We report on the first model of a thermal transistor to control heat flow.
Like its electronic counterpart, our thermal transistor is a three-terminal
device with the important feature that the current through the two terminals
can be controlled by small changes in the temperature or in the current through
the third terminal. This control feature allows us to switch the device between
"off" (insulating) and "on" (conducting) states or to amplify a small current.
The thermal transistor model is possible because of the negative differential
thermal resistance.Comment: 4 pages, 4 figures. SHortened. To appear in Applied Physics Letter

Baowen Li

Giulio Casati

Lei Wang

Pokrovsky V. L.

English

arXiv

Crossref

Negative differential thermal resistance and thermal transistor

BW LI

L. WANG

CASATI, GIULIO

Archivio istituzionale della ricerca - Università dell'Insubria

APPLIED PHYSICS LETTERS 88, 143501 2006Negative differential thermal resistance and thermal transistorBaowen LiaDepartment of Physics, National University of Singapore, Singapore 117546, Republic of Singapore;Center for Computational Science and Engineering, National University of Singapore, Singapore 117542,Republic of Singapore; Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University,Nanjing 210093, People’s Republic of China; and NUS Graduate School for Integrative Sciencesand Engineering, Singapore 117597, Republic of SingaporeLei WangDepartment of Physics, National University of Singapore, Singapore 117546, Republic of Singaporeand Center for Computational Science and Engineering, National University of Singapore,Singapore 117542, Republic of SingaporeGiulio CasatiDepartment of Physics, National University of Singapore, Singapore 117546, Republic of Singapore;Center for Computational Science and Engineering, National University of Singapore, Singapore 117542,Republic of Singapore; International Center for Nonlinear and Complex Systems, Universita’ deglistudi dell’Insubria, 22100 Como, Italy; Istituto Nazionale di Fisica della Materia, Unita’ di Como,22100 Como, Italy; and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, I-20133 Milano, ItalyReceived 9 January 2006; accepted 7 March 2006; published online 3 April 2006We report on the first model of a thermal transistor to control heat flow. Like its electroniccounterpart, our thermal transistor is a three-terminal device with the important feature that thecurrent through the two terminals can be controlled by small changes in the temperature or in thecurrent through the third terminal. This control feature allows us to switch the device between “off”insulating and “on” conducting states or to amplify a small current. The thermal transistor modelis possible because of the negative differential thermal resistance. © 2006 American Institute ofPhysics. DOI: 10.1063/1.2191730The invention of the transistor1 and other relevant de-vices that control the electric charge flow has led to an im-pressive technological development. In recent years some in-teresting progress has been made also in the control of theheat current. Indeed a theoretical model of a thermal rectifierhas been recently proposed2 in which the heat can flow pref-erentially in one direction. More recently, new models ofthermal diode have been devised3,4 which allow to improvethe efficiency by more than three orders of magnitude.In this Letter we propose a model for a thermal transistorwhich can control the heat flow in analogy to the usual elec-tric transistor for the control of the electric current. The rea-son why such a device can, in principle, operate is groundedin the phenomenon of negative differential thermal resis-tance NDTR which, as shown below, can take place innonlinear lattices.Our model of thermal transistor consists of three seg-ments, D, S, and G see Fig. 1a for its configuration. Thenames D, S, and G follow from the ones used in a metal-oxide-semiconductor field-effect transistor MOSFET. Eachsegment is a Frenkel-Kontorova FK lattice.5,6 Segments Sand D are coupled usually weakly via a spring of constantkint, while segment G, the control segment, is coupled to thejunction particle O between segments S and D via a spring ofconstant kintG. Temperature TG is used to control temperatureTO at the junction O which determines the heat currentfrom D to S. The total Hamiltonian of the model writesaElectronic mail: phylibw@nus.edu.sg0003-6951/2006/8814/143501/3/$23.00 88, 14350Downloaded 08 May 2006 to 193.206.162.158. Redistribution subject toH = HS + HD + HG + Hint + HintG = i=1NS 12x˙S,i2 +12kSxS,i− xS,i−12 −VS22cos 2xS,i + i=1ND 12x˙D,i2 +12kDxD,i− xD,i−12 −VD22cos 2xD,i + i=1NG 12x˙G,i2 +12kGxG,i− xG,i−12 −VG22cos 2xG,i +12kintxS,NS − xD,ND2+12kintGxS,NS − xG,NG2. 1Here xD,i, xS,i, and xG,i are the particles’ displacementsfrom their equilibrium positions in segments D, S, andG, respectively. Fixed boundary conditions are taken, i.e.,xD,0=xS,0=xG,0=0.In the following we will present first the results of ournumerical simulations, then the theoretical explanation. Inour numerical simulations the lattices are coupled, at theirfirst particles, with heat baths at different temperatures TD,TS, and TG, respectively. We use Langevin heat baths and wehave checked that our results do not depend on the particularheat bath realization e.g., Nose-Hoover heat baths. The lo-cal heat flux is defined by Jnkx˙nxn−xn−1 and the localtemperature is defined as Tn= x˙n2. The simulations are per-formed long enough to allow the system to reach a nonequi-librium steady state where the local heat flux is constant ineach segment.© 2006 American Institute of Physics1-1 AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp143501-2 Li, Wang, and Casati Appl. Phys. Lett. 88, 143501 2006As illustrated in Figs. 1 and 2, the model described byHamiltonian 1 can exhibit different useful functions de-pending on the parameters values.Thermal switch. We first demonstrate the switch functionof our transistor, namely, we show that the system can actlike a good heat conductor or an insulator depending on thecontrol temperature TG. This is shown in Fig. 1b, where weplot JG, JS, and JD vs TG. When TG increases from 0.03 to0.135, both JD and JS increase. In particular, at three pointsTG	0.04, 0.09, and 0.135, we have that JD=JS and thus JGis exactly zero. These three points correspond to “off,”“semi-on,” and “on” states, at which JD is 2.410−6,1.210−4, and 2.310−4, respectively. The ratio of the heatcurrents at the on and off states is about 100; hence ourmodel displays one important function—switch—just likethe function of a MOSFET used in a digital circuit.Thermal modulator/amplifier. As demonstrated above,the heat current from D to S can be switched between differ-ent values. However, in many cases, such as in an analogcircuit, one needs to continuously adjust the current in a widerange. In Fig. 2 we demonstrate this “modulator/amplifier”function of our transistor. Here it is seen that in the tempera-ture interval TG 0.05,0.135, the heat current through thesegment G remains very small −10−5–10−5, within theshadow strip in Fig. 2, while the heat currents JS and JD arecontinuously controlled from 510−5 to 210−4.FIG. 1. Color online a Configuration of the thermal transistor. b Heatcurrent vs the control temperature TG. Parameters are TD=0.2, VD=1.0,kD=0.2, kint=0.05, TS=0.04, VS=5, kS=0.2, VG=5, kG=1, and kintG =1. No-tice that in a wide region both JS and JD increase when the temperature TGis increased.FIG. 2. Color online Heat current vs the control temperature TG. HereTD=0.2, VD=1.0, kD=0.2, TS=0.04, VS=5, kS=0.2, kint=0.05, VG=5,kG=1, and kintG =0.1. The shadow region is the range of variation of JG inthe temperature interval TG 0.05,0.135.Downloaded 08 May 2006 to 193.206.162.158. Redistribution subject toLet us now turn to the discussion of the underlyingphysical mechanism which allows the transistor to function.In all cases considered here, the heat resistance of segmentG has been taken small enough so that TO	TG. For fixedthermal bath temperatures TS and TD, we would like now tounderstand how the heat currents JS and JD depend ontemperature TO. This dependence is determined by thedifferential thermal resistances, RS= JS /TOTS=const−1 andRD=−JD /TOTD=const−1. In typical situations the heat currentincreases with the increase of the thermal gradient and there-fore both RS and RD are positive. As a consequence, thecurrent amplification factor  JDJG =  RSRS + RD  1, 2namely, in order to induce a change JD, the control heatbath has to provide a larger JG. This means that the “tran-sistor” can never work!In order to get the amplification effect, namely, 1,either RS or RD must be negative. Thus we must have anegative differential thermal resistance NDTR. In such casethe possibility arises that, when TO changes both JS and JDchanges simultaneously in the same way. Therefore the con-dition JS=JD can be achieved for several different values ofTO as shown in Fig. 1. Analogously, the condition JS	JDcan be verified in a wide range of TO as shown in Fig. 2. Inthese situations heat switch and heat modulator/amplifier arepossible. In the ideal, limiting case of RS=−RD which, inprinciple, can be obtained by adjusting parameters, the tran-sistor works perfectly.Figure 3 illustrates the mechanism of NDTR. In Fig.3b we plot the heat current versus temperature TL see Fig.3a for fixed TR, fixed lattice size, and different couplingconstants kint. In Fig. 3c instead the heat current is plottedfor different lattice sizes with fixed coupling constants kint.As it is clearly seen, in a wide range of system param-FIG. 3. Color online a Configuration of the model exhibiting the nega-tive differential thermal resistance see definition in the text. b Heat cur-rent vs temperature TL at fixed TR=0.2 for different coupling constants kint,with lattice size NL=NR=25. The system parameters are VL=5, VR=1,kL=1, and kR=0.2. c Same as b but for different system sizes andkint=0.05. Notice that when TL0.1 the heat current increases with decreas-ing the external temperature difference.eters, for a fixed temperature TR, there exists a temperature AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp143501-3 Li, Wang, and Casati Appl. Phys. Lett. 88, 143501 2006interval in which a larger value of TLTR, corresponding toa smaller thermal gradient, can induce a larger heat current.The phenomenon of NDTR can be understood from themismatch between the phonon bands of the two interfaceparticles. This is illustrated in Fig. 4 where we plot the powerspectra of the two interface particles at different temperaturesTL for a fixed TR=0.2 Fig. 3a. As it is seen, the overlap ofthe power spectra increases as TL increases, i.e., as the ther-mal gradient decreases. Therefore, by increasing TL we are inthe presence of two competing effects: from one hand thethermal gradient decreases and on the other hand the bandoverlap increases. The beavior of the heat current depends onwhich of these two effects prevails. In particular, the phe-nomenon of NDTR can take place.The physical mechanism of such behavior is due to thefact that the variation of the band position with temperaturedepends on the strength of nonlinearity. This general prop-erty is at the base of the thermal diode models.3,4 In particu-lar, for the model discussed in this Letter, as the temperatureTL increases, the kinetic energy of the left interface particlebecomes large enough to overcome the on-site potential bar-rier and low frequency modes appear.Finally, we would like to discuss the possibility of build-ing a thermal transistor in real physical systems and the pos-sible applications. Due to the fast developing semiconductortechnology, it is now possible to fabricate several low dimen-sional devices, such as nanotubes, nanowires, and thin films,in nanometer scale.7 A nanotube with small diameter or ananowire is basically a one dimensional 1D heat conductor.For the purpose of our model, which is indeed one dimen-sional, one can fabricate a T-shaped nanotube or nanowire.FIG. 4. Color online The power spectra of the particles at the right Rside and the left L side of the interface for different temperatures TL withfixed TR=0.2. The system parameters are VL=5, VR=1, kL=1, kR=0.2,kint=0.05, and N=50. Notice that the overlap of the spectra increases as thetemperature TL increases, which results in the increase of the heat currentwith decreasing the external temperature difference when TL0.11.Downloaded 08 May 2006 to 193.206.162.158. Redistribution subject toTo build the on-site potential in the FK model, one may putdifferent segments of nanotubes/nanowires on the surface ofdifferent substrates.Among the possible applications we would like to men-tion, as an example, that in our model, by adjusting the tem-perature of the third terminal, one can amplify the currentthrough the other two terminals. This can be used as an ef-ficient heat pump to dissipate heat from a microelectronic ornanoelectronic device where a large amount of redundantheat is produced due to the fast operational speed of thedevice. This could be very important since, as it is known,the redundant heat, if it is not dissipated immediately, maydeteriorate the device.In conclusion, we have built a theoretical model for athermal transistor. The model displays two basic functions ofa transistor: switch and modulator/amplifier. The crucial ele-ment of the thermal transistor is the negative differentialthermal resistance. Although at present this is no more thanan abstract model, we believe that, sooner or later, it can berealized in a nanoscale system experiment. After all theFrenkel-Kontorova model used in our simulations is a verypopular model in condensed matter physics.5,6 For instance,it has been used to model crystal dislocations, epitaxialmonolayers on the crystal surface, ionic conductors andglassy materials, an electron in a quasi-1D metal below thePeierls transition, charge density waves, Josephson junctionchains, and dry friction. More recently, this model has beenemployed to study transport properties of vortices in easyflow channels8 and strain-mediated interaction of vacancylines in a pseudomorphic adsorbate system.9This project is supported in part by FRG of NUS and theDSTA Singapore under Project Agreement POD0410553B.L.. One of the authors L.W. is supported by DSTASingapore under Project Agreement POD0001821. Anotherauthor G.C. is also partially supported by EU Contract No.HPRN-CT-2000-0156 QTRANS and by MURSTPrin 2003, Ordine e caos nei sistemi estesi non lineari: strut-ture, stocasticita’ debole e trasporto anomalo.1J. Bardeen and W. H. Brattain, Phys. Rev. 74, 230 1948.2M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 0943022002.3B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 2004.4B. Li, J.-H. Lan, and L. Wang, Phys. Rev. Lett. 95, 104302 2005.5V. L. Pokrovsky and A. L. Talapov, Theory of Incommensurate Crystals,Soviet Scientific Reviews Supplement Series, Physics Vol. 1 Harwood,New York, 1984.6O. M. Braun and Y. S. Kivshar, Phys. Rep. 306, 1 1998.7D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar,H. J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phys. 93, 793 2003.8R. Besseling, R. Niggebrugge, and P. H. Kes, Phys. Rev. Lett. 82, 31441999.9S. C. Erwin, A. A. Baski, L. J. Whitman, and R. E. Rudd, Phys. Rev. Lett.83, 1818 1999. AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

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