Radiative Exchange of Heat Between Nanostructures - a Quantum Story
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1 Radiative Exchange of Heat Between Nanostructures - a Quantum Story JB Pendry Imperial College, London 01 June 2001 page 1
2 REPORT DOCUMENTATION PAGE Form Approved OMB No Public reporting burder for this collection of information is estibated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burder to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports ( ), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA Respondents should be aware that notwithstanding any other provision of aw, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (FROM - TO) Workshop Presentations to TITLE AND SUBTITLE Radiative Exchange of Heat Between Nanostructures - a Quantum Story Unclassified 6. AUTHOR(S) Pendry, J. B. ; 7. PERFORMING ORGANIZATION NAME AND ADDRESS Imperial College xxxxx xxxxx, Londonxxxxx 9. SPONSORING/MONITORING AGENCY NAME AND ADDRESS Office of Naval Research International Field Office Office of Naval Research Washington, DCxxxxx 12. DISTRIBUTION/AVAILABILITY STATEMENT APUBLIC RELEASE 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER. PERFORMING ORGANIZATION REPORT NUMBER 10. SPONSOR/MONITORS ACRONYM(S) 11. SPONSOR/MONITORS REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES See Also ADM001348, Thermal Materials Workshop 2001, held in Cambridge, UK on May 30-June 1, Additional papers can be downloaded from: ABSTRACT Radiative Exchange of Heat Between Nanostructures 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: a. REPORT Unclassified b. ABSTRACT Unclassified c. THIS PAGE Unclassified 17. LIMITATION OF ABSTRACT Public Release 18. NUMBER OF PAGES NAME OF RESPONSIBLE PERSON Fenster, Lynn lfenster@dtic.mil 19b. TELEPHONE NUMBER International Area Code Area Code Telephone Number DSN Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39.18
3 Stefan's Law / WW> / ww*- «WW/H ; 7V QBB = <rr 2 1r 4 60H 3 CQ d 7i 4 - T 2 4 d» A T = ha 0 ^r 5 Stefan's Law neglects contributions from evanescent photon states which have large values of parallel momentum, k: E p =E 0p K + pex V (ik-r // -az) V2 2 2 k -co CQ 0/ Jw«e 200/ page 2
4 Critical Distance for Evanescent Waves to Dominate T(K) A j (microns) i.e. at distances of a few nanometres, radiative heat flow is almost entirely due to evanescent modes 01 June 2001 page 3
5 Evanescent waves play no role in heat loss from a hot dielectric surface to vacuum dielectric sub-critical rays -WvJ vacuum 01 June 2001 page 4
6 ... but evanescent waves can carry heat from a hot to a cold dielectric surface vacuum dielectric New evanescent contribution to heat flow Heat sink 01 June 2001 page 5
7 Resistive Media and Evanescent States (1) Resistive media also have evanescent waves outside their surfaces. In fact they are a much more potent source of evanescent waves because they support very short wavelength states not found near dielectrics. Dissipation implies charge fluctuations Resistive material f y" + p: ^ ^Z 4 ^- + charge fluctuations imply electric fields in the vacuum These fluctuations are the nanoscopic equivalent of Johnson noise. 01 June 2001 page 6
8 Resistive Media and Evanescent States (2) The energy density at distance d is proportional to, Im RJk,a>)Qxp(-kd) where R p (k,(ti) is the reflection coefficient of the surface at wave vector k and frequency co, and, ImR p (k,a>) «Im s(k,a>)-l (k 9 co) + l 2 (j/cdsf) _ CD ' v k» 4 + (<j/a> 0 ) 2 c 0 It is generally assumed that the conductivity is independent of (,co) over a wide range of values. The energy density is a maximum when, where we have substituted a frequency typical of temperature T. At room temperature the optimum electrical conductivity is 690(mQ) _1. 01 June 2001 page 7
9 The importance of short wavelength fluctuations At short distances, evanescent states dominate in phase space: propagating photon modes carry heat flux within the green circle, evanescent modes within the yellow circle. r-1 -+M- optimal heat flow depends on how many k-points are active which is determined by, ad ~ kd ~ 1 2 _ " 7ZK «-max 1 'k " hence ' n (itlf 4xd< provided that these modes exist in the solid. 01 June 2001 page >
10 Heat flux through evanescent modes q p (k,co) = 2ma>\ n Qxp(h\a)/k B T^j-l Qxp(h\o)\/k B T2)-l n m u li x Qxp[-2aa) dm -2 ad 1 - R\ p (k, co)r2p (k, co)e where p denotes the polarisation. The complete heat transfer from surface 1 to 2 via evanescent states is, QEv{ d ) = Yj\ [q p {k,co) + q s {k,co)dco k or most materials p-polarisation is dominant; s-polarisation is important only for magnetically active materials. Maximising heat flux through evanescent modes at fixed T x, T 2 Maximise, X = -lad (T (\mry r>\2 x_ R 2 e -2ad i.e. R} + Rl=e^. x = substituting, assuming only p-polarised modes are active. 01 June 2001 page 9
11 Connection to Quantum Information Theory {qualitative results) Transmitting information by modulating a stream of particles (photons, phonons, electrons ): nn / E In each time slice, T, one bit of information is provided by the presence or absence of a particle. Therefore the rate of flow of information is, / However in order to confine a particle in a time slice T, energy is required of order, -1 E «tlt~ Hence a flow of energy naturally and unavoidably accompanies the information, E ~ T E ^tlt '' In fact this is the minimum energy flow. E>*hi r2 hi a 01 June 2001 page 10
12 Connection to Quantum Information Theory see amongst other references: {quantitative results) Quantum Limits to the Flow of Information and Entropy, JB Pendry, J. Phys. A. 16, (1983). From very general arguments the flow of information in a channel is limited by, where E is the energy flow and / the information flow. Identifying the energy flow with heat flow, Q, 71 E = Q, Q i = kb T\n2 we have, hence as above, Q^ 7±2 BT 2 3h assuming only p-polarised modes are active. 01 June 2001 page 11
13 Measurement of the Quantum of Thermal Conductance K. Schwab, E.A. Henriksen, J. M. Worlock, and M. L. Roukes Suspended Mesoscopic Device - a series of progressive magnifications. I Sot ÜH cii«> f, V (a) Overall view of the ~1.0 x 0.8mm device, showing twelve wirebond pads that converge via thin film Nb leads into the center of the device. This central region is a 60mn thick silicon nitride membrane, which appears dark in the electron micrograph. 01 June 2001 page 12
14 (b) View of the suspended device, which comprises a 4 x 4 mm "phonon cavity" (center) patterned from the membrane. In this view the bright "c" shaped objects on the cavity are thin film Au transducers, whereas in the dark regions the membrane has been completely removed. The transducers are connected to thin film Nb leads that run atop the "phonon waveguides"; these leads ultimately terminate at the wirebond pads. (c) Close up of one of the catenoidal waveguides, displaying the narrowest region which necks down to ~200mn width. 01 June 2001 page 13
15 100 John Pendry - Radiative Exchange of Heat Between Nanostructures T ' ' "" ^^ /- so 04 GO «M Er 10- '" f" I ~7 - CO ' ># / r - er - "» ~ y _ T t - ) X -j r. ^^^^^^^^^_^_j j fr^ Temperature (mk) 6000 Thermal Conductance Data. We normalize the measured thermal conductance by the expected low temperature value for 16 occupied modes. For temperatures below T co - 0.8K, we observe a dramatic saturation in G th at a value near the expected quantum of thermal conductance. Measurement error is approximately the point size, except where indicated. 01 June 2001 page 14
16 Thermal Conductivity of Vacuum How close must two surfaces be to transmit heat as effectively as 1cm thickness of copper (thermal conductivity 400 Wm" K" ) at room temperature? Assume optimised surface reflectivity so that,,2 1 and > k (2/r) 2 Ami 2 hence, J»336Ä 01 June 2001 page 15
17 Rate of Cooling of a Nanosphere Rate of cooling of an isolated sphere in vacuo: QBB ~ ^-^~ Watts 63c^/z 5 <T where c is the conductivity of the sphere, assumed large. 2a d If we assume: a«d, d«xj then the rate of cooling of a sphere outside a surface conductivity c s : Q o 3 3, ArjA In a kßt 2 e Q 5dh 3 a s a 01 June 2001 page 16
18 Relative Efficiency of Black Body Versus Evanescent Modes Q SQCQTI 1 QBB 607r 2 k 2 B T 2 d 3 a s substituting, r = 300K d = 10" 8 m <T s = 10 5 (mq) -1 we have, ^ =1.6xl0 5 QBB 01 June 2001 page 17
19 Energy Density lnm from a Hot Surface u 1 0 E 2 = 0 kßt 2 24h d 3 cr If we choose, then, r = 300K, d = 10~ 9 m, ö- = 10 4 (mq) _1 compare the energy density for black body radiation at 300K: 1 2 ^0 E BB 2_ X2 4 r4 =L53xl0-6 Jm 60H 3 CQ June 2001 page 18
20 The STM as a Blowtorch a 'Nanoheater' applying heat to individual molecules Scanning tunneling microscope tip /\asordea molecule NB it is vital that the tip is manufactured from a material of optimum resistivity. 01 June 2001 page 19
21 Conclusions Photon tunnelling dominates radiative transfer between nanostructures. There is a simple formula for heat flow related to the dielectric properties of the nanostructures. Quantities entering the theory also occur in the theory of quantum friction. Photon tunnelling is a previously ignored source of heat flow and will have dramatic consequences for heat in nanostructures. Questions What experiments should we do to confirm the effect? Can we exploit the effect to design 'nano-coolers' or 'nanoheaters'? In what other situations are evanescent states important, possibly at lower temperatures and on different length scales? 01 June 2001 page 20
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