Chemical Physics 148 ( 1990) 209-2 17 North-Holland

A microscopic approach to the theory of polymer ablation

Jai Sir@ Research School, Faculty ofscience, Northern Territory University, P.O.B. 40146, Casuarina, NT 0811, Australia

and

N. Itoh Department of Physics, Faculty of Science, Nagoya University, Furo-cho, Chiku-saku, Nagoya 464, Japan

Received 27 November 1989; in final form 16 May 1990

A microscopic theory of polymer ablation is presented here. Our results suggest that the surface bonds of polymers are broken due to localisation of a pair of excitations on the same bond in a chain. A laser fluence that can create paired excitations on the same bond defines the threshold laser fluence which is observed experimentally. Other results of the present theory agree very well with experiments and the existing phcnomenological theory of polymer ablation.

1. Introduction

It is observed that the incident light pulses from short wavelength excimer lasers on the surface of a polymer etch holes in polymers [ l-31. The phenomenon is known as polymer ablation. This occurs due to photodecom- position of polymers yielding fragments of monomers. Although several mechanisms for laser ablation have been suggested [ 4-7 1, the basic nature of the phenomenon, for example the superlinear dependence of proba- bility of monomer release and the cause of the monomer decomposition, has not yet been explained. Garrison and Srinivasan [4] have suggested that the ablation occurs due to a volume explosion which would make the production of small molecules quite important. In order to explain the superlinear dependence, Sutcliffe and Srinivasan [ 5 ] have assumed a threshold laser fluence below which the polymer ablation is negligible, and Kiss and Simon [ 81 have assumed that ablation occurs when a monomer is hit by more than n photons, where n is about 10. These phenomenological assumptions still remain unexplained.

The superlinear dependence of the emission yield of neutral atoms from semiconductor surfaces, which is also entirely of a nonthermal origin, on the laser fluence has been reported recently [ 9- 171. The cause of the emis- sion has been suggested to be bond weakening due to the localisation of two holes on the same covalent bond [ 111. The localisation of two holes or hole pairing is considered to occur under dense electronic excitation because two excitons of which holes are paired have an energy much lower than two-paired excitons [ 15 1. The pairing of two holes on a bond is considered to break the bond, and the excess energy released due to formation of such states provides the loose neutral atom with kinetic energy to move out of the surface. According to this model the superlinear dependence has been attributed [ 121 to a superlinear increase in the pairing probability against screened repulsion between two holes.

The aim of this paper is to present a microscopic approach to understand the laser ablation of polymers. We consider that the polymer ablation takes place in a way similar to that of desorption of neutral atoms from semiconductor surfaces. High density excitons are created in the region of contact between the incident pulses and surface of the polymer. As excitons move close together two holes get paired on the same bond, since the

0301-0104/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

210 J. Singh, M. Itoh /Microscopic rheoty ofpolymer ~~i~i~on

pairing is energetically favourable, the bond gets broken, and then, as the interchain bonding is weak in polymers that segment of polymers becomes very loosely bound. It is then ejected out by the excess energy available due to pairing of the holes on the same bond. A theory is presented for a case where monomers are desorbed from the polymer although, as it will be clear later on, similar results would be obtained for other types of fragments as well. The high density exciton ,situation is presented here by co~ide~g an excited state of a polymer where a pair of excitons are localised on nei~~u~~ monomers. Like charge carriers of the two excitons are assumed to have paired spin for satisfying Anderson’s negative U criterion [ 14 ) . The energy eigenvalue of two excitons lying on neighbouring monomers is calculated taking into account the exciton-exciton and exciton-lattice in- teractions. The energy is then minimised with respect to molecular vibration, and it is found that the minimum of the energy is lower than that of free exciton states.

2. Theory

We consider a linear chain of polymers along the x-axis and the polymer surface parallel to the xy-plane. Interchain bonding is considered to be very weak, therefore all excitons are expected to move only along the chains. A one-~mension~ electronic Hamiltonian for polymers can be written as

where

G,,J@= ( l,j, cl -fi2V2/2m+ V(r) Im,j, a> ,

and

(1)

@a)

(2b)

In eq. ( 1 ), 1 and m denote the position of monomers on the x-axis and j represents the electronic states of monomers. al(a) is the creation operator of an electron at 1 in the electronic state j with spin o. The first term of eq. ( I ) is the single particle energy interaction operator and second one is the two particle energy interaction operator.

A polymer chain is shown in fig. 1. We assume that within the width of pulses, each monomer, denoted by dark circles, is excited by the incident laser pulses on the chain. These excitations remain locahsed on their original monomer, and we consider a situation when the monomers at 1 and l+a are excited, a being the inter monomeric separation along the x-axis. A ket vector representing a pair of such excitations can be written as

lncldent Radiation

Fig. 1. A polymer surfaa chain irradiated by laser pulses such that each of its monomers is excited in the irradiated region.

J. Singh, M. Itoh /Microscopic theory of polymer ablation 211

ILa> Tl =Bfcw7+,(W IO> 3 (3)

where B!(S) is the creation operator of an excitation localised on a monomer at 1 in spin state S. ( 0) is a vacuum state which has no excited monomers in the polymer. We consider that both excitations are in singlet spin state although a similar theory can be developed for two triplets as well. Like organic crystals, polymers have very weak inter monomeric overlap of electronic wavefunctions and electronic energy bands of polymers are quite narrow. The electronic ground and first excited state of a monomer will then be nearly the same as that of the polymer’s valence and conduction band respectively. Denoting g and f for the valence and conduction bands of a polymer respectively, we define the excitation of a monomer at I by exciting an electron in the state f and a hole in g. The pair of arrows in ( 3 ) represents that like charge carriers in two excitations at 1 and I + a have paired (opposite) spins. Denoting the creation operator of a hole created at 1, in state g and with spin Q by df,( a) one can write (3) in terms of Fermion operators as

ILa),,=i 5 tufr(~)~t,(-~)u7+~t(-~)~7+,(~)110) * (4)

As the polymer has translational symmetry along the x-axis, anyone of the monomers has equal probability of being excited Therefore a proper eigenvector with a pair of excitations localised on neighbouring monomers can be written as

la),,= T G(x,,-,Kv)I~,~),, 3 (5)

where the coefficient C, is a function of all monomeric displacement coordinates x1, . . . . x,,, along the x-axis. xl, -.., xN are different from the 1 values as the latter represent the equilibrium position coordinates of monomers on the x-axis. Using ( 5 ) and ( 1) we solve the following Schrijdinger equation [ 11,12,15 ] :

HIa> ,l =Hx, 9 ---, XN) la>,, 3 (6)

and then we get:

E(x, 7 -**, xN)c/(xI 3 **., XN) = [‘%(x,, ***, XN) +‘%+cdxl, ***, XN) + u:: + w(x, > -**, XN) 1 c/(x,, . . . . XN)

+ C’ ~/,,c?t+ C’ M+a,nCn , (7) m n

where E/(x,, . . . . x,) and EI+&, . . . . x,, ) are the energies of excitations localised on monomers at 1 and 1+ a respectively. U, is the electrostatic inter-cxciton interaction between all the charge carriers in two excitations, W(x,, -**, xN) is the total electronic energy of the polymer chain without any excitation, and Ml,,, represents the excitation transfer matrix element along the chain; l# m.

Eq. (7) is obtained with all energies as a function of displacement coordinates x1, . . . . xN along the chain. For getting the explicit dependence one can expand these energies in a Taylor series about the equilibrium position. However, U, and M,,,, being solely carrier-carrier interactions, will be little influenced by nuclear motions and hence will not be expanded. In the off-diagonal terms we will take into account only the nearest neighbour interactions, and then we assume that M,,,+p - T between any pair of nearest neighbours. We thus obtain:

E(x,, ..a, XN) c,(xl ~M~*xz, c,(-%~-~xN) m >

-~(C,+,+C/-,)-~(CI+*,,+CI) > (8)

where E,(O) and E,+,(O) represent the excitation energy at I and l+u in the position of equilibrium, i.e. when polymer is frozen, and the corresponding energy W(0) is set to be zero without the loss of any generality. /#I

212 J. Singh, M. itoh /Microscopic theory ofpolymer ablation

represents the force of vibration at position 1. The Taylor series expansion is terminated at terms linear in the displacement coordinates in eq. ( 8).

Applying the small polaron model approach by Holstein f 16 1, we obtain the mi~um energy of two excita- tions localised on neighbouring sites of the polymer as (see appendix)

E(xlO’, . ..) x.@))=2E(O)-4T-2(&-fU;), (9)

where

and

E B2 ab= (24T)-’ -.

(10)

(11)

if Of is the displacement coordinate (see eq. (A. 1) ) of the excited site at minimum energy, and EOb is the energy of exciton-polymer chain interaction.

If there was no exciton-polymer chain interaction, i.e. Eab= 0, the energy of two excitations localised on neigh- bouring monomers would be obtained from eq. ( 9 ) as

E(xfO’, . . . . x#‘~)=~E(O)+U;-~T, (121

which is the total energy of two excitations includ~g their electrostatic interaction energy U,. From eqs. (9) and ( 12), it is obvious that due to excitation-polymer chain interaction the energy of a pair of excitations localised on neighbouring sites is lowered by 2E& and the energy of each free excitation is lowered by Ed- f U,. Thus the paired excitation on neighbouring monomers is preferred as a more favourable excited state of a poly- mer than a free or a single monomer excitation state.

If only one monomer is excited its energy is also lowered due to excitation-polymer chain interaction, but in that case the exci~tion~h~n intemction energy is only half of & [ ii,1 5 3. The energy of a single excited state, including the excitation-chain interaction, is higher than that of a paired excitation state. If the intensity of light is weak, then of course, only single excitation state can be created. However, with intense laser pulses paired excitation states will preferably be excited.

Ablution case. We consider a situation in which each monomer unit is excited with a pair of excitations. It is very likely that when the intensity is very high a monomer, depending of course on its size, can have more than one excitation iocalised on it. That means in a linear chain each side of bonds between monomers will have localised excitations, as shown in fig 1. For this case we can write eq. ( 8 ) as

E(x, > **-, XN)G(& 9 **-9 XN) = (

2E(O)-2/3x,+Uj+& cMw2x: m >

C,(x,,...,x,)-2T(C,+,+C,_,). (13)

Now using the same technique as used for deriving (9) (see appendix), we get the minimum energy for a pair of excitations localised on the same monomer as:

E(xfm’, . . . . x&~))=~E,(O)-~T-~(~&&U;). (14)

x1@‘), the displacement coordinate corresponding to the minimum energy, is obtained as

X(oo) = 2BCj”)*Cfo) MO= - (15)

Comparing eqs. (A. 1) and ( IS), we find that xj 0~) = 2xf0) which means that the polymer chain gets distorted more severely at all those points where a pair of excitations is localised on the same monomer. Consequently

J. Singh, M. I&oh /Microscopic theory of polymer ablation 213

the bandings with neighbouring units are weakened because bonds on both sides of the monomer with a pair of excitations, will be enlarged due to large x1@“.

The excitation-chain interaction energy is doubled when a pair of excitations is localised on the same mono- mer, as it is obvious from eq. ( 14), and consequently the energy of such a pair of excitations is lowered by 4&z, - U,. In this situation three related events can take place at the monomer excited with a pair of excitations: ( 1) the bond lengths from both sides are enlarged, due to large xjoo’, (2) a pair of holes gets localised on each side of the bond, i.e. a covalent electron is removed from bonds on each side of the monomer which breaks both bonds, and (3) as the energy of such an excited state is lower than that of a free exciton state, the excess energy may be used in providing kinetic energy to the thus loosely bound excited monomer. Due to large xf”’ the monomer is pushed out of the chain as shown in fig. 2.

3. Results and discussion

Here we present a microscopic theory for the mechanism of polymer ablation which occurs due to high inten- sity ultraviolet laser pulses incident on a polymer surface. In the region of incident laser pulses high density excitons are excited at the polymer surface. Such excitations can be created either by exciting monomers or any other constituents of a polymer, which will of course depend on the wavelength of the exciting pulses. The excitation in polymers is regarded to be of the type of Frenkel excitons, which are created by exciting a pair of electron and hole on the same molecule.

Exciton-exciton interaction in organic crystals is very well studied [ 12,17 1. In some crystals the electrostatic potential U: (9) can be positive, and at low temperature if Ed < 1 U; then the total of exciton-exciton and exciton-lattice interaction energy will be less than zero, i.e. Eab - 1 U: ~0, and the pair of excitons may disso- ciate into free charge carriers [ 18 1. In other crystals, however, the exciton-exciton interaction potential is at- tractive, for example in anthracene U: = -0-l 2 eV [ 121 and naphthalene UC = - 0.17 eV [ 12 ] when excitons are localised on neighbouring molecules along the b-axis at 6 nm apart. In such cases Eab - f U: > 0, and the total energy of such paired excitons becomes lower than that of two free excitons. Excitons then may form a bound state. When the intensity of laser is very high, a pair of excitons can be excited on the same site. The electrostatic interaction energy U; in this case is calculated to be Uf = - 0.19 eV in anthracene and in naphthalene Uf = - 2.3 eV [ 12 1. In a mixed crystal of phthalazine in naphthalene we find Uf = - 5.9 eV [ 12 1. This indicates that the bound exciton state formed when both excitons are on the same molecule can have much lower energy than that of a free exciton state [ 15 1.

Exciton-chain (lattice) interaction energy can be estimated using an adiabatic potential for the excited state energy. In the case of a polymer chain intermonomeric vibrations can be assumed parallel to the chain because of the inter-chain bonding being weaker than the bonding of monomers along the chain. The excited state energy of the polymer with one excited monomer can be written in terms of adiabatic potential as

INCIDENT RADIATION

llllll Fig. 2. Effect of a pair of excitations localised on both bonds of a monomer irradiated by laser pukes. The monomer is pushed out of the surface as the bond length gets increased.

214 J. Singh, M. Itoh /Microscopic theory ofpolymer ablation

E=EZ + f T MwZ(x, -x0)* ) (16)

where E, is a constant independent of x,, and x,, is the position of the minimum of the adiabatic potential of the excited state. The force constant /I can be estimated from eq. ( 16) as

aE -- B- axI x,=o

=MU2~. (17)

Substituting eq. ( 17) into eq. ( 11) we obtain:

EIlb= (24T)-‘(MW*X;)* , (18)

from which Eab can be calculated. Let us, for example, consider the case of PMMA ( poly-methylmethacrylate ) polymer in which ablation has been observed. It consists of the monomer C5Hs02 which has a mass of 1.66 X 1 O-25 kg. We assume that the intermonomer vibration along the linear chain has frequency w= 50 cm-’ which is about half the Debye cut-off frequency [ 19 ] (90 cm-’ ) for anthracene. x0 Iy -0.3 nm which is the length between the centres of mass of neighbouring monomers along the chain. The intermonomer excitation transfer energy Tmay be taken equivalent to the intermolecular excitation transfer energy in anthracene crystals which is 50 cm-’ [20]. Using these (16) we obtain Ed- -0.3 eV. Instead of taking w= 50 cm-‘, however, if one takes it 90 cm-’ (the Debye cut-off frequency) one gets Eab= 3.1 eV.

When a pair of excitations occupies the same monomer the energy of each excitation is lowered by 2E,- 1 U, see eq. (14). Taking U,= - 1.9 eV [ 12 ] as for anthracene, and Ed- -0.3and3.1 eV,weobtain2Eti-tU/=1.6 and 7.2 eV respectively. Thus in PMMA an excited state with two excitations localised on one monomer can be below that with one excitation locahsed on a monomer by 1.55 to 7.2 eV.

From above estimates it may be conclusively suggested that the ablation in polymers takes place due to exci- tation of many monomers, some of which may become doubly excited, by intense incident laser pulses on a polymer surface. As the doubly excited monomer state is favoured energetically, single excitations on neigh- bouring monomers move to form doubly excited monomer state of polymers. In a situation when, for example, three consecutive monomers are doubly excited on the surface chain the two bonding covalent electrons are excited to the conduction band of the polymer and may localise near any other atomic site away from the bond forming the chain, as shown in fig. 3. As the paired excitation on each monomer has an energy lower than a single isolated excited state, the holes thus excited and exposed will get paired on the bonds of the chain. Con- sequently, the covalent electrons will be replaced by holes and the bond would be broken, resulting into frag- mentation of the polymer. It is very important to note that such paired excited states are not stable states and cannot be observed. What one observes is the resulting fragmentation of polymers (polymer ablation). As the x(O”) ( 15) is larger for doubly excited state the excited monomer in the middle will move out of the surface chain (see fig. 2) which will become distorted. The available excess energy 2E&- f U, will provide kinetic energy to the free monomer thus created and it will move away from the surface with a certain speed that can be estimated. For PMMA polymer the ejection speed v of monomer may be estimated to be 5% [ 2(4E,*- U,)M] ‘I*, M being the mass of the monomer, and we obtain I+ 2.4x lo3 m s-i for E,,=O.3 eV and lJ,= 1.9

m a 43 fa m 63

6 Ait AA A& +- ih ii

pJ @J w w Fig. 3. Schematic illustration of localisation of two holes (from two excitons) on a bond where their uairing electrons remain lo-

Surface Chain calised elsewke within the monomtbs. -

J. Singh, M. Itoh /Microscopic theory ofpolymer ablation 215

eV, and Vja 5.3 x 1 O3 m s- ’ for Ed= 3.1 eV and U,= 1.9 eV, which are about an order of magnitude higher than the speed of sound in air. One should however remember that this is the maximum speed a monomer may acquire, because a part of the excess energy may be lost as heat in the process.

After the front layer is ejected the polymer surface layer cannot rebuild itself because at a time several mono- mers may be ejected from the region of incidence of laser pulses. The next sequence of incident pulses will affect the next interior layer thus exposed due to ablation. Therefore the depth of ablation or incremental etch length will be proportional to the laser fluence and this is in agreement with experiments [ 2 1 ] and recent phenome- nological theory [ 61. Incident pulses etch first the top layer, then the next layer only after the top layer is already ablated. In this situation the depth of ablation will not depend much on the absorption coefficient of the polymer as long as the layer by layer ablation occurs. This is also in agreement with experiment [ 2 I ] where it is observed that the incremental etch depth does not depend directly on the absorption coefficient. However, it may be expected that the etch depth may depend on the duration of pulses. Only when paired excited states are formed due to intense laser pulses that ablation will be observed. This explains the existence of a threshold laser fluence for ablation as observed experimentally.

Monomers are usually large. When they are doubly excited or multiply excited, it is possible that pairing of excitations can occur within a monomer as well exactly in the same way as in a polymer. Thus bonds may be broken within a monomer and its fragment may be ejected from the surface. This also agrees with experiments quite well as more than one type of fragments are observed to be ejected in polymer ablation.

Although further more work needs to be done for providing an adequate explanation to the superlinear release of polymer fragments in ablation, the availability of excess energy following the formation of each paired exci- tation that breaks a bond seems to play the key role. Under the condition of dense excitation, the accumulated excess energy can be absorbed by an ejected fragment of polymers to dissociate it further into smaller fragments. Consequently, the observed number of ejected fragments will be superlinear like in a chain reaction. It may also be attributed to the superlinear increase in the pairing probability against screened repulsion between two holes, which is expected to be a possibility for superlinear desorption of neutral atoms from semiconductor surfaces.

4. Conclusion

A microscopic theory for the process of ablation in polymers is developed. It is suggested that the ablation of polymers is a mechanism similar to the sputtering of neutral atoms from non-metallic compound surfaces. Abla- tion occurs because of the replacement of covalent electrons by holes in a covalent bond which has got a localised pair of excitations excited by intense laser pulses. Results of present theory are in agreement with those from experiments and phenomenological theories developed earlier.

Note added in proof

A discussion with Professor H. Baessler revealed that in most organic polymers the evidence of strong exci- ton-lattice interaction is not found. Therefore the pairing of excitons takes place due primarily to attractive U, the electrostatic interaction between two excitons.

Appendix. Derivation of eqs. (9) and (10)

Multiply (8) by Ct and sum overall 1, and then set

aE(xl, ..-7 XN) =. ax, 9

216 J. Singh, h4. Itoh /Microscopic theory ofpolymer ablation

to get the displacement coordinate ~1’) corresponding to minimum of the energy as

xp = /!12cp* cp MO: .

(A.11

Here C/O) corresponds to x , =xfO) Substituting eq. (A. 1) in eq. ( 8 ) , one obtains:

E(x{O’, . . . . xW’)CfO’ = (

2&(O) - “‘;L: I2 + B;C$’ I2 +v:+t c Mc.&xz, I ,+a m >

CfO’

- T( cg,, + cp;> - T( cj$, + CfO’ ) . (A.21

E,( 0) is independent of the location of a monomer in the chain, and w, would also be the same for vibrations of all monomers along the chain. Within the region of impact with laser pulses the polymer may be expected to have all its monomeric sites excited, and within such a region it may be assumed that the coefficient C’, is nearly the same as that corresponding to neighbouring sites, C,+o or C,_,.

Eq. (A.2 ) then reduces to:

E(xjO’, . . ..xfvo’)C.= (

28lGl’ 2E(O)- w +v:+t c Mw2x2, m >

c,-2T(C,+,+C,_,) ) (A.3)

where the superscript of C, is dropped for convenience. Following Holstein [ 16 ] we may solve (A.3) as follows: Define

-Go=E(x{'), . . . . xW’)-2E(O)-US-4 CMwZX$?2+4T (A-4) m

to obtain from (A.3):

a31G12 ___ -Go Mw2

C,+2T(C,+,+C,_,-2C,)=O.

Eq. (A. 5 ) can be solved using continuous approximation [ 16 ] for coefficients C, as

c _c*ac,+l% ,+a- I a1 2 iI2 .

Using (A.6) and the normalising condition I, 1 C, 1 2 = 0, we obtain:

E(x{O', . . . . xgP,)=2E(0)+U;-4T-2E,,

as given in eq. ( 9 ) .

(A.51

(-4.6)

(A-7)

References

[ 1 ] R. Srinivasan and V. Mayne-Banton, Appl. Phys. Letters 41 ( 1982) 579. [ 21 Y. Kawapura, T. Toyoda and S. Namba, Appl. Phys. Letters 40 (1982) 374. [ 3) R. Srinivasan, in: Interfaces under Laser Irradiation, eds. L.D. Daude, D. Bauerle and M. Wautelet (Nijhof, Dordrecht, 1987) p.

359. [4] B.J. Garrison and R. Srinivasan, J. Appl. Phys. 57 (1985) 2090. [5] E. Sutcliffe and R. Srinivasan, J. Appl. Phys. 60 (1986) 3315. [6] G.D. Mahan, H.S. Cole, Y.S. Liu and H.R. Phillipp, Appl. Phys. Letters 53 (1988) 2377. [ 71 T. Keyes, R.H. Clarke and J.M. Isner, J. Phys. Chem. 89 (1985) 494. [8] D.B. Kiss and P. Simon, Solid State Commun. 65 (1988) 1253. [ 91 T. Nakayama, N. Itoh, T. Kawai, K. Hashimoto and T. Sakata, Rad. Effects Letters 67 ( 1982) 129;

N. Nakayama, Surface Sci. 133 (1983) 101.

J. Singh, M. Itoh /Microscopic theory of polymer ablation 217

[IO] J.M. Moison and M. Benoussam, J. Vat. Sci. Technol. 21 (1982) 315; Surface Sci. 126 (1983) 294. [ 111 N. Itoh and T. Nakayama, Phys. Letters A 92 ( 1982 ) 47 1;

J. Singh and N. Itoh, Appl. Phys. A, in press. [ 121 J. Sin&N. Itoh and V.V. Troung, Appl. Phys. A 49 (1989) 631. [ 131 H.S. Cole, Y.S. Liu and H.R. Phillipp, Appl. Phys. Letters 48 (1986) 76. [ 141 P.W. Anderson, Phys. Rev. Letters 34 (1975) 953. [ 151 J. Sin& Chem. Phys. Letters 149 (1988) 447. [ 161 T. Holstein, Ann. Physik 8 (1958) 325. [ 171 J. Sin& The dynamics of excitons, in: Solid State Physics, Vol. 38, eds. D. Tumbull and Eherenreich (Academic Press, New York,

1984) p. 295. [ 181 J. Sin&. J. Phys. C 13 (1980) 3639. [ 191 G.C. Morris and M.G. Sceats, Chem. Phys. 1 (1973) 376. [20] W.L. Greer, J. Chem. Phys. 64 (1976) 1407. [21] H.R. Phillipp, H.S. Cole, Y.S. Liu and T.A. Sitrik, Appl. Phys. Letters 48 ( 1986) 192.