Basics of non-equilibrium Electrodynamics on the nano-scale

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1 Basics of non-equilibrium Electrodynamics on the nano-scale C. Henkel Institute of Physics and Astronomy Universität Potsdam, Germany Heat Transfer and Conduction on the Nanoscale WE Heraeus workshop 613 Bad Honnef April 2016

2 Non-equilibrium... Light Messenger of Stars (Venice 1610) Hubble ultra deep field (NASA 2004)

3 Fundamental Interactions focus on condensed matter electric forces spin exchange "potential" electromagnetic ("collective") forces chemical bonds

4 Electrodynamics matter = sources matter = medium response mesoscopic sources average response + fluctuations ("Rytov split")

5 Fluctuation Electrodynamics [Rytov & al >1950s] source: thermalised matter random source (Langevin force) Johnson & Nyquist + magn. fluctuation dissipation relation [Callen & Welton, Phys Rev 1951]

6 whose Temperature? em field does not equilibrate matter thermalises (local eq = LTE) conservation laws and entropy production

7 non-equilibrium... non-lte? fast physics excitation with short laser pulse (few fs) plasmon dephases hot electrons (10 ps) equilibrate with lattice phonons concept: weak (thermal) contact slow physics (= this workshop) local thermal equilibrium (LTE) (steady) heat flux heat diffusion toy model [... Barton JPCM 2015]

8 nano-scale "local temperature", coarse grain mesoscopic electrodynamics ("Rytov split") thermal fluctuations "collective" averaged field ("local"?) local Ohm: dx mfp (few nm) non-local: "anomalous skin effect" [Lindhard] [Singwi & Sjölander 1968]

9 nano-scale spirit of 2nd FDT: local sources local approx fluctuation dissipation relation [Callen & Welton, Phys Rev 1951] Rytov-Maxwell thermal radiation (Poynting v. etc) how to work with this (numerical schemes)

10 painless Green solve radiation by point source (Green tensor) homogeneous medium

11 Mode Densities "LDOS" occupation / mode mode density blackbody radiation [Planck 1900] FD relation "1st kind" [Callen & Welton 1951] [Eckhardt Opt Commun 1984] local mode density LDOS = Im G, r-dependent global vs local temperature(s)

12 "thermal Photons" in Metal S j HwL K, kt t = 1.67 current noise 2.0 = = 0.23 frequency w t spectrum in bulk (Drude)

13 "thermal Photons" near SiC S j HwL K, kt t = 1.67 = 1.0 = 0.23 current noise frequency w t spectrum in bulk I (ω, z c ) I( ω, z b ) I ( ω, z a ) (a) (b) (c) Ts = 300 K 1 mm 2 µm 100 nm ω, s 1 log I (ω, z a ) log I (ω, z b ) log I (ω, z c ) Shchegrov & al [Phys Rev Lett 2000] near field spectrum surface plasmon

14 Photons v Phonons T = 300K: wavelengths 2-10µm ~ 1nm elastic field theory, heat capacity, local temperature, defects, kinetic theory (Boltzmann) ballistic v diffusive: conduction "heat transfer across vacuum" = "ballistic wire" 2 T d 1 T T +! T T photon tunnelling phonon shooting

15 Boundaries connection rules: Maxwell fields, surface charges/currents 1 2 fluctuates, too matter currents (add'l boundary cond s ABC ) genuine surface response example: plasmon dispersion [Flores & al SSC 1972] [Feibelman PRB 1989] [Horovitz & H EPL 2012] double layer ω / Ω p Kc/ Ω p

16 Regimes of Heat Transfer T1 Stefan-Boltzmann T2 polariton tunnelling & induction heating "phonon tunnelling" phonon conduction Kapitza resistance expts: A. Kittel, T. Kralik, Pramod S. Reddy, Y. de Wilde... theory: A. Rodriguez, K. Sasihithlu, S. Volz...

17 Challenges heat transfer across 1 10nm gap ( xnf ) LETTER group & comp MT Homer Reid RESEARCH expt P Reddy GeNFRHT, SiN SiN (pw K 1) data ata 500 SiN Si TR 200 c d If ) Au SiN SiO 2 Tp Ts SiO2 TR f data Rg Ts 0 Cr VTC Tp OK μm Rp SiO2 GapV size (nm) d PD Photodiode 10 μm K Laser Experimental data Computed data 310 K 400 b 40 Heater thermometer tip ~ 400 nm diam 50 Au 100 Experimental data data Au set-up and SEMComputed images of SThM probes and 80 Figure 1 Experimental Kim & al [Nature 2015] b TR a nm Au coating 50 μm Au measurements, the outer SiO2 coating is appropriately substitu

18 Challenges heat transfer across 1 10nm gap ( xnf ) expt P Reddy group & comp MT Homer Reid OK expt A Kittel group & comp AW Rodriguez setup, keeping in mind that i level. Now, within this appr 0.25 nonlocal eﬀects using the L 0.2 (see Suppl. Mat.), and we fi cal eﬀects increase the heat factor 1/ they turn out to be relative too low distances. Hence, we find th 0.1 scopic model of heat transfer 0.05 heat flux found in our expe wrong phasize that the above calc approximation and fully acc d (nm) finite el t mesh surface plasmon polaritons, t AW Rodriguez ligible for gold at room tem & alconsider [arxiv: ] lows, we a number o Figure 3: Theoretical results of the transfered heat Kloppstech P (nw) 0.3

Challenges heat transfer across 1 10nm gap ( xnf ) expt P Reddy group & comp MT Homer Reid OK expt A Kittel group & comp AW Rodriguez 2 wrong a typical working pressure of 10 10 mbar.

19 Challenges heat transfer across 1 10nm gap ( xnf ) expt P Reddy group & comp MT Homer Reid OK expt A Kittel group & comp AW Rodriguez 2 wrong a typical working pressure of mbar. based on a commercial scanning tunneling STM). As depicted in Fig. 1 (a) and (b), de STM probes consist of a platinum wire, glass capillary, pulled sharp with a pipette factor 1/1000 e then coated too low with 100 nm of Au by means aporation ex situ. At the point where the rates from the Pt-core, a thermocouple is probe design allows for local heat flux meafinite 19]. el t mesh addition to its STM ability [18, The AW Rodriguez pled into the tip apex drains towards the Figure 3: Theoretical results of the transfered heat heflux. tip Sketch holder causing temperature of the consideredageometry (righthand diﬀerside) and numerical results finally, using exactisnumerical calculations for the them which, generating a therspherical tip and the cone-like protruding part. We have dyed A diﬀerent scanning microscope (SEM) parts ofelectron the probe (see inset) with the same colors h. the used for the lines in plots. The parameters of the tip are a as probe is depicted in Fig. 1 (c). The prothe following: the foremost part is modeled by a sphere of P (nw) d (nm) setup, keeping in mind that it overestimates the heat flux Au coated tip, heat power level. Now, within this approximation we have included 300K nonlocal eﬀects using the Lindhard-Mermin model [29] (see Suppl. Mat.), and we find that while these nonlocal eﬀects increase the heat flux, as expected [28, 29], 30nm they turn out to be relatively weak for the considered distances. Hence, we find that the conventional macroau, 120K scopic model of heat transfer greatly underestimates the heat flux found in our experiments. We note and emphasize that thetunnel above calculations in Fig. 3 make no approximation and fully account for flux mediated by surface plasmon current polaritons, though these tend to be negligible for gold at room temperature [29]. In what follows, we consider a number of currently accepted models of phonon (conductive) transfer and argue that they too cannot explain the above mentioned enhancement: (A) Prunnila and Meltaus [1] have studied the tunneling of acoustic phonons between piezoelectric materials. 3 Kloppstech & al [arxiv: ] They report an approximate 1/d distance dependence. Making the same estimation for the eﬀective tip area A as

20 Sketch of a "theory" Idea: proximity force (Deryagin) approximation R "snap in" d flux plate-plate distance

21 Teasers radiation near metals is mainly magnetic (LMDOS) [J. D. Jackson] [K. Joulain, PRB 2003] electrodynamics & relativity: medium edyn & general relativity [U. Leonhardt & Th. Philbin, Progr Opt 2009] relative motion is a non-eq setting (4-vector field) similar to temperature gradients [G. Neugebauer] [Zh. Ch. Wu, EPL 2009] quantum aspects / multiple-scale simulations friction (Cherenkov) forces: I. Nefedov, A. Volokitin...

22 Summary

23 Summary fluctuation EDyn (Rytov theory) = Maxwell + Langevin matter thermalises to T(r), equilibrates the e.m. field via absorption / fluctuation (FD relations 1+2) super-planck heat transfer: "activate" matter degrees of freedom (SPP, phonons) n-scale: non-sharp boundaries, non-local response

arxiv: v2 [cond-mat.mes-hall] 18 Mar 2011

arxiv: v2 [cond-mat.mes-hall] 18 Mar 2011 Near-field thermal imaging of nano-structured surfaces A. Kittel, U. F. Wischnath, J. Welker, O. Huth, F. Rüting, and S.-A. Biehs Institut für Physik, Carl von Ossietzky Universität, D-6111 Oldenburg,