MNC THE ROLE OF THERMAL EXCITATION OF D BAND ELECTRONS IN ULTRAFAST LASER INTERACTION WITH NOBLE (Cu) AND TRANSITION (Pt) METALS

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1 Procdings of MNC007 MicroNanoChina07 January 10-13, 007, Sanya, Hainan, China MNC THE ROLE O THERMAL EXCITATION O D BAND ELECTRONS IN ULTRAAST LASER INTERACTION WITH NOBLE (Cu) AND TRANSITION (Pt) METALS Zhibin Lin Dpartmnt of Matrial Scinc and Enginring Univrsity of Virginia Charlottsvill, VA 904 zl5c@virginia.du Lonid V. Zhigili Dpartmnt of Matrial Scinc and Enginring Univrsity of Virginia Charlottsvill, VA 904 lzn@virginia.du ABSTRACT Th tmpratur dpndncs of th lctron hat capacity and lctron-phonon coupling factor for nobl (Cu) and transition (Pt) mtals ar invstigatd basd on th lctron dnsity of stats (DOS) obtaind from ab initio lctronic structur calculations. or Cu, d band lctrons could b thrmally xcitd whn th lctron tmpratur xcds ~3000 K, lading to a significant incras, up to an ordr of magnitud, in th lctron-phonon coupling factor and strong nhancmnt of th lctron hat capacity away from th linar dpndnc on th lctron tmpratur, which is commonly usd in most of th currnt computational and thortical invstigations of ultrafast lasr intractions with mtals. Opposit to th cas in Cu, th thrmal xcitation of d band lctrons in Pt lads to a monotonic dcras of th lctronphonon coupling factor and contributs to significant ngativ dviations of th lctron hat capacity from th linar dpndnc in th rang of lctron tmpraturs that ar typically ralizd in ultrafast lasr matrial procssing applications. Strong and drastically diffrnt tmpratur dpndncs of th thrmophysical proprtis prdictd for Cu and Pt point to th importanc of th lctron DOS ffcts and th ncssity of full considration of thrmal xcitation of d band lctrons for ralistic modling of short puls lasr intraction with nobl and transition mtals. 1. INTRODUCTION Ultrafast lasr irradiation of a mtal targt can driv th targt matrial into a highly nonquilibrium stat whr th lctron tmpratur could b transintly brought up to vry high valus, comparabl to th rmi tmpratur, whil th lattic rmains rlativly cold. or mtals, th tim volution of th lattic and lctron tmpraturs, T l and T, is typically dscribd with th two-tmpratur-modl (TTM) [1]. Th TTM modl consists of two coupld non-linar diffrntial quations that account for th lasr xcitation of th conduction band lctrons and subsqunt nrgy rlaxation procsss, i.. nrgy transfr from hot lctrons to th lattic vibrations du to th lctron phonon coupling and th lctronic hat diffusion from th irradiatd surfac to th bulk of th targt: T r C ( T ) = [ K( T ) T ] G( T )( T Tl ) + S(, t) (1) t Tl Cl ( Tl ) = G( T )( T Tl ) () t whr C and K ar th hat capacitis and thrmal conductivitis of th lctrons and th lattic as dnotd by subscripts and l, G(T ) is th lctron-phonon coupling factor rlatd to th rat of th nrgy xchang btwn th lctrons and th lattic, and S ( r, t) is a sourc trm dscribing th lasr nrgy dposition. Th succss of th application of TTM for th quantitativ dscription of th tmporal and spatial volution of th tmpratur and nrgy in an irradiatd targt rlis in a big part on th ralistic choic of tmpratur dpndnt thrmophysical paramtrs in th TTM, Eqs. (1) and (). At 1 Copyright 007 by ASME

2 high lctron tmpraturs th thrmophysical proprtis can b snsitiv to th dtails of th lctronic structur of th targt matrial. It has bn shown, in particular, that th lctron hat capacity in Au [,3] and Pt [4] can b dirctly affctd by th thrmal xcitation of d band lctrons. A systmatic analysis of th tmpratur dpndnc of th thrmophysical proprtis of Au, rportd in Rf. [5], suggsts that at high lctron tmpraturs (>3000 K) th thrmal xcitation of d band lctrons rsults in a significant incras of both th lctron-phonon coupling factor and th lctron hat capacity. Ths changs hav strong implications on th prdictions of TTM calculations. Thrfor, th approximations commonly usd in most of th currnt TTM calculations, namly, a linar tmpratur dpndnc of th lctron hat capacity obtaind from th Sömmrfld xpansion, which is valid only at vry low lctron tmpraturs [6], and a constant valu for th lctron-phonon coupling factor, ar inappropriat for dscribing th matrial proprtis undr conditions of strong ultrafast lasr xcitation. In this papr w invstigat th ffcts of thrmal xcitation of d band lctrons on th tmpratur dpndncs of th lctron hat capacity and lctron-phonon coupling factor in Cu and Pt, two mtals with vry diffrnt lctronic structurs, typical for nobl and transition mtals. Th lctron DOS obtaind from lctronic structur calculations for Cu and Pt ar discussd in Sction. Rsults from th analysis of th tmpratur dpndnt thrmophysical proprtis ar prsntd in Sction 3 and summarizd in Sction 4.. ELECTRONIC STRUCTURE CALCULATIONS In ordr to invstigat th ffct of th lctron DOS on th high tmpratur thrmophysical proprtis of Cu and Pt, w prform lctronic structur calculations basd on th dnsity functional thory using th Vinna Ab-initio Simulation Packag (VASP) [7]. Th Projctor Augmntd Wav (PAW) potntial [8] is usd in th calculation, whr th xchang corrlation trm is tratd within th Gnralizd Gradint Approximation (GGA). Th calculations ar don for Cu and Pt at th quilibrium lattic constants of Å and 3.9 Å, rspctivly. Th lctron DOS for Cu and Pt obtaind from VASP at T =0 K ar shown in ig. 1. As a mmbr of th nobl mtal family, Cu has a filld d band and an lctronic structur similar to Au [5,6]. In th cas of Pt, th rmi lvl cuts through th partially filld d band, lading to a vry high dnsity of lctron stats at th rmi lvl, typical for transition mtals. By xamining th lctron DOS for Cu and Pt togthr with th rmi distribution functions at various lctron tmpraturs, on can s that for Cu at lctron tmpraturs lss than 0.1 V (~10 3 K) only lctrons around th rmi lvl ar xcitd, whil at T ~ 1 V (~10 4 K) or highr, th numbr of xcitd d band lctrons can b significant and should b takn into account. or Pt, th d band lctrons at nrgy lvls around th rmi nrgy can b asily xcitd vn at low lctron tmpraturs, shifting th rmi lvl to highr nrgis and altring th thrmophysical proprtis of th mtal in a way diffrnt from what would b xpctd for th fr lctron modl with a parabolic lctron DOS [6]. 3. TEMPERATURE DEPENDENCE O THERMO- PHYSICAL PROPERTIES In this sction, th lctron DOS obtaind in Sction is usd to invstigat th tmpratur dpndncs of th lctron hat capacity and lctron-phonon coupling in Cu and Pt. Th rsults ar compard with approximations commonly usd in most of th currnt TTM calculations. 3.1 Elctron hat capacity C (T ) In gnral, th tmpratur dpndnc of th lctron hat capacity in mtals is dscribd as [6]: a. b. igur 1. Elctron DOS obtaind in lctronic structur calculations prformd with VASP (solid lins) for (a) Cu and (b) Pt. Th rmi distribution functions ar also shown for thr diffrnt valus of th lctron tmpratur. Th nrgy is shown with rspct to th rmi nrgy at zro tmpratur, ε. Copyright 007 by ASME

3 igur. Chmical potntial for Cu and Pt as a function of th lctron tmpratur togthr with th chmical potntial prdictd by th fr lctron modl (solid lin). Rsults ar shown with rspct to th rmi lvl at zro tmpratur, ε. f ( ε, µ, T ) C ( T ) = ( ε ε ) g( ε ) dε (3) T whr g(ε) is th lctron DOS at th nrgy lvl ε, f ( ε, µ, T ) is th rmi distribution function, dfind -1 as f ( ε, µ, T ) = {xp[( ε - µ )/ kbt ] + 1}, ε is th rmi nrgy at zro tmpratur, and µ is th chmical potntial at T. Th valuation of th lctron hat capacity from Eq. (3) rquirs th knowldg of th chmical potntial µ as a function of th lctron tmpratur. Th chmical potntial can b found by stting th rsult of th intgration of th product of DOS and th rmi distribution function at T ovr all nrgy lvls to b qual to th total numbr of lctrons [6]. Whn th lctron tmpratur is much lowr than th rmi tmpratur, for th fr lctron modl th chmical potntial can b approximatd by th Sömmrfld xpansion for th lctronic fr nrgy: π k BT µ ( T ) = 1 ε (4) 1 ε ig. shows th chmical potntials for Cu and Pt calculatd at various lctron tmpraturs using DOS from VASP, togthr with th chmical potntial prdictd by th fr lctron modl, i.. Eq. (4). or Cu, whn T is blow ~3000 K, µ roughly follows th fr-lctron-lik dscription, consistnt with th good agrmnt of th shap of th DOS around th rmi lvl with th prdiction of th fr lctron modl, ig. 1(a). As th lctron tmpratur xcds ~3000 K, th thrmal xcitation from th high dnsity of stats at th dg of th d band lads to th incras of th chmical potntial. In th cas of Pt, th xcitation from high dnsity of stats at nrgy lvls blow ε to small dnsity of stats abov ε occurs at low lctron tmpraturs, lading to th dviation of th chmical potntial from th Sömmrfld xpansion vn at low lctron tmpraturs. By incorporating th rsults for th chmical potntial into Eq. (3), w obtain th lctron hat capacity of Cu and Pt as a function of th lctron tmpratur, ig. 3. Th rsults ar shown togthr with th linar approximation C ( ) T = γt obtaind from th Sömmrfld xpansion, which is valid only at low lctron tmpraturs [6]. Th diffrnc in th tmpratur dpndnc of th lctron hat capacity prdictd for Cu and Pt can b undrstood from th following analysis. or Cu, blow ~3000 K, th rgion of th lctron DOS affctd by thrmal xcitations ( k BT ) is similar to th on prdictd within th fr lctron modl. As a rsult, th lctron hat capacity follows th linar dpndnc, with th thortical valu of th cofficint γ calculatd with th numbr dnsity of s lctrons in Cu, n = 1.0 atom -1, as γ π n k = / ε = 71 Jm -3 K -1 [6], B bing in a good agrmnt with th xprimntal valu of 98 Jm -3 K - [9]. At lctron tmpraturs xcding 3000 K, th xcitation from th larg numbr of d stats in Cu rsults in a positiv dviation of th hat capacity from th linar tmpratur dpndnc. or Pt, th high dnsity of lctron stats at th rmi lvl nsurs that th d lctrons can b asily xcitd to th s band. Th s band has a much smallr dnsity of stats and th shift of th chmical potntial to highr nrgis, ig., lads to th ngativ dviation of th hat capacity from th linar dpndnc, ig. 3(b). Larg dviations of th lctron hat capacity from th linar dpndnc at high lctron tmpraturs imply that th us of th linar approximation in TTM calculations can ithr significantly ovrstimat or undrstimat th transint valus of th lctron tmpratur during th tim of th lctronlattic nonquilibrium. Th lctron thrmal conductivity is rlatd to th lctron hat capacity through th Drud modl rlationship, κ ( T ) = vc ( T ) τ ( T ) / 3 whr v is th rmi vlocity and τ ( T ) is th total lctron scattring tim with othr lctrons and th lattic [,10]. Thrfor, larg positiv or ngativ dviations of th lctron hat capacity from th linar tmpratur dpndnc, shown in ig. 3, would also affct th lctron thrmal conductivity, spcially at th arly stag of th lctron-phonon quilibration, whn th lctron tmpratur is clos to its maximum. 3. Elctron-phonon coupling factor G(T ) Th nrgy rlaxation procss btwn th xcitd hot lctrons and th lattic in th irradiatd targt procds through th lctron-phonon scattring procsss, whr th rat of th nrgy xchang is charactrizd by th lctronphonon coupling factor [11,1]. Dspit th fact that a constant valu for th lctron-phonon coupling factor is 3 Copyright 007 by ASME

4 a. b. igur 3. Elctron hat capacity of (a) Cu and (b) Pt as a function of th lctron tmpratur calculatd with DOS obtaind from VASP (solid lins) and using C = γt with (a) γ = 98 Jm -3 K - and (b) γ = 740 Jm -3 K - [9] (dashd lins). a. b. igur 4. Elctron-phonon coupling factor of (a) Cu and (b) Pt as a function of th lctron tmpratur calculatd with DOS obtaind from VASP. Data prsntd in this figur is accssibl in tabulatd form from [15]. commonly usd in most of th currnt computational and thortical invstigations of ultrafast lasr intractions with mtals, growing xprimntal vidnc suggsts that th application of th constant lctron-phonon coupling may b inappropriat for strong lasr xcitation [,13,14]. In ordr to xplicitly account for th thrmal xcitation of d band lctrons in nobl and transition mtals, w adopt th xprssion for th tmpratur dpndnt lctron-phonon coupling factor obtaind in Rf. []: πhkbλ ω f G( T ) = g ε dε g ε ( )( ) (5) ( ) ε whr λ is th lctron-phonon coupling constant dfind in th suprconductivity thory, <ω > is th scond momnt of th phonon spctrum dfind by McMillan [16], and g(ε ) is th lctron DOS at th rmi lvl ε. At low lctron tmpraturs, f / ε rducs to a dlta function cntrd at th rmi lvl ε and Eq. (5) rcovrs th xprssion for th lctron-phonon coupling constant proposd by Alln in Rf. 11: G0 = πhk B λ ω g( ε ). At lvatd lctron tmpraturs, howvr, f / ε at nrgy lvls away from ε bcoms nonzro and th lctron DOS g(ε) at ths nrgy lvls nds to b takn into account in Eq. (5). or Cu, with th valu λ<ω >=9±4 mv masurd in Rf. [17] and DOS shown in ig. 1(a), w can dtrmin th room tmpratur lctron-phonon coupling constant from Alln s xprssion, G 0 = (5.5±0.7) Wm -3 K -1. Th litratur valus of G 0 for Cu vary btwn 1 and Wm -3 K -1 [18,19,0]. or Pt, du to absnc of th xprimntal data for λ<ω >, a valu of λ=0.66 [ 1 ] from rsistivity masurmnts and an approximation of <ω >=1/θ D [], whr θ D =40 K is th Dby tmpratur for Pt [9], can b usd to stimat th room tmpratur lctron-phonon 4 Copyright 007 by ASME

5 coupling constant for Pt to b G 0 ~ Wm -3 K -1. This valu is of th sam ordr of magnitud as th on valuatd from low tmpratur rflctivity masurmnts for Pt, (1.09±0.5) Wm -3 K -1 [3]. As th vrification of th valu of λ<ω > for Pt is byond th scop of this papr, and considring that th xprimntal valu of th lctron-phonon coupling constant in Rf. 3 is masurd at rlativly low lctron tmpraturs, from room tmpratur up to ~465 K, w st th coupling factor at T =300 K in Eq. (5) to b qual to Wm -3 K -1 and obtain λ<ω > = 14.5 mv. This valu is thn usd in Eq. (5) to calculat th tmpratur dpndnc of th lctron-phonon coupling factor in a broad rang of lctron tmpraturs as shown in ig. 4(b). Th lctron-phonon coupling factors of Cu and Pt calculatd with Eq. (5) using th lctron DOS obtaind from VASP calculations ar shown in ig. 4 [15]. or Cu, th lctron-phonon coupling factor rmains clos to a constant valu blow ~3000 K, similar to Au [5]. Howvr, as th lctron tmpratur xcds ~3000 K, th incras of th coupling factor obsrvd for Cu is significantly stpr as compard to Au. This obsrvation can b rlatd to th diffrncs in th lctron DOS of th two nobl mtals, ig. 1(a) and ig. 1 of Rf. [5]. Th width of d band for Cu, ~3.5 V, is much smallr than th on for Au, ~6 V, rsulting in a highr dnsity of stats at th high-nrgy dg of th d band in Cu as compard to Au. As a rsult, thrmal xcitation of d band lctrons in Cu lads to a mor significant incras in th lctron-phonon coupling factor. In particular, th lctronphonon coupling factor for Cu xcds th room tmpratur valu by a factor of 9.5 at T = K, ig. 4(a), compard to 5.8 for Au at th sam lctron tmpratur [5]. In th cas of Pt, th strngth of th lctron-phonon coupling dcrass monotonically as th lctron tmpratur incrass. This could b xplaind from th prspctiv of th rol of thrmal xcitation of d band lctrons in nobl and transition mtals. or Pt, as th thrmal xcitation of d band lctron from blow th rmi lvl to highr nrgy lvls lads to th incras in th chmical potntial, ig., th contribution to th lctron-phonon coupling from d band lctrons is rducd, whras for Cu th high valu of DOS from d band nsurs th significant contribution from d band lctrons to th coupling, vn though th chmical potntial xhibits a tmpratur dpndnt bhavior similar to Pt whn th lctron tmpratur is highr than ~3000 K. It should b notd that th trnd of th dcas in th strngth of th lctron-phonon coupling at high lctron tmpraturs for Pt, ig. 4(b), is consistnt with th xprimntal valu of Wm -3 K -1, obtaind from fitting th rsults of TTM calculations to th thrshold lasr fluncs for th onst of surfac mlting in Pt films of diffrnt thicknss [4]. rom ig. 4(b), th lctron-phonon coupling factor for Pt dcrass with rspct to its room tmpratur valu by a factor of.8 at T = K, by a factor of 3.9 at T = K, and by a factor of 5.1 at T = 10 4 K. Th larg changs in th strngth of th lctron-phonon coupling with incrasing lctron tmpratur, illustratd in ig. 4, suggst that th commonly usd assumption of th tmpratur-indpndnt lctron-phonon coupling can lad to ithr significant undrstimation or ovrstimation of th lctron-phonon nrgy xchang rat following a short puls lasr irradiation. An adquat dscription of th tmpratur dpndnc of th lctron-phonon coupling factor is important for a rliabl quantitativ analysis of practically important charactristics of th lasr-matrial intraction, such as thrshold fluncs for th onst of mlting and ablation, strngth of th lasr-inducd strss wav, and dpth of th mlting and/or hat-affctd zon. 4. SUMMARY Th tmpratur dpndncs of th lctron hat capacity and lctron-phonon coupling factor ar invstigatd for a nobl mtal Cu and a transition mtal Pt basd on th lctronic structur calculations prformd within th dnsity functional thory. or Cu, thr is no significant thrmal xcitation of d band lctrons blow ~3000 K, and th lctron hat capacity follows a linar dpndnc on th lctron tmpratur whil th lctron-phonon coupling rmains constant. As th lctron tmpratur xcds ~3000 K, thrmal xcitation of lctrons from th high-dnsity dg of th d band lads to a stp and larg (up to an ordr of magnitud) incras of th lctron-phonon coupling factor and a significant positiv dviation of th lctron hat capacity from th linar dpndnc. or Pt, th shift of th chmical potntial away from th larg dnsity of stats around th rmi lvl rsults in a monotonic dcras of th lctron-phonon coupling factor with incrasing lctron tmpratur and ngativ dviations of th lctron hat capacity from th linar dpndnc. Th contrasting rsults obtaind in this work for Cu and Pt suggst that th ffct of thrmal xcitation of d band lctrons on th thrmophysical proprtis is snsitiv to th structur of th lctron DOS and a dtaild analysis of th DOS ffcts is rquird for ach targt matrial. ACKNOWLEDGMENTS inancial support of this work is providd by th National Scinc oundation through th Thrmal Transport and Thrmal Procsss Program of th Chmical and Transport Systms Division (Award No. CTS ). REERENCES [1] S. I. Anisimov, B. L. Kapliovich, and T. L. Prl'man, Elctron mission from mtal surfacs xposd to ultrashort lasr pulss, Sov. Phys. JETP 39, 375 (1974). [] X. Y. Wang, D. M. Riff, Y. S. L, and M. C. Down, Tim-rsolvd lctron-tmpratur masurmnt in a 5 Copyright 007 by ASME

6 highly xcitd gold targt using fmtoscond thrmionic mission, Phys. Rv. B 50, 8016 (1994). [3] A. N. Smith and P. M. Norris, Numrical solution for th diffusion of high intnsity, ultrashort lasr pulss within mtal films, Procdings of 11 th Intrnational Hat Transfr Confrnc 5, 41 (1998). [4] T. Tsuchiya and K. Kawamura, irst-principls lctronic thrmal prssur of mtal Au and Pt, Phys. Rv. B 66, (00). [5] Z. Lin and L. V. Zhigili, Thrmal xcitation of d band lctrons in Au: implications for lasr-inducd phas transformations, Proc. SPIE 661, 6610U (006). [6] N. W. Ashcroft and N. D. Mrmin, Solid Stat Physics (Holt, Rinhart and Winston, Nw York, 1976). [7] G. Krss and J. Hafnr, Ab initio molcular dynamics for liquid mtals, Phys. Rv. B 47, 558 (1993). [8] G. Krss and D. Joubrt, rom ultrasoft psudopotntials to th projctor augmntd-wav mthod, Phys. Rv. B 59, 1758 (1999). [9] Amrican Institut of Physics Handbook, Third dition, (McGraw-Hill, Nw York, 198). [10] M. Kavh and N. Wisr, Elctron lctron-scattring in conducting matrials, Adv. Phys. 33, 57 (1984). [11] M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Rlaxation btwn lctrons and crystallin lattics, Sov. Phys. JETP 4, 173 (1957). [1] P. B. Alln, Thory of thrmal rlaxation of lctrons in mtals, Phys. Rv. Ltt. 59, 1460 (1987). [13] J. P. Girardau-Montaut and C. Girardau-Montaut, Thory of ultrashort nonlinar multiphoton photolctric mission from mtals, Phys. Rv. B 51, (1995). [14] J. K. Chn and J. E. Braun, Modling of ultrashort lasr ablation of gold films in vacuum, J. Opt. A: Pur Appl. Opt. 5, 168 (003). [15] Elctron tmpratur dpndncs of thrmophysical proprtis of Cu and Pt and svral othr mtals ar accssibl in tabulatd form from -coupling/ [16] W. L. McMillan, Transition tmpratur of strong-coupld suprconductors, Phys.Rv. 167, 331 (1968). [17] S. D. Brorson, A. Karoonian, J. S. Moodra, D. W. ac, T. K. Chng, E. P. Ippn, M. S. Drsslhaus and G. Drsslhaus, mtoscond room-tmpratur masurmnt of th lctron-phonon coupling constant λ in mtallic suprconductors, Phys. Rv. Ltt. 64, 17 (1990). [18] P. B. Corkum,. Brunl, N. K. Shrman, and T. Srinivasan-Rao, Thrmal rspons of mtals to ultrashortpuls lasr xcitation, Phys. Rv. Ltt. 61, 886 (1988). [19] H. E. Elsayd-Ali, T. B. Norris, M. A. Pssot, and G. A. Mourou, Tim-rsolvd obsrvation of lctron-phonon rlaxation in coppr, Phys. Rv. Ltt. 58, 11 (1987). [0] Z. G. Wang, C. Dufour, E. Paumir, and M. Toulmond, Th S snsitivity of mtals undr swift-havy-ion irradiation: a transint thrmal procss, J. Phys.: Condns. Mattr 6, 6733 (1994). [1] P. B. Alln, Empirical lctron-phonon λ valus from rsistivity of cubic mtallic lmnts, Phys. Rv. B 36, 90 (1987). [] G. Grimvall, In Slctd topics in solid stat physics, Vol. 16, ditd by E. P. Wohlfahrt (North Holland, Amstrdam 1980). [3] A. P. Caffry, P. E. Hopkins, J. M. Klopf, and P. M. Norris, Thin film non-nobl transition mtal thrmophysical proprtis, Microscal Thrmophys. Eng. 9, 365 (005). [4] J. Hohlfld, S.-S. Wllrshoff, J. Güdd, U. Conrad, V. Jahnk, and E. Matthias, Elctron and lattic dynamics following optical xcitation of mtals, Chm. Phys. 51, 37 (000). 6 Copyright 007 by ASME