Nonlinear electron dynamics in metallic nanostructures

Transcription

1 Nonlinar lctron dynamics in mtallic nanostructurs Giovanni MANFREDI Institut d Physiqu t Chimi ds Matériaux d Strasbourg Strasbourg - Franc Giovanni.Manfrdi@ipcms.u-strasbg.fr Mastr Lctur 1 1

2 Plan of th lcturs 1. Introductory rmarks on mtallic nanostructurs Rlvant quantitis and typical physical paramtrs Applications 2. Linar lctron rspons: Mi thory and gnralizations 3. Nonlinar rspons Survy of various modls from N-body to macroscopic Man-fild approximation (Hartr and Vlasov quations) 4. Byond th man-fild approximation Hartr-Fock quations Tim-dpndnt dnsity functional thory (DFT) and local-dnsity approximation (LDA) 5. Macroscopic modls: quantum hydrodynamics Linar thory and comparison of various modls 6. Spin dynamics: xprimntal rsults and rcnt thortical advancs 7. Illustration: th nonlinar lctron dynamics in thin mtal films Mastr Lctur 1 2

3 Suggstd rading mastr-mc.u-strasbg.fr 2 m anné Support ds cours Manfrdi F. Calvayrac t al. Nonlinar lctron dynamics in mtal clustrs, Physics Rports 337, (2000). U. Kribig and M. Vollmr, Optical proprtis of mtal clustrs, Springr sris in matrials scinc (1995). E.K.U. Gross, J.F. Dobson, M. Ptrsilka, Dnsity functional thory of timdpndnt phnomna, in Topics in Currnt Chmistry n 181 (Springr, 1996). K. Burk, Th ABC of DFT, G. Manfrdi, How to modl quantum plasmas, Filds Institut Communication Sris (2005), quant-ph/ G. Manfrdi, P.-A. Hrviux, Y. Yin, and N. Crousills, Collctiv Elctron Dynamics in Mtallic and Smiconductor Nanostructurs, Lctur Nots in Physics (Springr, 2009). Mastr Lctur 1 3

4 Nanotchnologis from th past Lycurgus cup Lat Roman, 4th cntury AD, British Musum, London Grn in scattrd light and rd in transmittd light Silvr-gold nanoparticls ( nm) mbddd in th glass Mastr Lctur 1 4

5 Staind-glass windows Mastr Lctur 1 5

6 Applications to biology A clustr of gold nanoparticls (d 50nm) can crat a much largr cratr ( 20µm) in an ic sampl. Gold nanoparticls can act as powrful localizd hat sourcs: possibl biomdical applications. Rsonant absorption at th surfac plasmon (Mi) frquncy Mastr Lctur 1 6

7 Physics at th nano-scal 1 nm Avrag distanc btwn atoms d A 0.15 nm (ordr of magnitud of Bohr s radius a 0 = nm) N 300 atoms N ~ R 3 R=1 nm Avrag distanc btwn lctrons: Wignr-Sitz radius r s π r s =n 3 Avrag volum occupid by an lctron Mtallic dnsitis n m -3 For gold (Au): r s = nm Mastr Lctur 1 7

8 Quantum or classical? D Brogli thrmal wavlngth: λ B = h mv th = m h k B T Thrmal spd: man-squar vlocity from thrmal random motion in a classical gas with a Maxwll-Boltzmann distribution V = k T / m th B At T = 300K, λ B = 1.7 nm ; at T = 10K, λ B = 9.4 nm Quantum ffcts bcom important whn λ B > r s ~ n 1/3 For gold: r s = nm r s Elctrons in mtals ar in th quantum rgim vn at room tmpratur λb Mastr Lctur 1 8

9 Quantum or classical? Frmi statistics Quantum ffcts bcom important as: T < T F Frmi tmpratur: T F / T (n λ B 3 ) 2/3 whrλ B is th thrmal d Brogli wavlngth T < T F λ B > n-1/3 ~ r s = avrag distanc btwn lctrons For gold: T F = K E F = 5.53 V Frmi-Dirac distribution E F Mastr Lctur 1 9

10 Elctron dynamics: som dimnsional analysis Hypothss: Elctrostatic (Coulomb intractions) btwn th lctrons Fixd ions Zro tmpratur (T << T F ) Th rlvant quantitis ar: m,, ε 0, n Combin ths paramtrs to obtain a quantity with th dimnsions of an invrs tim (frquncy): plasma frquncy Th plasma frquncy rprsnts th typical timscal for th lctron dynamics in a mtallic nanostructur For gold: 2πω p fs Mastr Lctur 1 10

11 Dimnsional analysis th plasma frquncy Look for a quantity with th dimnsions of a tim Mastr Lctur 1 11

12 Typical vlocity and lngth scals Classical Thrmal spd Vth = kbt / m Dby lngth (classical scrning lngth) V λ D = ω th p Frmi spd Quantum vf = 2kBTF / m Thomas-Frmi scrning lngth V λ F TF = ω p For gold at T = 300 K: v F = m/s (<< c) v th = m/s λ TF = 0.1 nm (NB: sam ordr as r s ) Mastr Lctur 1 12

13 Plasma frquncy Typical frquncy of lctrostatic oscillations in an lctron gas. 2 ωp Slab with n = ni δx σ = surfac charg σ E = ε 0 Harmonic oscillator with frquncy ω p NB: in this simpl approximation thr is no damping of th oscillations. σ E = = lctric ε 0 fild cratd by a plan capacitor Mastr Lctur 1 13

14 Scrning lngth Positiv charg is surroundd by ngativ lctrons Scrning of th lctrostatic fild + q + + Poisson s quation + n i n / λd λ D V / ω = Dby = th p lngth Quantum cas: Frmi-Dirac distribution: λ TF V / ω = Thomas - Frmi scrning lngth = F p Mastr Lctur 1 14

15 Coupling paramtr Dimnsionlss paramtrs: important to dtrmin th physical rgims Using th quantitis:, m, n, k B T,ε 0 on can construct only on (classical) dimnsionlss paramtr Coupling paramtr : Can b writtn as th ratio of th kintic nrgy ovr th potntial (Coulomb) nrgy Quantum coupling paramtr: Bohr radius: g << 1 : collctiv ffcts dominat man fild g 1 : binary collisions bcom important Mastr Lctur 1 15

16 A (classical) thought xprimnt Lt s imagin w can cut an lctron in two /2, m/2 Various quantitis thn transform as follows /2 ; m m/2 v v n 2n, but: ρ = n ρ T T/2, but: p = nt p What happns to th classical tim, spac, and vlocity scals : /2, m/2 ω p ω p λ D λ D Thy rmain invariant! v th v th Classical coupling paramtr: g C g C / 2 Aftr N>>1 cuttings, w nd up with a continuous distribution of mass and charg Only th man fild rmains, which is thus charactrizd by: Classical tim, spac, and vlocity scals : ω p, v th, λ D Coupling paramtr: g C 0 Mastr Lctur 1 16

17 Typical mtallic paramtrs Gold Sam ordr of magnitud Mastr Lctur 1 17

18 Mtal vs. smiconductor nanostructurs Mtals Smiconductors Mastr Lctur 1 18

19 Log n log T diagram W hav found thr dimnsionlss paramtrs (which dpnd on n and T ): T / T F : classical vs. quantum g C : collctiv vs. collisional (classical) g Q ~ r s /a 0 : collctiv vs. collisional (quantum) T / T T F F log( T / T logt = 1 = K n = F / 3 ) = logt logn + logk 2 logn logk 3 Mastr Lctur 1 19

20 Log n log T diagram I. Man-fild classical II. Non-man-fild classical III. Non-man-fild quantum IV. Man-fild quantum II I III IV TOK: tokamak xprimnt (magntic confinmnt fusion) CORONA: solar corona IONO: ionosphr DISCHA: lctric discharg SPACE: intrstllar spac ICF: inrtial confinmnt fusion DWARF: whit dwarf star JUP: Jupitr s cor MET: mtals Mastr Lctur 1 20

21 Timscals byond th plasma priod (τ p ~1 fs) Initial quilibrium: T = Ti 300K Th lasr puls dposits som nrgy in th lctron gas nonquilibrium distribution τ p Th lctron gas thrmalizs via - collsions: T >> 300K τ - Coupling to th ion lattic (phonons) : th lctron gas rturns to its initial tmpratur τ -ph Mastr Lctur 1 21

22 Collisional timscals lctron-lctron collisions Pauli principl rducs th - collision frquncy For T=0: no collisions, as all quantum stats ar occupid ( Pauli blocking ). For T>0: only lctrons with nrgy comprisd in a rang of k B T around th Frmi surfac (E F ) can undrgo collisions. Thir collision rat is: Th avrag collision rat is obtaind by multiplying ν by th numbr of lctrons availabl for collisions, which is of th ordr T/T F : Using dimnsionlss quantitis: ~ 10 5 which would yild τ 100 ps far too much!! Mastr Lctur 1 22

23 Elctron-lctron collisions But th prvious rasoning is corrct only at thrmal quilibrium Rcall that th lasr puls brings th lctrons strongly out of quilibrium Th kintic nrgy acquird by th lctrons in this arly transint corrsponds to an ffctiv tmpratur T 3000 K Using this valu w obtain, for gold τ - 80 fs Consistnt with obsrvations Mastr Lctur 1 23

24 Elctron-phonon collisions: two-tmpratur modl C C i dt 2 ( T ) = κ T dt dti = GT ( Ti ) dt GT ( T ) + i P( t) C C i ( T G 10 π ) = n W/(m Sodium k J/(m 3 T B TF 3 K) K) 96.6T J/(m 3 K) Ti (K) T (K) Tim (ps) Mastr Lctur 1 24

25 Summary of tim scals τ p << τ - τ -ph τ p 0 50 fs τ - 100fs 0.5ps τ -ph 1 ps 5 ps Mastr Lctur 1 25

26 What w (might) hav larnt from lctur #1 Transition from classical to quantum at th nanomtr scal λ B > r s ~ n 1/3 T < T F Typical tim, vlocity, and lngth scals Plasma frquncy Thrmal spd / Frmi spd Dby lngth / Thomas-Frmi scrning lngth Dimnsionlss coupling paramtr (classical and quantum) Dfins th man fild approximation Various rgims in th log n log T plan Timscals byond th plasma priod Elctron-lctron collision tim Elctron-phonon collision tim Mastr Lctur 1 26