system of field equations (4), (9)— (11) has an exact solution of the following form: ~9constant,

~0,

Nnn 0

2a2/mc2b. (12)

=e

The vector potential A (x, t) corresponding to the solution (12) is of the form A(x,t) =A(°)(x,t) a{ a exp[i(qx

—

wt)J + a’ exp[—i(q’x

—

wt)]} (13a)

2

—

0)(Vn) (7)

,

(11)

—

=

mnu~ (h2/8mn

n0)

0)~n+m~+ep+~mu2 bnO.

l~(x,t) is a scalar dynamical variable,

provided that q and &~satisfy the dispersion law Cr0.)2

=

c2q2 + 0

(13b)

where w 0 is the electron plasma frequency,

2n

—

(e/mc)A

,

(8)

A and p are the electromagnetic potentials associated with the electromagnetic fields E and B by the formulae 92

w~= 4ire

0/m.

In elementary particle theorists’ jargon one would say that photons have acquired a mass by interacting with an electron—hole plasma ground state. The solution

Volume 80A, number 1

PHYSICS LETTERS

given by the relations (12), (13) is associated with the ground state of our system. The energy density corresponding to the ground state is 1

express each of the field operators in terms of the For instance eq. (9b) gives the operators 3k and result

4.

2

~lCo= w bn0study collective elementary excitations Next0we will of our system by applying a method similar to that of Arponen and Pajanne [6]. These authors have treated an electron gas interacting with impurities as a system of collective boson-like elementary excitations. We do this by noting that the lagrangian density (7) gives us the canonical momentum — -~

O,C/O~Y=—m(n_IV’)iz—mp(x,t)

(14)

with respect to the canonical coordinate ~(x, t). Quantum field theory requires the commutation rela-

4lre/er) p(x,t)=( + exp [—i(kx

4

Pk k—2 {ak exp[i(k.x —

—

~2kt)]}.

~kt)] (19)

In other words the individual field operators associated with their subsystems can be expressed in terms of single pairs of annihilation and creation operators of collective elementary excitations. Next we linearize eq. (4) by ignoring the electromagnetic radiation of the electron—hole plasma. Then we get the equation ôp/Ot + n 0~ = 0,

tion [p(x,t), i9(x’,t)] = (ih/m)iS(x.—x’) to be satisfied. Thecreation last condition is satisfiedoperators by introducing boson-like and annihilation and ak respectively as given by

4

p(x,t)E pk{akexp[i(kx— ~kt)l k + ak exp[—i(kx ~kt)]}, —

~(x, t) = —i ~

10 November 1980

i~

—4 exp[—i(kx

{a~exp [i(k• x

—

&Tlkt)l}

—

—

~k (15)

flkt)] (16)

,

which implies the relation 2 ~ = 0. (20) &~3k n0 k and (20) determine the parameters Pk Eqs. (18) and ~ as given by

where ~2k is an unspecified frequency.

(hnok2/2mVflk)’12

~k

(h&~k/2mnOk2V)1!2 (21) We are now prepared to evaluate the total hamiltonian H of our system in the linear approximation ignoring the electromagnetic radiation of the electron— hole plasma produced by the laser. By straightforward calculations using relations (15), (16), (19) and (21) -

we get the hamiltonian in the following form: H=

f ~Cd3x

=

(w

E

0 ~ b4) v + hIlk(4 ak + V (22) with the energy spectrum of the collective elementary excitations given by —

Here the operators a~tand a~obey the commutation relations 8kk’ Eak, ak’] = E4~ = 0, (17) ~ak = and the expansion coefficients ~k and i~kare subject to the condition

,4’]

4’]

Ii {(k2 + ~JC2) Pk~k=h/2mV,

(18)

where V is the volume of our system. One symmetrizes the energy density (3) and field equations (4)—(1 1) so that any field operator entering them becomes hermitian. The symmetrized relations (4)—(1 1) become the constraints and equations of motion for the field operators in the Heisenberg plcture. By solving these equations one can, in principle,

2)/[k2 +

(23)

7(k

where (k2) (h2k2/4m2)(k2 + ~(‘2) (bn /m)(k2 + ~2) —

+ (w~/e~), ~2

(m/Mu2)(w~/er).

93

Volume 80A, number 1

PHYSICS LETTERS

10 November 1980

The form of the hamiltonian (22) and the commutation relations (17) imply that our system, consisting of three different subsystems which are mutually interacting, can be considered in the lowest approximation as an ideal Bose gas of quasiparticles with the interesting energy spectrum given by eq. (23). Hence we have the following mean occupation numbers for the

above the threshold value 1~‘o > 1~produces a gap of forbiddenmomenta k as is depicted by curve (III) in fig. 1. For the laser intensities I~ 1~the mean occupation numbers of the quasiparticles in the states e(k) = 0 become divergent quantities. The quasiparticles collapse to zero energy states and the system under-

quasiparticles:

goes a phase transition similar to Bose condensation. 1

—

{exp[e(k)/knT] 1} (24) The energy spectrum of the quasiparticles depends significantly on the energy flux 1o = IS0I produced by =

—

.

the laser and passing through the system. In the (e,k) plane eq. (23) represents a parametrical set of curves of the parameter Io. Four such curves are depicted in fig. 1. For ~ = 0 the energy spectrum (23) reduces to the energy of a free electron, 2k2‘2 m. = h The same result follows from (23) in the classical limit, h -÷0,hk -~p,as determined by the choice of the coefficient h2/(8mn 0) in front of the fourth term of relation (3). By increasing the power of the laser the energy spectrum (23) is modified in such a way as is shown by the curves (I), (II), (III) in fig. 1 . There is a threshold intensity ‘c which implies the existence of quasiparticle states with zero energy, e(k) = 0, but for nonvanishing momenta k. The energy spectrum of the quasiparticles corresponding to the threshold intensity is depicted by curve (II) in fig. 1. A laser intensity

// /

~

/

/

~ 0

/

/ k

Fig. 1. The quasi-particle energy spectrum for four different values of the laser intensity ~

94

distribution (15) and the scalar potential (19) will contam macroscopic parts described by time independent and periodic functions of the coordinate x. For the numerical value b l0~~ J m3 the described phase transition takes place if the energy flux i~ 100 MW/ 2 which is consistent with the experimental obsercm vations [1]. By the theory presented in this letter we have obtamed [7] the theoretical explanation for two additional effects associated with the experimental observation of the optical stopping effect on guided laser light in As—S amorphous thin films [8] and with the laser induced transmittance oscifiation in GeSe 2 thin films [9].

References

/ /

/

a~= (~j~1/2 k ‘. ki This implies that the electron—hole plasma is “crystallized”, for its quantum mechanical fluctuations have acquired macroscopic scales. The electric charge =

III)

/

/

a k

/

(II)

/ II)

The creation and annihilation operators a~and a corresponding to zero energy states become classical (macroscopic) variables

The author is very grateful to Prof. Stig Stenholm, the director of the Institute for Theoretical Physics at the University of Helsinki for hospitality and stimulating discussions. He wishes to express his sincere indebtedness and gratitude to Prof. C. Cronström for many enlightening discussions on problems encountered in constructing the theory presented in this letter.

(0)

/

~‘

[1] G. Vitali, M. Bertolotti, G. Foti and E. Rimini, Ph~s. Lett. 63A (1977) 351; J.C. Wang, R.F. Wood and P.P. Pronko, Appi. Phys. Lett. 33 (1978)455; C.M. Surko, A.L. Simons, D.H. Auston, J.A. Golovchenko and R.E. Slusher, Appl. Phys. Lett. 34 (1979) 363. [2] I.B. Khaibullin, E.I. Shtyrkov M.M. Zaripov R.M.Bayazitov and M.F. Galjautdinov, Radiat. Eff. 36

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PHYSICS LETTERS

[3] J.A. Van Vechten, R. Tsu, F.W. Saris and D. Hoonhout, Phys. Lett. 74A (1979) 417. [4] J.A. Van Vechten, R. Tsu and F.W. Saris, Phys. Lett. 74A (1979) 422. [5] See e.g. L.D. Landau and E.M. Lifshitz, Statistical physics (Nauka, Moscow, 1976) p. 270 [6] J. Arponen and E. Pajanne, J. Phys. C12 (1979) 3013; 12 (1979) L161; Ann. Phys. (NY) 91(1975)450; 121 (1979) 343.

10 November 1980

[7] M. Nagy and M. Noga, in preparation. [8] M. Kikuchi and A. Matsuda, in: Proc. VIth Intern. Conf. on Amorphous and liquid semiconductors (Leningrad, 1975) p. 35; A. Matsuda et al., AppL Phys. Lett. 24 (1974) 314. [9] J. Hajtô, G. Zentai and I. Kosa-Somogyi, Solid State Commun. 23 (1977) 401.

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