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International Journal of Thermal Sciences 50 (2011) 648e662

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Review

A review of thermal rectification observations and models in solid materials

N.A. Roberts, D.G. Walker*

Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37325, USA

a r t i c l e i n f o

Article history:Received 8 September 2010Received in revised form22 November 2010Accepted 1 December 2010Available online 15 January 2011

Keywords:Thermal rectificationNon-equilibrium transportNanostructured materials

* Corresponding author.E-mail address: greg.walker@vanderbilt.edu (D.G.

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.12.004

a b s t r a c t

Thermal rectification is a phenomenon in which thermal transport along a specific axis is dependentupon the sign of the temperature gradient or heat current. This phenomenon offers improved thermalmanagement of electronics as size scales continue to decrease and new technologies emerge by havingdirections of preferred thermal transport. For most applications where thermally rectifying materialscould be of use they would need to exhibit one direction with high thermal conductivity to allow forefficient transport of heat from heat generating components to a sink and one direction with lowconductivity to insulate the temperature and heat flux sensitive components. In the process of under-standing and developing these materials multiple mechanisms have been found which produce ther-mally rectifying behavior and much work has been and is being done to improve our understanding ofthe mechanisms and how these mechanisms can be used with our improved ability to fabricate at thenanoscale to produce efficient materials which have high levels of thermal rectification.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Thermal rectification is a phenomenon inwhich thermal transportalong a specific axis is dependent upon the sign of the temperaturegradient or heat current. Thermal rectification has been a topic ofinterest dating back to its initial experimental observation in 1936 byStarr [1]. Since then, there have been several experimental andtheoretical studies performed in an attempt to understand whatmechanisms cause thermal rectification. With an improved under-standing of how thermal rectification is achieved, devices like thermaltransistors, thermal logic circuits and thermal diodes could bedeveloped and utilized in micro/nanoelectronic cooling as well asthermalmemoryand computations. In a recentmeetingof the thermalmanagement community it was mentioned that devices likethermal switches will be necessary for our ability to cool electronicseffectively in the future as device sizes become smaller andas we move toward stacked chip designs and more complicatedthermal management problems [2]. Currently several mechanismsfor thermal rectification have been proposed including surfaceroughness/flatness at material contacts, thermal potential barrierbetween material contacts, difference in temperature dependenceof thermal conductivity between dissimilar materials at a contact,nanostructured asymmetry (ie. mass-loaded nanotubes, asymmetric

Walker).

son SAS. All rights reserved.

geometries in nanostructures, nanostructured interfaces), anhar-monic lattices (typically 1D) and quantum thermal systems. Each ofthese mechanisms will be examined in detail.

Since the first known observation of thermal rectificationactivity in the area has been sporadic with a very large increase ininterest in the last decade. Fig. 1 shows the number of publicationsrelated to thermal rectification since the original observation in1936. In the 1960s and 1970s interest in this area was advancedbecause of the studies of composite materials that were of greatinterest in the aerospace industry [3]. In the 2000s we see anexplosion of interest mostly due to the 1D non-linear latticesinitially presented by Terraneo et al. [4] and the experiments ofnon-uniformly mass-loaded nanotubes by Chang et al. [5]. Theseworks along with our improved ability to model, synthesis andcharacterize systems at the nanoscale have led to an increasingtrend in thermal rectification research in the latter part of the2000s and into the current decade which is shown in the inset ofFig. 1.

At first look, thermal rectification may appear to violate the 2ndLaw of Thermodynamics, but we can provide common simpleexamples where rectification is indisputable. A difference in theNusselt number is observed in natural convection when twoparallel plates are oriented horizontally relative to gravity andseparated by a gas as shown in Fig. 2. When the lower plate isheated, the heat transfer is driven by buoyancy induced flows and isgoverned by RayleigheBernard convection in addition to conduc-tion if the Rayleigh number is greater than the critical value ofRacr ¼ 1708 which is given by [6]

0

5

10

15

20

25

30

35

40

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

num

ber o

f pub

licat

ions

decade

02468

10 12 14

2000 2002 2004 2006 2008 2010

num

ber o

f pub

licat

ions

year

Fig. 1. Number of publications on thermal rectification per decade since 1936.

0

1

2

3

4

5

6

7

8

9

10

10 20 30 40 50 60 70 80 90 100

Nus

selt

num

ber

temperature difference (K)

air upper plate heatedair lower plate heated

water upper plate heatedwater lower plate heated

Fig. 3. Level of rectification in the natural convection system with air and water.

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662 649

RaLhgbðTH � TCÞL3

an; (1)

where g is the gravitational constant, b is the volumetric thermalexpansion coefficient, TH is the temperature of the bottom plate, TCis the temperature of the top plate such that TC < TH, a is thethermal diffusivity and n is the fluid viscosity. We can then calculatethe Nusselt number using [7]

NuL ¼ 0:069Ra1=3L Pr0:074; (2)

for 3�105< RaL<7� 109 where NuL is the average Nusselt number,RaL is the Rayleigh number and Pr is the Prandtl number. If theupper plate is heated there is no fluid motion and the transport isgoverned by conduction through the gas and the Nusselt number is1 [6]. In general the transport in the bottom-heated case will begreater than the top-heated case for the same temperature differ-ence as long as the Rayleigh number is greater than 1708. Fig. 3shows the Nusselt numbers for the two configurations in thenatural convection system when the participating fluid is air andwater. Thermal rectification is possible in this system because thegravitational acceleration breaks the symmetry.

Another demonstration of thermal rectification can be observedin radiation exchange between two emitters separated by a mate-rial that absorbs and re-emits radiation of all wavelengthse a blackbody. A schematic of this system is shown in Fig. 4. If both emittersare black then their emitted radiation is exactly the same when thetemperature difference is reversed. If the emitting surfaces are notblack and have different emissivities with respect to wavelengththen rectification is possible when these emitters are separated bya thermal coupler, in this case a black body. In Fig. 4 the emittingsurface on the left has an emissivity of one for low wavelengthphotons while the emitting body on the right has the opposite. The

Fig. 2. Schematic of rectification in natural convection.

material between the two bodies is a black body. Based on thisscenariowewould expect to see all the radiation emitted from bothemitters to be absorbed and re-emitted by the black body. The re-emitted radiation from the black body will either be absorbed,reflected or a combination of absorbed and reflected, depending onthe wavelength of the radiation and the absorptivity of the surface,from the emitters in which case it would be re-absorbed by theblack body. To calculate the rectification in this case we perform anenergy balance on the black body to give [8]

El þ Eref ;l � Eb ¼ Er þ Eref ;r; (3)

where El and Er are the incident radiation from the left and rightemitters, respectively, Eref,l and Eref,r are the reflected radiationoriginally emitted from the black body and Eb is the radiationemitted from the black body. Fig. 5 shows the level of rectificationfor the system in Fig. 4 where the temperatures of the emitters are1000 K and 500 K with opposite emissivities separated by a singleblack body. This system exhibits thermal rectification because thesymmetry is broken by the difference in emitter properties and iscoupled by the black body in between. This system exhibits similar

Fig. 4. Schematic of rectification in radiation heat transfer.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0 2 4 6 8 10 12 14 16 18 20

leve

l of r

ectif

icat

ion

cutoff wavelength (μm)

1000 K/500 K

Fig. 5. Level of rectification as a function of the cutoff wavelength for the systemshown in Fig. 4 with the temperatures of the emitters at 1000 K and 500 K.

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662650

characteristics as some of the solid systems that will be discussed inthis paper, specifically the nanostructured interfaces in x2.2.3 andthe 1D non-linear chains in x2.2.4.

A thermal rectifier that exhibits high transport properties in onedirection and insulating properties in the opposite would bebeneficial for electronics cooling applications specifically at thenanoscale if they could be incorporated into the design of thecomponent. Imagine a device that was composed of materials thatwere capable of transporting heat along a specific path withextremely high transfer rates (high conductivity) while at the sametime insulating the components along that same path that aresensitive to high temperatures and heat flux. Thermally rectifyingmaterials exhibit these properties and are therefore the idealcandidate material for future electronics cooling applications.Researchers have also proposed the use of thermally rectifyingmaterials in the design of thermal logic circuits, transistors andmemory [9].

This paper will discuss the different known mechanisms ofthermal rectification in solid systems. This paper will also look backat the literature and attempt to support or reject the hypothesis/mechanisms proposed in the past with our modern knowledge ofthermal transport. This paper begins by looking at bulk thermalrectification mechanisms followed by nano and atomic scalethermal rectification mechanisms. Finally this paper will report thelevel of rectification in theoretical and experimental studies foundin the literature and propose a direction for thermal rectificationresearch based on these results.

2. Thermal rectification mechanisms

Thermal rectification has been observed experimentally andpredicted theoretically in several different systems, and differentmechanisms are responsible for thermal rectification depending ongeometry, materials, surface preparation and contamination to lista few.

In this paper the level of rectification will be defined as

3 ¼ kþ � k�

kþ þ k�; (4)

where kþ is the thermal conductivity in the forward direction andk� is the thermal conductivity in the reverse direction. In thisdefinition of rectification we assume that kþ > k� meaning weassume that the conductivity in the forward conduction is greater

than the conductivity in the reverse conduction direction. Similarlywe can define the rectification as the ratio of the difference in heatflux between the two directions over the sum of the two heat fluxesas long as the applied temperature difference is equivalent for theforward and reverse cases. We define the level of rectification thisway similarly to some other researchers [5,10]. In the case of norectification we have no difference in any of the parameters (eg.kþ ¼ k�) and the level of rectification is zero (3 ¼ 0). This definitionalso gives a value of unity for an ideal rectifier where one directionwould be a perfect insulator (k ¼ 0) or a conductor of infiniteconductivity (k ¼ N). Other definitions of rectification have beena ratio of the larger heat flux or thermal conductivity to the smallerheat flux or thermal conductivity (kþ/k� or qþ/q�). This definitionleads to a value of one when the values of each of these are equaland no rectification occurs which could be confusing.

2.1. Bulk mechanisms

The first observations of thermal rectification were in bulksystems typically composed of metals or metal oxides. In thesesystems the mechanism was debated and appeared to be verydependent on surface characteristics (ie. roughness and flatness) aswell as bulk material properties.

2.1.1. Metal/insulator effectThe first observation of thermal rectification was by Starr in

1936 [1]. He was investigating electrical rectification characteristicsof Copper/Cuprous Oxide interfaces and also noticed thermalrectification. The measurements showed that thermal conductancewas greater in the direction of greater electrical conduction,which was from the metal to oxide (copper to cuprous oxide). Starrattributed this effect to asymmetric electron transport in thesystem resulting in asymmetric thermal transport, which isexpected from the electron theory of heat conduction. The level ofthermal rectification in this system was a maximum of 3 ¼ 0.39,which suggests that the forward conductivity is about 130% greaterthan the reverse direction.

In these systems the thermal energy is transported primarily byelectrons in the metal and phonons in the insulator or oxide. Sincethese two materials transport thermal energy with differentcarriers there must be transfer between the electrons and phonons.Near the interface in the metal, the electrons must scatter withphonons which can then be transmitted into the insulator orreflected back into themetal at the interface. This electronephononscattering and the phonon transmission lead to an effective contactresistance at the interface which is given by

Rc ¼ Re�p þ Rpt ; (5)

where Re� p is the resistance due to electronephonon scatteringand Rpt is the resistance due to phonon transmission/reflection. Inthese systems the materials are assumed to be in perfect contact,but because of the localized resistances, a temperature jump existsat the interface. The resistance due to electronephonon scatteringin metal is governed by scattering rates that are not in equilibrium,which means that the energy transfer from electrons to phononscan be different than that from phonons to electrons. This suggeststhat thermal rectification is possible in these systems. In 2006Walker [10] developed a non-equilibrium transport model basedon Majumdar and Reddy’s work [11] and scattering rates fromLundstrom [12], which is presented here. Fig. 6 shows thetemperature distributions achieved when the direction of thethermal bias is switched. Although the electronephonon system isin non-equilibrium, Boltzmann moments with an energy exchangeterm are assumed to be adequate to model the non-continuum

q"

q"

profilestrainedA

B

rs

cr

Fig. 7. Change in thermal contact area due to strain or warping.

Tp

Te

Te

Tp

heat flux

y x

T

heat flux

y x

Tinsulator

metal

metal

insulator

Fig. 6. Electron and phonon temperature distributions through a metal/insulatorsystem with reverse temperature gradients.

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662 651

effects [10]. This assumption leads to a system of two coupledFourier-like carriers given by

ked2Tedy2

� G�Te � Tp

� ¼ 0; (6)

kpd2Tpdy2

þ G�Te � Tp

� ¼ 0; (7)

where ke and kp are the electron and phonon thermal conductivitiesin themetal and G is related to the rate of energy transport betweenelectrons and phonons. At the interface the metal and insulator arecoupled such that the phonon heat flux is matched and the electronheat flux is zero. These conditions are given by

kpdTpdy

����y¼0

¼ �kdTdx

����x¼0; (8)

kedTedy

����y¼0

¼ 0: (9)

Additionally a temperature boundary condition must exist forthe temperature jump at the interface. This boundary condition isgiven by

kpdTpdy

����y¼0

¼ h�Tpðy ¼ 0Þ � Tðx ¼ 0Þ�: (10)

Finally, wemust also assume that away from the interface in themetal, equilibrium exists and Te ¼ Tp. If we assume that ke [ kp,which is true for metals, we obtain a solution for the interfaceconductance given by [11]

h ¼ he�phpthe�p þ hpt

; (11)

where he�p ¼ ffiffiffiffiffiffiffiffiGkp

p. In this case G is dependent on the

electronephonon scattering rate, usually expressed as G ¼ C/swhere C is the electron specific heat and s is the electro-nephonon scattering rate.

This model is, in general, not dependent on the direction ofthermal transport through the interface. The difference in direc-tional conductance comes from the energy dependence of theelectronephonon scattering rates. The scattering rate from Fermi’sGolden rule is given by [12]

1s¼ pD2

o2ruo

�N þ 1

2H12

gCðE � ZuÞ (12)

where the electron density of states gC is dependent onwhether thephonon is emitted (þ) or absorbed (�), Zu is the phonon energy, r is

the material density and Do is the optical deformation potential.The energy relaxation rate is found by scaling the scattering rate as

1sE

¼ Zu

EðpÞ1s: (13)

The fact that s varies depending on the direction of transportsuggests that G is also dependent on direction. Therefore theGregorian model allows for thermal rectification.

2.1.2. Thermal strain/warping at interfacesInterfaces that are composed of materials that exhibit properties

with different temperature dependencies will result in differenteffective contact areas when the temperatures at each side of theinterface change. Experimental and theoretical results have shownthat surface roughness and flatness can result in thermally recti-fying interfaces when the roughness or flatness is of a specific typeand the materials possess the desired relationship. An exaggeratedstrain or warping is shown in Fig. 7 which is described by [13]

dr0dq

¼ G

3

"1

r1r2ðr1 þ r2Þ2#2=3

r22dr1dq

þ r21dr2dq

�; (14)

where r1 and r2 are the radii of the two smooth spheres in contact.The load and the mechanical properties of the two spheres arecontained in the factor G. The possibility of thermal rectification,which is represented by the sign of dr0/dq, lies in the last term of theequationwhere the change of the radii of either sphere is a functionof the thermoelastic properties of the constituent materials, butwill change sign depending on the direction of the heat flux. Asshown in Fig. 7, a heat flux entering a sphere will cause the radius todecrease [14], but a heat flux in the opposite directionwill cause thesurface to flatten thereby increasing its radius of curvature.Experimental results have been difficult to predict using modelsbased on thermal strain andwarping at an interface, but with awellknown surface roughness and flatness some predictability has beenachieved for the boundary conductance and rectifying effect.

In 1955 Barzelay et al. performed measurements of the thermalcontact resistance between similar and dissimilar metals underdifferent loads [15]. Their results showed that the thermal contactresistance decreased with increased pressure with a large changeoccurring at lower pressures but a leveling off at higher pressures.Their results also showed that the boundary resistance was smaller

d

metal B

dielectric

metal A

FA

FB

Eo

− φB

φA

φ−

Fig. 8. Energy band diagram of two dissimilar metals in contact via a dielectric.

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662652

in the direction of aluminum to stainless steel. Barzelay et al. alsoshowed that the rectification increased with increasing pressure inthe ranges they studied (0e400 psi) in which the boundaryconductance from the stainless steel to the aluminum did notchange greatly with pressure while the boundary conductancefrom aluminum to stainless steel increased roughly linearly in thispressure range (see Figure 9 of ref. [15]). The authors attributedthese results to differences in warping of the interface when theheat current is reversed because of a difference in the temperaturedependence of the mechanical properties (ie. stiffness, thermalexpansion coefficient) of the materials making up the interfaceresulting in area changes. The level of rectification observed in thisstudy was approximately 0.67. The explanation of the rectificationobserved was later improved and elaborated by Wheeler [16].

In 1961 the work of Rogers argued that the effect seen by Bar-zelay et al. was a result of a thermal potential barrier and not theresult of the change in the contact resistance with the direction ofheat flow [17]. In a response to the work of Rogers, Williams [18]stated that the pressures under which Rogers conducted hisexperiments were too low to eliminate any effects of the changingcontact resistance and that his results were not sufficient todiscount a change in contact area due to the difference of materialproperties at the interface [18]. In 1962 Powell et al. followed theproposed improvement by Williams and found that with theincreased pressure loading, a specified contact geometry anda controlled film thickness the directional effect seen by Rogers waseliminated which supported the theory of Williams [19]. Powellet al. also stated that the mechanism proposed by Barzelay et al.could possibly result in rectification when the contacting surfacearea was large. The rectification in all of these works, excludingRogers, was believed to be a result of a change in contact resistancedue to a change in contact area, which was dependent on thedirection of heat flow because of thermal warping at the interface.A schematic of this effect is seen in Fig. 7.

In 1966 Clausing discovered an additional mechanism forthermal rectification due to a difference in material properties at aninterface [20]. Clausing developed a macroscopic model to explainthe rectification being observed, which he argued was a result ofthermal strain at the interface in the material with lower thermalconductivity. In his experiments and model, the interface wascomposed of aluminum and stainless steel with a spherical endwith a large radius of curvature in which the contact area couldeasily be defined. With his model rectification would only exist ifthe heat flux was large enough and the thermal conductivity of oneof the materials was low enough, otherwise there would not bethermal strain and the effect would vanish. The experiments con-ducted by Clausing showed the opposite results of previousexperiments with the same materials [15]. His measurementsshowed an increased contact conductance or a decreased contactresistance from the stainless steel to the aluminum by a factor of 3or a level of rectification of 0.5. This directional effect being theopposite of previous works was attributed to thermal strain in thestainless steel when large heat currents were used. The directionaleffect disappeared when the heat flux was reduced because thereduction of thermal strain in the stainless steel. To eliminate anyeffect of the surface roughness in these experiments, the surfaces ofeach sample were polished.

In 1967 Lewis and Perkins conducted experiments in order todetermine which effect, either the surface warping or thermalstrain, was responsible for the directional effect that was observedand why differences have been reported between Barzelay et al.[15] and Clausing [20] where the experiments were both conductedwith aluminum and stainless steel [21]. They performed experi-ments with samples that had varying levels or surface roughnessand surface flatness. They presented two models for the rectifying

effect. The first model, which wewill call the Thermal StrainModel,was based on the mechanism of thermal strain at the interfaceproposed by Clausing and the second model, which we will call theThermal Warping Model, was based on the mechanism of warpingat the interface proposed by Barzelay et al. In the cases where thesurfaces were flat and the roughness was the same level as theflatness, the results followed that of the Thermal Warping Model,which was based on warping at the interface and conductance wasgreater from the high thermal conductivity to the low thermalconductivity material (aluminum to stainless steel). In thesesamples the rectification was eliminated when the surface rough-ness was decreased by high speed buffing. The rectification effectwas then reversed when the surfaces were lapped which resultedin smoother, but rounded surfaces which followed the ThermalStrainModel. In all of these cases the boundary conductance in bothdirections increased with increasing pressure loading. The resultsof Lewis and Perkins showed that both the ThermalWarpingModeland the Thermal Strain Model can result in thermal rectification,but in opposite directions. In order to predict the level of rectifi-cation in these systems the surface conditions of the samples mustbe known. In 1976 Hudson also concluded that spherical capsurfaces can result in non-uniform isothermal surfaces leading tothermal strain at the interface in the material with lower thermalconductivity [22]. He showed theoretical evidence that this straincould result in thermal rectification. In 1987 Somers et al. devel-oped a model for thermal rectification that accounted for thechange in contact area of two spherical surfaces in contact [13]. In1991 Stevenson et al. developed a more detailed model that couldbe applied to similar and dissimilar spherical metal contacts forseveral different metals [23]. Their data matched experimentalresults reasonably well and showed thermal rectification, whichgenerally increases with loading.

In 1971 Jones et al. performed experiments on a system of 100stainless steel discs in series with an aluminum expansion element[24]. Their experiments showed thermally rectifying behavior dueto the expansion of the stainless steel discs. The authors claimedthat their system was more robust, showed a higher and morepredictable level of rectification and was more useful for lowerthermal resistance systems than the rectification systems previ-ously discussed in this section, though the authors did not justify allof these conclusions.

2.1.3. Thermal potential barrier at interfacesElectronic effects at interfaces are not only observed in Metal/

Oxide interfaces, but also they are seen inMetal/Metal interfaces. In1961 Rogers [17] argued that the rectifying effect he observed inaluminum/steel interfaces was due to a difference in the workfunctions of the twomaterials. A schematic of this effect is shown inFig. 8. Rogers found that materials with differences in the workfunction (aluminum and steel) produced a thermally rectifyingeffect. To eliminate the possibility of this being a result of a ther-moelectric phenomena Rogers also tested a similar system whichconsisted of two materials which had very different thermoelectricpotentials (Chromel and Alumel) and found no rectifying effect.

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662 653

Rogers also devised an experiment that he felt would test the val-idity of the argument of warping at the interface due to non-uniform temperature distributions over the cross-section. He didthis by replacing the aluminum with copper which results ina greater difference in thermal conductivity between the twomaterials. His theory was that a larger difference in thermalconductivity should produce greater warping (if that is the mech-anism) and therefore result in greater rectification than in thealuminum/steel system. The copper/steel system produced onlylittle rectification, below that of the aluminum/steel system whichweakens the argument for warping from non-uniform temperaturedistribution because of a difference in thermal conductivity. Thisresult could also support the hypothesis that a contact potentialbetween the two materials results in asymmetric electron trans-port. In the copper/steel system an oxide layer in the copper couldresult in the elimination of this potential and therefore thermalrectification. This potential barrier that exists is believed to beassociated with the work function of the materials in contact. Totest this hypothesis further Rogers repeated the measurementswith the aluminum and steel with a thin section of mica insertedbetween them to act as a barrier for electrons, similar to the oxidelayer assumed to exist in the copper, in both directions. Thesemeasurements resulted in no rectifying effect which supported thehypothesis that the heat transport due to phonons in this systemwas symmetric and the heat transport due to electrons in thesystem of just aluminum and steel was asymmetric due to thedifference in work function of the two materials. Their measure-ments of the aluminum/steel system showed greater transportfrom the aluminum to the steel by about 20% or a level or rectifi-cation of 3 ¼ 0.1. This work was criticized because the low pressureloading of the interface which could lead to rectifying effects due toa change in surface geometry [18] which was discussed in x2.1.2.

In 1962 Moon and Keeler developed a theoretical model basedon Rogers work which showed that a difference in material workfunctions could result in thermal rectification [25]. Their modelpredicts greater transport in the direction of decreasing workfunction (step down) as opposed to increasing work function (stepup) when a thin oxide layer is present in between the two metals(Fig. 8). Their model for rectification is given by

qe12qe21

¼ Aa12

Aa21

T12T21

T2AT2B

expEw1 � Ew2

kTA � TBTATB

�; (15)

where Aa12 and Aa21 are the interface area and usually close to beingequal, T12 and T21 are the interface temperatures and usually closeto being equal, TA and TB are the temperatures of the reservoirs anddifferent and Ew1 and Eo are the material work functions of the firstmetal and oxide, respectively [25]. For the systems studied in thiswork, which were the same as those investigated by Rogers [17],a level of rectification of 3 ¼ 0.13 was predicted which compareswell to the value measured by Rogers.

In 1970 Thomas and Probert [26] conducted a series of experi-ments between similar and dissimilar materials and observedthermal rectification. They stated that the mechanism likelyresponsible for the results in their specimen pair labeled C, whichwas composed of steel and aluminum, was likely the difference inwork function [25], though the lack of knowledge of material workfunctions of contacting surfaces was an obstacle [26]. Also in 1970O’Callaghan et al. [27] performed measurements of systemscomposed of many interfaces. Based on their measurements ofdirectional effect over a full range of pressure loading they foundthat the directional effect would increase under low loading forfreshly assembled stacks up to a certain point and then the direc-tional effect would decrease with increased loading until at somecritical value there would be no directional effect. These

experiments could be repeated after a re-cleaning process to resultin the same measurements. They also observed that if the systemwas not loaded to the critical value and the loadingwas reduced thedirectional effect would return. Based on their results the authorsargued that this directional effect in these systems was a result ofthe formation and buildup of oxide layers and the elimination ofthe directional effect was due to the compression of the oxide andthe welding of the existing contacts of the metals. The electricalrectification of these systems was also measured, but the levelsobtained with those measurements were very small therefore theauthors suggested that the thermal rectification was due to thephonon contribution to the thermal transport which is contrary toMoon and Keeler’s hypothesis [27].

This mechanism has possibly been verified experimentally andhas a strong basis in theory, but could also be affected by many ofthe parameters which affect the level of rectification seen in thegeometrical models discussed in 2.1.2 where the contact area andpressure at the interface are important. The results that have beenpresented to support this mechanism for thermal rectificationcould also support the geometric model of either strain or warpingof the interface resulting in changes of contact area.

2.1.4. Temperature dependence of thermal conductivity at interfacesIn the mechanisms discussed thus far there has always existed

an asymmetry in the system in the form of either dissimilarmaterials or a geometric asymmetry. In this section we present anadditional mechanism for thermal rectification in bulk materials.The asymmetry in this mechanism is due to the difference in thetemperature dependence of thermal conductivity between twomaterials composing an interface. For these systems to result inthermal rectification a distinct difference in the temperaturedependence of the thermal conductivity is necessary (ie. material Ahas low thermal conductivity at low temperature and high thermalconductivity at high temperature while material B is the opposite).This mechanismwas initially proposed byMarucha et al. when theyperformed experiments on inhomogeneous GaAs and founda directional dependence of the thermal conductivity [28]. Mar-ucha et al. later analyzed the system using the one-dimensionalFourier equation and found that this temperature dependence ofthe thermal conductivity would result in thermal rectification [29].The experimental and theoretical work by Marucha et al. showedpromise for thermally rectifying systems, but later work by Hudsonin 1976 showed that their results may not entirely be a result of theeffect of the thermal conductivity and may be enhanced by non-uniform isotherms over the cross-section of the specimen [22]. In1978 Jezowski and Rafalowicz performed experiments of quartzand graphite (differing temperature dependence of thermalconductivity) in series [30]. Unlike the inhomogeneous GaAssystem investigated by Marucha et al. this system possessed onlytwo sections of the differing materials instead of four. Quartz andgraphite were chosen due to the different signs of the slope of thethermal conductivity. Jezowski and Rafalowicz found a differencein heat flow of 70% or a level of rectification of 3 ¼ 0.26 when thetemperature differencewas 40 K. Greater heat flowwas observed inthe graphite to quartz direction. The graphite has a high thermalconductivity at high temperatures and a lower thermal conduc-tivity at low temperatures while the quartz has a high thermalconductivity at low temperatures and a lower thermal conductivityat higher temperatures. Also in 1978 Balcerek and Tyc performedexperiments and theoretical studies of tin and a-brass systems [31].Their experimental results agreed extremely well with the theo-retical prediction based on Fourier’s law, which showed that therewas no effect due to the interface geometry. Their results alsoshowed that larger thermal rectification was possible and wasdependent on the properties of the materials composing the

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662654

interface. In this case greater transport was observed from a-brassto tin. The rectifying effect increased with temperature differencewhere the maximum observed rectification was around 0.4 ata temperature difference of 9 K. The theoretical calculationsshowed a rectifying effect of 0.59 at a temperature difference of40 K. In 1984 Kokshenev et al. developed a model for thermalrectification in solidesolid systems based on the AcousticMismatch Model [32]. Their model showed that when theboundary or interface resistance was small rectification could existin systems in which the materials exhibited different temperaturedependent thermal conductivity, specifically different signs of theslope of the thermal conductivity with respect to temperature.Kokshenev et al. found good qualitative agreement with the workof Jezowiski and Rafalowicz [30] and Balcerek and Tyc [31].

In 1985 Hoff showed theoretical results that a system witha temperature and spatial dependent thermal conductivity like aninhomogeneous material was a necessary condition for thermalrectification based on this mechanism [33]. In 1993 Hoff verified hisown work through experiments and found excellent agreement ina system composed of aluminum and brass where transport wasgreater in the direction of brass to aluminum [34].

More recently there have been some more rigorous models forthe prediction of thermal rectification in systems with materialsthat have different temperature dependence of thermal conduc-tivity. In 2001 Sun et al. developed a model which suggested thatwhen the thermal conductivity of two contacting solids is differentthermal rectification will occur [35]. Previous models have sug-gested a difference in temperature dependence is needed, but donot show a difference in the value will result in thermal rectifica-tion. Their model is based on Fourier’s law and the interfacetemperature. The model presented is flawed however because itassumes the system has an overall effective thermal conductivitywhile the interface is calculated assuming each component of thematerial has it’s own thermal conductivity. Assuming the materialsthey use maintain a constant thermal conductivity thermal recti-fication will not occur. In 2006 Hu et al. proposed a design fora macroscale thermal rectifier based on the same mechanism thathas been discussed in this section [36]. They use non-equilibriummolecular dynamics (NEMD) simulations to demonstrate therectifying behavior of a system of quartz and diamond. In 2009Dames presented a single algebraic equation for thermal rectifica-tion in systems of dissimilar materials with different thermalconductivities [37]. His model made the assumption that thematerial thermal conductivities followed the power law in order tomaintain it’s simplicity. This work investigated many combinationsof materials which had opposite signs of the power (ranging from�3.5 to þ5.4). The power law matched experimental measure-ments very well in the temperature ranges used in this work. In thelow-bias limit the level of rectification was shown to be propor-tional to the dimensionless thermal bias DT/To, the difference inpower law exponents n1 � n2 and a simple function of the thermalresistance ratio r. The expression for rectification presented byDames is given by

g ¼ ðn1 � n2Þ�r1=2 þ r�1=2

�2D; (16)

where D ¼ 2DT/(TH þ TC), r ¼ R0,2/R0,1 and R0,i is the thermalresistance of leg i at uniform temperature T0. Beyond the low-biasregion (D > 0.1) a second order theory showed to be extremelyclose to the full theory that also fit experimental data from Jezowskiand Rafalowicz [30] very well. This model also showed goodagreement with Balcerek and Tyc [31]. The work of Dames not onlyhighlights the ability of this mechanism to result in thermal recti-fication, but also points out a limitation. In order to achieve larger

thermal rectification (greater than 3 ¼ 0.5) larger temperaturegradients are required (of the order of the average temperature).Later in 2009 Kobayashi et al. experimentally demonstratedthermal rectification in a system of two materials with differenttemperature dependent thermal conductivities which fit well witha theoretical model based on Fourier’s law [38]. Their results alsoshowed that conduction was greater in the direction of high to lowthermal conductivity.

In 2010 Go and Sen stated that a necessary condition for thermalrectification in a material or a structure is that the thermalconductivity must be a function of both temperature and space andthis function must be non-separable [39]. Their results showed thatrectification did not always occur when the thermal conductivitywas a function of temperature and space and non-separable, butthat this property of the thermal conductivity was required for thepossibility for the material or system to exhibit thermal rectifica-tion. This property of thermal rectification can potentially be ach-ieved in a single material, but this work is mentioned here becauseit can more thought of more easily as a composite system.

2.2. Molecular mechanisms

The previous section discussed the major mechanisms forthermal rectification in bulk materials. Recently, as was shown inFig. 1, there has been a large increase in interest in thermal recti-fication in nanostructured materials. There have been very fewexperimental reports of thermal rectification in nanostructures, butthere has been a large increase in theoretical studies recently thatshow promising results for future thermally rectifying devicestaking advantage of nano and atomic scale asymmetries. The lack ofexperimental results of thermal rectification in nanostructuredsystems is a result of our lack of understanding about thermallyrectifying mechanisms as well as our inability to fabricate devicesor structures with the tolerances necessary to produce rectificationat this time. Increased experimental investigations of nano-structure thermal rectifiers are expected as our abilities areimproved in both of these areas.

2.2.1. Non-uniform mass loadingIn 2006 Chang et al. observed thermal rectification in the

measurements of non-uniformly mass-loaded carbon and boronnitride nanotubes [5]. The nanotubes were non-uniformly loadedexternally with Trimethyl-cyclopentadienyl platinum (C9H16Pt)along the length of the tube and the thermal conductivity wasmeasured along each direction using the system designed by Shiet al. [40]. The system resulted in a level of rectification of 3 ¼ 0.01in the carbon nanotube and a maximum of 3 ¼ 0.034 in the boronnitride nanotube. In both of these cases the greater thermalconductivity was observed in the direction of high-mass to low-mass. The authors concluded that the effect was not a result ofphonon transport or due to an impedance mismatch at thecontacts, but actually due to presence of solitons which are non-perturbative solutions of non-linear systems. They result in recti-fication because they are particle-like entities that can collide witheach other without changing shape giving rise to asymmetric heatflow in an inhomogeneous material [5]. However, rigorous analysishas not been performed an additional finding in this work was thatthe rectification was not only a result of decreased thermal trans-port in one direction, but also an increase in the higher conductivitydirection. An alternative hypothesis was suggested by Otey et al.[41] who pointed to surface polaritons as a likely candidate.Although they computed a maximum rectification effect of R ¼ 0.4,the never defined what R is so we have not included these data inTable 2.

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In nearly all of the other mechanisms of thermal rectificationthat have been discussed now and in later sections the rectifyingeffect is due entirely to the reduction of thermal conductivity in onedirection and no increase in the other direction.

In 2009 Alaghemandi et al. performed non-equilibrium molec-ular dynamics simulations of carbon nanotubes where an asym-metry was present due to a non-uniform mass distribution [42]. Inthe first case the mass of the carbon atoms was varied along thelength resulting in a mass or density gradient along the tube. In thesecond case additional carbon atoms were added to the exterior ofthe tube also adding mass to one end of the tube. In this work theMuller-Plathe method [43] was used for the calculation of thermalconductivity in both directions. This method restricts kineticenergy exchange to atoms with the same mass therefore the hotand cold baths were of the same mass, which added an additionalinterface between the heavy and light carbon atoms in the non-uniform mass distribution case. This restriction did not impact theexternally loaded cases, but the added atoms were also of carbonunlike the work of Chang et al. [5]. This work produced results ofthe same magnitude as that of Ref. [5], but with the oppositepreferred direction of transport. The non-equilibrium moleculardynamics simulations showed that transport was greater in thedirection of low-mass to high-mass with an effective rectificationbetween 0.015 and 0.074 depending on temperature and the massgradient. In 2010 Alaghemandi et al. performed additional non-equilibrium simulations of non-uniform mass distributed carbonnanotubes and found the same result as their previous work [44]. Inthis work they found that the addition of anharmonic effectsperpendicular to the longitudinal phonon modes acted to increasethe contribution of the longitudinal modes to the vibrationalenergy and transport which does not occur in symmetric systems ofthe 1D systems that will be discussed in x2.2.4.

2.2.2. Asymmetric nanostructured geometryThe recent work involving carbon and boron nitride nanotubes

have led researchers to look theoretically at other nanomaterials forpossible rectifying behavior. Materials that have become extremelypopular in addition to the carbon nanotube include asymmetricgraphene sheets and carbon nanocones or nanohorns formed fromgraphene sheets [45e49].

In 2008 Roberts and Walker performed a theoretical study ofnanowires with an asymmetric boundary roughness using theMonte Carlo method for phonon transport [45,50]. This work wasmotivated by the Monte Carlo work performed by Saha et al. andMoore et al., which investigated phonon transport in nanowiresthat had a sawtooth boundary roughness resulting in phononbackscattering [51,52]. Roberts and Walker found that deviceswhich had an atomically smooth boundary in one direction anda rough boundary in the opposite were thermally self-biasing andrectifying. To achieve a device like this an asymmetric sawtoothgeometry was assumed where phonons traveling in one directionobserved a rough boundary while phonons traveling in the oppo-site would observe a smooth boundary. The phonons scatteringwith the boundary in the rough direction would reflect diffuselywhile phonons scattering with the boundary in the smooth direc-tion would reflect specularly. This difference in phonon boundaryscattering resulted in a collection of phonons at the boundary inwhich emitted phonons would travel in the rough direction. Whenthermalizing boundaries were included in this work high levels ofrectification would result at low temperatures, but as the temper-ature of the device increased multi-phonon scattering wouldeliminate the observed rectification effect. In a similar study Milleret al. performed ray-tracing simulations [53] and later used theLandauer-Buttiker formalism constrained by the 2nd Law of Ther-modynamics along with Monte Carlo simulations [54] to

investigate thermal rectification in devices with nanostructuredasymmetric inclusions and voids. These voids were of pyramidalgeometry. They found rectification levels of around 0.43 due to theasymmetric phonon scattering at the interface of the host materialand the voids. The level of rectification in this work increased withthe applied temperature bias, though extremely large thermal biaswas necessary for rectification levels above 0.025.

In 2009 Yang et al. performed molecular dynamics simulationsof asymmetric graphene ribbons and found a rectifying effect ofaround 0.64 [48]. Their simulations involved graphene sheets withtwo different angular asymmetries. The first of these asymmetrieswas a triangular shape where the graphene sheet had a triangularshape and the width of the sheet decreased linearly along thelength of the sheet. The second asymmetry was a rectangular shapewhere the graphene sheet maintained a different, but uniformwidth on each half of the sheet. Their results showed rectification inboth systems with the triangular system having a greater rectifyingeffect. In both cases transport was greater from the larger section tothe smaller section. In the case of the triangular geometry therectifying effect increased with the angle of width reduction anddecreased with the length of the device even when the angle washeld constant. The change in the level of rectification as a functionof the length of the device appears to be a result of phononconfinement because at lengths greater than about 10 nm the levelof rectification remained at approximately 0.33 (a doubling ofthermal conductivity). At smaller device lengths a change inrectification is occurring because increased length allows foradditional longer wavelength phonon modes to contribute to thethermal transport. In 2009 Hu et al. also performed moleculardynamics simulations of graphene nanoribbons of single anddouble layer triangular and trapezoidal geometry [49,55]. They alsoincluded verification simulations of rectangular geometries andasymmetric void geometries. Their results showed that the recti-fication was maximized in the single perfect triangle simulations.The rectification effect and thermal conductivity were reduced atshort device lengths in the single layer triangle with the addition ofa roughness at the boundary which resulted in diffuse phononscattering at the boundary. At longer device lengths the rectifica-tion of the single perfect interfacewas reduced drastically while thesingle rough triangle remained essentially constant. The conduc-tivity of the double layer triangle was greater in both directions,which resulted in the reduced rectification even though thedifference in the conductivities remained the same as in the singlelayer case because of how the level of rectification is defined inequation (4). Rectification still occurred in the trapezoidal case, butwas much lower than that of either of the triangular cases due toa reduced level of asymmetry in the system. No rectification wasobserved in any of the symmetric geometries. In these systems therectifying effect is a result of the constant change in the phononspectra along the length of the ribbon in the triangular and thetrapezoidal cases and an abrupt change in the phonon spectra inthe rectangular ribbons. An interesting point to note about therectification in the graphene nanoribbons is that the rectification isnot necessarily limited by the length of the ribbon as with inter-faces which will be discussed in x2.2.3 and x2.2.4.

In 2007 Wu and Li performed molecular dynamics simulationsof carbon nanotube intramolecular junctions where two nanotubesof differing diameters were joined at an interface in the center ofthe system [56]. In these systems there were defects present at theinterface between the two nanotubes. By looking at the powerspectra on each side of the interface when the temperaturegradient was in both directions the authors noticed a change in theoverlap of the power spectra. This result is similar to what has beenreported in theoretical studies of 1D non-linear lattices which willbe discussed in x2.2.4. By looking at the overlap of the phonon

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spectra in the work by Wu and Li we see that the largest changesoccur in the optical phonon modes with low wave vector. Shiomiand Maruyama showed that optical modes with low wave vectorwere shown to contribute significantly to non-Fourier heatconductionwhen the length of the tubes was finite [57]. Shiomi andMaruyama also presented the phonon dispersion for the carbonnanotube superlattice structures which showed the phonon groupvelocity for the optical branches had comparable group velocities tothe acoustic branches. Wu and Li showed that the phonons in thefrequency range of 400e1000 cm�1 (optical phonons) had a muchgreater contribution to transport and thermal rectification becausethey had relatively large group velocities and a directional depen-dence. The phonons above that frequency range had much smallergroup velocities reducing their contribution to transport and thephonons below that frequency range did not exhibit a directionaldependence. The lower frequency phonons are not changed in thissystem because of the relative small size of the connecting regionthat was used in the simulation when compared to their wave-length. With this knowledge of how the interface results in recti-fication the authors performed additional simulations to optimizethe rectifying effect. Through their optimization simulations theyfound that when the tube radius increases to a sufficiently largevalue the vibrational density of states approaches that of a gra-phene sheet (curvature effects are becoming less obvious) thereforewhen the diameter is large in both of the tubes there is littledifference between the phonon spectra resulting in little rectifica-tion. At small tube diameters the thermal conductivity becomeslarge and the relative difference between the two can be large, butwhen normalized by them the percent change becomes smaller.The authors in this study found an optimized rectification effectwhen the tubes had a diameter ratio of 2 with non-axial alignment.They also investigated the effect of defects at the interface andfound the maximum rectification occurred when a minimum ofdefects existed at the interface (cannot have 0 defects because ofthe change in diameter). Since the authors only found a differencein conductivity of around 20% between the two directions, or3 ¼ 0.1, they performed simulations where the tubes were strainedwith tensile and torsional stresses. They found that torsional loadshad no impact on the rectifying effect while tensile loads resulted inan enhancement of the thermal rectification up to approximately35% or 3 ¼ 0.15. The tensile stress results in an elongation of thetube, which reduces the potential energy between atoms andtherefore a reduced thermal conductivity in both directions. Again,based on how the level of rectification is defined, with the samedifference in directional thermal conductivity, a greater level ofrectification will result in a system with lower directional thermalconductivities. In all the cases reported the system exhibitedgreater transport in the direction of larger to smaller tube diameter.

In 2008 Wu and Li also performed molecular dynamics simu-lations of deformed carbon nanohorns which are carbon nanotubeswhich have a varying diameter along the length of the tube [46].Carbon nanohorns or nanocones are formed by taking a graphenesheet, cutting a slice of a specified angle out of it and connecting theends of the removed section which will form a cone or a horn. Thiswork by Wu and Li investigated the angle of the nanohorn, thetensile load applied to the nanohorn along the axis of transport andthe temperature gradient applied to the nanohorn. In all cases thenanohorn exhibited a preferred direction of transport in thedirection of decreasing radius (and also mass) which was similar towhat has been seen in other nanotube and graphene studies[47,49,55,56]. Under no tensile loading the rectificationwas greaterwith a larger horn angle when the length of the horns was roughlyequivalent. One device was simulated with the larger angle, butonly half the length which resulted in a lower level of rectification.Under tensile loading the rectification effect is initially increased

due to the decreased force between atoms and the resultingdecrease in thermal conductivity, but once the tensile loadingreaches a critical value which is dependent on the angle and thelength of the horn the effect will be reduced. This change in recti-fication can be explained in a similar way as in Ref. [56] where itwas shown that a difference in phonon spectra is the mechanismfor thermal rectification. In this work the end of the horn witha large radius resembles a graphene sheet while the end of the hornwith a small radius resembles a small carbon nanotube inwhich thephonon spectra of the two ends becomes very different. As thetensile load is increased the thermal conductivity is decreasedresulting in an increase in rectifying effect. If the tensile stress isincreased the difference in phonon spectra begins to be reduced asthe angle of the horn is reduced resulting in a reduced rectifyingeffect. Wu and Li also showed that the rectifying effect was alsoincreasedwith increasing temperature gradient applied. This is alsoexplained by the difference in phonon spectra at the two endsbecause of the temperature dependence of the phonon spectra.When the applied temperature gradient becomes very large (on theorder of the system average temperature) a phenomenon callednegative differential thermal resistance is observed in which theresulting net heat flux is reduced [58]. This effect results ina reduced thermal rectifying effect. This effect is extremely inter-esting because it is essential for thermal transistors which arementioned, but not discussed here [59]. The maximum thermalrectification reported by Wu and Li in the work involving nano-horns was approximately 0.55 with the largest horn angle at a hightemperature gradient (DT ¼ 0.8T0) and under no tensile loading.

In 2008 Yang et al. performed molecular dynamics simulationsof carbon nanocones to investigate their thermally rectifying effectsas a function of the applied temperature gradient, average systemtemperature and the length of the nanocone [47]. In this worka fixed cone anglewas usedwhichwas greater than that of Ref. [46].The rectifying effect was enhanced with increasing temperaturegradient with a maximum rectification of about 0.44 undera temperature gradient normalized by the average systemtemperature of 0.7 at 300 K. This work did not report results withtemperature gradients greater than 0.8; therefore no negativedifferential thermal resistance was reported. The rectifying effectwas shown to decrease with increasing temperature gradient dueto the author’s definition of the level of rectification. While thedifference in resulting heat flux was effectively the same for thetemperature range studied, the heat fluxof each direction increasedwith increasing temperature which resulted in a reduction of therectifying effect. Neither the rectifying effect nor the heat fluxchanged greatly with varying cone length. The authors attributedthis to ballistic energy transport in the system in which the trans-port is not dependent upon length. Since transport is ballistic in thenanocone and nanohorne unlike systems which contain interfacese the nanocone and nanohorn show the desirable characteristics ofthermal rectifiers since longer device lengths do not eliminate therectifying effect.

In 2009 Noya et al. performed heat pulse molecular dynamicssimulations of Y junction carbon nanotubes [60]. These Y junctionnanotubes are single nanotubes where one end has a specificdiameter and at the other end has two nanotubes which havea smaller diameter. These Y junction nanotubes have already shownelectrical current rectification both experimentally [61,62] andtheoretically [63e65]. In the heat pulse simulations atoms near theend of one side of the nanotube is heated to 800 K during a 0.1 psfollowed by 0.8 ps of free equilibration of the system and finishedwith a 0.1 ps quenching of the heated atoms. The heat pulse is thenmonitored throughout the system for 4 ps in addition. Their resultsshowed asymmetric thermal transport that was dependent on thelattice structure of the nanotubes. No rectification was observed in

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the (10,10) to (5,5)X(5,5) system, but transport was preferred fromthe single tube (stem) to the two tubes (branches) in the (14,0) to(7,0)X(7,0).

In 2010 Alaghemandi et al. performed molecular dynamicssimulations of asymmetric systems composed of carbon nanotubesconnected by a single graphene layer normal to the axial directionof the carbon nanotubes [66]. The asymmetry of the system camefrom having different numbers of carbon nanotubes on either sideof the graphene layer. Their results showed that the 1D systemscould not be mapped to larger, more realistic systems. Their resultsalso showed a preferred direction of transport from the highermassside to the lower mass side. Similarly in 2010 Varshney et al. per-formed molecular dynamics simulations of pillared-graphene-CNTnanostructures and mention the possibility of using structures ofthis architecture as improved thermally rectifying devices [67].

Another asymmetric nanostructure that has shown thermalrectification is a CNT with geometric variations of doped nitrogenintroduced by Chein et al. [68]. They showed thermal rectificationfrom molecular dynamics simulations of CNTs with an asymmetricdistribution of doped nitrogen along the axial direction of the tube.Rectification only occurred when both a geometric and massdistribution asymmetry existed within the system. The preferreddirection of transport in these systems was in the direction ofincreasing atomic density of nitrogen and increasing geometric sizeof the doped nitrogen.

2.2.3. Nanostructured interfacesThermal transport at solidesolid interfaces has been a topic of

great interest because of the necessary understanding in designingsmaller and smaller electronics devices because of the increasingeffect of the interfaces [69,70]. In 1959 Little developed the firstmodel for predicting the thermal transport at a solidesolid inter-face at low temperatures called the Acoustic Mismatch Model(AMM) [71]. The AMM is only valid for perfect interfaces attemperatures below 7 K and the transmission is based on theacoustic impedance of the two materials. In 1989 Swartz and Pohlextended the temperature range for which transport between twodissimilar solids could be predicted with the Diffuse MismatchModel (DMM) [72]. The DMM transmission is based on the ratio ofthe materials’ density of states and the phonon group velocity. Aspecial case of the DMM known as the Phonon Radiation Limit(PRL) is the case in which the transmission probability is unityresulting in perfect transmission at the interface. The PRL isassumed to be the maximum boundary conductance at an interfacewhen inelastic scattering is not present. Since then a few additionalmodels that take into account inelastic scattering have beendeveloped and will be discussed later.

Phonon transport at nanoscale interfaces in thin films or nano-wires has also become interesting for thermal rectification.Materialsthat have different phonon spectra can exhibit phonon transmissionin one direction and reflection in the opposite due to the availabilityof phonon frequencies in thematerials. This effect has also been seenin one-dimensional systems and asymmetric carbon nanotubes andgraphene sheets as well (see x2.2.4). Although a rectifying effect hasbeen seen at interfaces in bulk systems where the temperaturedependence of thermal conductivity of the twomaterials is different,the spectral effect is different. In the works reported here theresearchers minimized the impact of rectification due to differenttemperature dependencies of thermal conductivity by usingsmall temperature gradients in comparison to the average systemtemperature and also using materials with similar temperaturedependent thermal conductivities in the temperature rangesinvestigated.

In 2008 Roberts and Walker performed molecular dynamicssimulations of planar, diffuse and asymmetric interfaces between

two dissimilar materials and showed the possibility of thermalrectification [73]. Their results showed a nearly 30% difference intransport, or 3 ¼ 0.13, which favored the argon to krypton directionover the krypton to argon direction at 25 K in the perfect planarinterface system. The rough or diffuse interfaces resulted ina reduced rectification effect as a result of increased inelasticscattering at the interface. The asymmetric boundaries in this workalso resulted in a reduction of the rectifying effect as a result ofinelastic scattering. The authors still believe the asymmetricboundaries can enhance the rectifying effect, but in the simulationsthe finite size of the asymmetric shape resulted in a roughnesssimilar to the rough or diffuse interfaces. In 2010 Roberts andWalker continued their investigation of thermal rectification atinterfaces, which involved simulations of longer devices and ata range of temperatures [74]. The authors found that inelasticscattering at the interface was important for thermal rectification.At low temperatures and perfectly planar interfaces no inelasticscattering and a small level of rectification was shown witha preferred direction of transport being from the lower frequencymaterial to the higher frequency material because the limitedcontribution of phonon modes in the higher frequency material. Athigher temperatures or slightly diffuse interfaces some inelasticscattering occurred allowing additional modes in the higherfrequencymaterial to contribute to the overall transport and resultsin higher levels of rectification with the preferred direction oftransport now being from the higher frequency material to thelower frequency material. At even higher temperatures or verydiffuse interfaces no rectification was observed because of highlevels of inelastic scattering. Their results showed that inelasticscattering can help to produce thermal rectification though highlevels will eliminate the rectifying effect.

In thermal transport at interfaces, inelastic scattering allows forthermal rectification by allowing phonons in the higher frequencymaterial at frequencies that are not present in the lower frequencymaterial by scattering them into multiple lower frequencyphonons, allowing them to transmit. Inelastic scattering at inter-faces is shown to increase with temperature and imperfections (ie.diffused atoms) at interfaces. The thermal interface conductance/resistance models mentioned previously (AMM and DMM) do notconsider inelastic scattering, but they assume no anharmonicinteractions and only consider the lower frequencymaterial in theirformulation. These assumptions eliminate the possibility of pre-dicting thermal rectification in solidesolid interfaces betweendissimilar materials. In 2008 Hopkins and Norris developed theJoint Frequency Diffuse Mismatch Model (JFDMM) which partiallyconsidered the effects of inelastic scattering by assuming that theinterface possesses joint frequency modes which are weightedfractions of each of the two materials creating the interface whichwould allow frequencies above the maximum frequency in thelower frequency material to be transmitted into that material.Similar to the PRL, the JFDMM has a special case in which allphonons transmit which is called the Inelastic Phonon RadiationLimit (IPRL) in which all phonons regardless of their frequencytransmit at the interface which is the upper limit for thermalboundary conductance possible. Though these models have beenshown to predict well known interfaces somewhat accurately, theyare not able to predict thermal rectification due to the assumptionof no anharmonic or non-linear interaction [75]. To develop a betterunderstanding of the rectifying effect observed in the non-equi-librium molecular dynamics simulations Roberts and Walker per-formed phononwave-packet simulations [76] of perfect planar anddiffuse interfaces in which a single frequency wave-packet isformed by displacing that atoms at one end of the system andallowing it to propagate through the device, interact with theinterface and propagate to either end of the device while observing

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the wave-packet [77]. Roberts and Walker observed asymmetricphonon transmission probabilities in the argon and krypton systemand observed increased inelastic scattering at the interface with anincrease in the diffusion region. From the transmission probabilitiesobtained from the wave-packet simulations the conductance wasthen calculated and a temperature and interface roughnessdependence was observed for thermal rectification.

In 2008 Hu et al. performed molecular dynamics simulations ofsilicon-amorphous polyethylene interfaces and observed thermalrectification due to a mismatch of phonon density of states [78].They found that a mismatch of phonon vibrational properties of thematerials forming the interface resulted in asymmetric phonontransport across the interface. This mismatch resulted in a direc-tional dependence on the thermal boundary resistance. In all oftheir simulations they found greater transport in the direction ofthe amorphous polyethylene to silicon with a maximum differenceoccurring at the largest applied heat flux giving a level of rectifi-cation of 0.18. The rectifying effect is increased with heat flux in thiscase because the increased heat flux results in a greater differencebetween the vibrational properties of the two materials in thereverse direction while maintaining similar vibrational propertiesin the forward direction. In 2009 Hu et al. performed moremolecular dynamics simulations of water-functionalized silicainterfaces and found a rectifying effect of 0.21 [79]. This work isreported here because of the level of rectification and the mecha-nism of rectification, but is outside the scope of this paper as this isa liquidesolid interface.

2.2.4. Anharmonic/non-linear lattices in 1D chainsIn 2002 Terraneo et al. presented a model for a thermal rectifier

using a non-linear 1D lattice connecting two thermal reservoirs atdifferent temperatures [4]. The lattice is a chain of N particles witha harmonic coupling and a Morse on-site potential. The Morsepotential is a highly anharmonic soft potential in which thefrequency decreases drastically with increasing amplitude. Usingidealized materials their theoretical study showed that they couldcontrol the heat flow from insulator to a good conductor just byvarying the Morse potential constant from 0.5 to 1.2. By taking thislattice and inserting it between two weakly anharmonic regionsa thermal rectifier can be devised. Terraneo et al. showed thata doubling of conductivity could be achieved from the parametersinvestigated in their theoretical study of the thermal rectifier. In2005 Casati performed a similar study where he found that if theseparation of the two phonon bands was larger than twice theharmonic constant in the middle region there would be no trans-port [80].

In 2004 Li et al. performed a theoretical study similar to that ofTerreano et al. except in their work they used two non-linearsegments coupled together by a constant harmonic spring usingthe Frenkel-Kontorova (FK) model [81]. This work again involvedtwo idealized materials in series which were optimized to producelarge levels of rectification by having phonon spectra overlap in onedirection of heat current and no phonon spectra overlap in theopposite. In their system they choose material A so that it wouldonly exhibit a high frequency spectra at low temperatures anda broad spectra high temperatures while material B had only a lowfrequency spectra at low temperatures and a broader, but still lowfrequency spectra at high temperatures. Their results showeda conduction direction that was 100 times as large as the oppositeinsulating direction. When a temperature gradient was appliedfrom material A to material B good conducting behavior wasobserved because of the large phonon spectra overlap. When thegradient was reversed (material B to material A) there was almostno phonon spectra overlap because of the narrow range and highfrequency spectra in material A which resulted in insulating

behavior. For a thermal rectifier to be realized experimentally usingthis mechanism, materials must be selected that possess similarproperties though it is unlikely to findmaterials whichwould resultin levels of rectification near that of the idealized system. In 2005Casati proposed that a prototype of this device could be a disor-dered harmonic lattice. In 2005 Hu and Yangmodified the design ofLi et al. [81] with the FK model and showed a change in conduc-tivity of 10,000 times [82]. This modification was that the two FKsections had the same interparticle potential, but different externalpotentials. Their results also showed a reduced rectifying effectwith increasing device length.

In 2005 Segal and Nitzan presented an analytical study witha simplified model from that of Terraneo et al. which showed thata nonlinearity and a structural asymmetry results in thermalrectification [83]. Their model was a generalized spin-bosonmodel where the non-linear lattice is coupled to two differentthermal baths at different temperatures with different couplingstrengths. An additional model presented by Segal and Nitzan thatimplied nonseperable coupling also resulted in thermal rectifica-tion. The authors stated that asymmetric coupling at the thermalbaths could be originated from chemical bonding, by usingreservoirs with different Debye temperatures or from differentspatial organization of vibrational states if one were to designa thermal rectifier for an experiment. Segal and Nitzan extendedthis work by highlighting the features of these two models andpresented results of rectification in a system of identical anhar-monic oscillators connecting asymmetrically two heat baths atdifferent temperatures using classical Langevin dynamics andinternal asymmetry of anharmonic molecular models [84]. In theoriginal model (the generalized spin-boson model where the non-linear lattice is coupled to two different thermal baths at differenttemperatures with different coupling strengths) the authorsreported that (a) in the deep quantum limit rectification decreasesexponentially with frequency and in the inverse temperatures, (b)in the highly classical limit the asymmetry increases linearly withthe square of the temperature difference and frequency, (c) in theintermediate regime rectification decreases with frequency and(d) the current is larger when the bridge links more strongly tothe colder reservoir than when it links more strongly to the hotterone. The maximum rectification observed with this model was0.05. The major difference observed in the second model pre-sented (nonseperable coupling) was that strong dependence onthe coupling strength was observed. The maximum rectificationobserved with this model was 0.27. The study of the anharmonicsystems showed that rectification through asymmetric chains wasa general property and the level of rectification increased with anincrease in anharmonicity. The rectification was also shown toincrease with system length in the range investigated in thisstudy. The maximum rectification observed with this model was0.04. In the internal asymmetry of anharmonic molecular modelswork showed rectification levels up to nearly 0.15. In all of thiswork rectification was observed in asymmetric systems althoughthe conductivity in either direction remained low which is notideal for most applications that could be exploited with a thermalrectifier.

In 2006 Peyrard proposed a slightly different model fora thermal rectifier where he proposed a system composed of tworegions which exhibited different temperature dependent thermalconductivities (like the bulk mechanism presented in x2.1.4) [85].To create a region with this temperature dependence Peyrard usea method similar to Terraneo et al. [4] for a composite material ofalternating layers which exhibited a low thermal conductivity atlow temperature and a high thermal conductivity at high temper-ature. The changing of thermal conductivity of the compositematerial was due to the overlap (high thermal conductivity) or

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separation (low thermal conductivity) of the phonon spectra of thedifferent layers.

In 2006 Hu et al. performed a theoretical study of a 1D latticeusing the FK model and found that the direction of greater trans-port in a thermal rectifier composed of two FK segments wasdependent on the properties of the interface and the system size[86]. As the harmonic spring constant coupling between the two FKchains becomes larger, their results agree with those from Li et al.[81], but when it is small the phonon band overlap theory no longerapplies due to a mixing of phonon bands because of interactionsbetween phonons of the two sections induced by the nonlinearityin the chain. The role of the harmonic spring constant here issimilar to what would be observed with an impurity in the chain,which could drastically impact the level and the direction ofrectification. This is an interesting finding which is similar toa problem at the bulk level when the surface characteristics of twomaterial samples forming an interface varied [21]. Hu et al. alsoshowed that length plays a role in the rectification level anddirection as well. As the length of the system increases, the impactof the harmonic spring constant (treated as an impurity) becomesless significant and the results are what is expected from thephonon band overlap theory. Inweakly coupled systems the resultsshould agree with that predicted by phonon band overlap theory,but when the segments are well coupled and the device is suffi-ciently long, phonon band mixing will become important andopposite to the prediction from phonon band overlap theory.

In 2007 Lan et al. performed a theoretical study of thermalrectification in a 1D lattice composed of two segments, one an FKmodel and the other a Fermi-Pasta-Ulam (FPU) model [75]. Resultsshowed that a difference in transport of around 2000 times wasobservedwith the preferred direction being the FK to FPU direction.

In 2008 Zeng and Wang investigated the influence of phononfrequency, asymmetry and non-linear interactions relating tothermal rectification [87]. They studied the impacts of each witha 1D chain of atoms. Their results showed a difference in thetransmission probability in one direction and not the oppositewhen comparing a constant mass system to one in which the leftand right boundaries differed considerably in mass. Additionallythe authors investigate asymmetric thermal transport at an inter-face. This is similar to what was discussed earlier in the nano-structured interface section, but is located here because of thedimensionality of the system. The authors argue that non-linearinteractions at the interface allow higher frequency phonons(phonons with frequency higher than the maximum frequency ofthe lower frequency material) to contribute to transport andwithout this contribution no rectification could occur. These non-linear interactions result in inelastic scattering which, as wediscussed previously, has a contributory effect on thermal transportat interfaces due to the contribution of additional phononfrequencies.

In 2008 Li et al. performed simulations of an asymmetric latticecomposed of one segment of an FK non-linear lattice and onesegment of a harmonic lattice of equal lengths weakly coupled atthe center without a constant thermal bias [58]. Instead ofa constant thermal bias the bath temperatures are varied over time.By varying the bath temperatures a ratcheting effect is shown tooccur which can reverse the heat flux, cause heat to flow from coldto hot against an average thermal bias and turn a regime witha negative differential thermal resistance into one with a positivedifferential thermal resistance. The ratcheting system was able toproduce a non-uniform temperature distribution with no net fluxapplied which is a self-biasing device. The device did not exhibitthis when constant bath temperatures were used.

In 2009 Hopkins and Serrano performed a non-equilibriumGreen’s function study of asymmetric harmonic chains [88]. The

asymmetry in this work comes from a doubling of the mass(doubling of the number of silicon atoms) at the right bath of thesystem which results in a different phonon distribution from theleft bath. In small systems no rectification and no difference inthe phonon spectra are observed from the mass loading of one ofthe baths. By adding an impurity in a location other than the centerof the lattice rectification can be produced. This occurs by makingthe lattice inhomogeneous and acts to manipulate the FabryePerotphonon oscillations. The rectification here requires both an asym-metry at the contacts or baths as well as the impurity asymmetry inthe lattice. The maximum rectification reported was about 0.29 ina device that was only 10 atoms long. Rectification in a system likethis would likely not be observed in longer devices because theeffects of the asymmetric impurity would become negligible(similar to the effect seen in interface rectification).

In 2010 Pereira performed an analytical study of a quantummass graded system [89]. His results showed that thermal rectifi-cation occurs in an inhomogeneous quantum model if a phononscattering mechanism or effective anharmonicity is present. Thiswork does not exclude the rectifying effect that is observed in realanharmonic potentials through numerical studies, but do state thatthese graded mass systems could be constructed and could beexperimentally reliable.

These 1D systems have shown incredible rectification propertiesin simulations, but realizing these devices experimentally will bechallenging and will result in a greatly decreased rectifying effectfromwhat is seen from simulations. Fabrication of a 1D chain withmaterials that exhibit the exact properties is not possible and evenif it were, the systems would need to be scaled up, which has beenshown to decrease the rectification in many of the studies. Thiswork is still important at this time because it allows us to peak atthe possibilities that may become available down the road as weimprove our ability to manipulate and synthesize materials at theatomic level. My proposal for the researchers working in this area isnot only to continue to study the idealized systems, but also followthe lead of Hopkins and Serrano [88] where they used silicon as thematerial composing the lattice and used impurities and massloading to construct a 1D thermal rectifier. Dual path studies of thisnature will inspire researchers to think and study the possibilitiesand push the limits of what we can do while the grounded studiesof current capabilities allow us to improve and possible discovereffects that could have large impacts in the near future.

2.2.5. Quantum thermal systemsModels for thermal rectifiers as well as thermal circuits which

exhibit rectification have also been studied. In this section we willdiscuss some of the quantum thermal rectifiers and systems likeswitches that have recently been explored. These rectifiers aredifferent fromwhat we have discussed to this point. Some of thesedevices require external control (similar to gate control) for opera-tion which gives rise to thermal circuits and other devices like that.

In 2006 Eckmann and Mejia-Monasterio [90] showed thermalrectification in a billiardlike system where two differently sizeddisks in a left and right cell of a device are used to drive the particlesinto collisions at the junction between the two cells. In this systemthe transport in the preferred direction was shown to be up to 3times that of the lower transport direction (3 ¼ 0.5).

In 2008 Segal discussed the control of energy flow betweenelectronic conductors in a single mode heat rectifier [91]. In thissystem, which is similar to the 1D chain systems, electron transferis considered where one of the metals is required to have a non-linear dispersion for thermal rectification to result. The systemwasdesigned to model energy transfer through vibrating link andradiative heat transfer. The rectifying effect in this system isincreased when the density of states varies strongly with energy or

Table 2Table of reported rectification using equation (4) from theoretical studies.

Investigators System Mechanism 3

Terraneo et al. [4] 1D chain Non-linear lattice 0.33Li et al. [81] 1D chain Non-linear lattice 0.98Hu et al. [82] 1D chain Non-linear lattice 1.0Lan et al. [75] 1D chain Non-linear lattice 1.0Hopkins and

Serrano [88]1D chain Asymmetric mass

loading and defect0.29

Yang et al. [48] Graphene Asymmetric ribbons 0.6Hu et al. [49,55] Graphene Asymmetric ribbons 0.47Alaghemandi

et al. [42,44]CNT Non-uniform

mass loading0.17

Hu et al. [79] Water/silica Interface transmission 0.21Roberts and Walker [74] Argon/krypton Interface transmission 0.13Wu and Li [46] CNH Asymmetric CNT 0.55Yang et al. [47] CNC Asymmetric CNT 0.46Wu and Li [56] CNT CNT interface 0.39

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the system-bath couplings are energy dependent. This effect isinevitable in energy transport between metals and dielectricsurfaces due to the anharmonic interactions, but likely small due toelectron tunneling at high voltages or thermal gradients. In 2009Wu and Segal continued this study by looking at two specific cases[92]. These cases were onewhere the baths were dissimilar and onewhere the baths were identical and they were coupled by unequalstrengths and an anharmonic lattice. In this work they were able toachieve a sufficient condition for thermal rectification in hybridstructures in an open quantum system. Their conditions are thatthe baths must be dissimilar or their statics must differ from thecoupling of the two baths.

Quantum dots have also been given some interest as thermalrectifiers. In 2008 an experimental study by Scheibner et al. showedthermally rectifying behavior in quantum dots when high in-planemagnetic fields are present as a result of the presence of a highorbital momentum state in the dot [93]. Their analysis showeda stronger coupling to the drain than the source when not in theground state. A theoretical study by Chen et al. was in agreementwith Scheibner et al. when they showed the thermally rectifyingeffect was dependent on the coupling between the quantum dotand the electrode [94]. In 2009 Ruokola et al. investigated thermalrectification in non-linear quantum thermal circuits where thecoupling between the baths and the non-linear resonator circuitwere mutual inductances [95]. Their results showed levels ofrectification up to 0.11. In their system they found a strongdependence on the bath coupling as well as the temperature rangein which the device operated. The operating temperature couldeffect the direction of preferred transport in the device. In a similarstudy Ojanen also found rectification in a quantum system wherethe selection of bath coupling and the properties of the centralquantum system were key to the result and level of rectification[96]. Zhang et al. also showed a change in the preferred direction oftransport in a quantum system, but their dependence was on theoperating temperature alone [97]. The sign of the level of rectifi-cation in their study was dependent on the magnetic field in thechain, temperature and the level of anisotropy in the spin chain.

In each of these works there have been mentions of thermaldevices, but the design of such a device has never really beeninvestigated for an actual application. In 2006 Li et al. againdescribed the effect of the negative differential thermal resistanceand how it could be used in creating a thermal transistor [98,99].This thermal transistor is a three terminal device which consists ofa source, a drain and a gate that is temperature controlled. At lowgate temperatures the switch is off resulting in no heat currentbecause of the insulating properties, but at higher gate tempera-tures the switch acts like a conductor and is therefore on whichallows heat current to flow. This switch also has a semi-on regionwhere the heat current is about half that of the on state. Thetemperature dependence of the gate region uses the same effectthat has been discussed in the previous section (x2.2.4).

Table 1Table of reported rectification using equation (4) from experimental studies.

Investigators System Mechanism 3

Starr [1] Cu/Cu2O Electronic 0.39Barzelay [15] Al/SS Thermal warping 0.67Rogers [17] Al/SS Thermal potential barrier 0.1Powell et al. [19] Al/SS Thermal warping 0Clausing [20] SS/Al Thermal strain 0.2Lewis and Perkins [21] Al/SS Thermal warping 0.41O’Callaghan et al. [27] Varied Thermal warping 0.13Stevenson et al. [23] Varied Thermal warping 0.21Chang et al. [5] CNT and BNNT Non-uniform mass loading 0.034Kobayashi et al. [38] Varied Bulk thermal conductivity 0.18

The reader may notice the dearth of experimental evidence ofnanostructured systems. Chang et al. [5] is essentially the onlyreport that makes thermal rectification claims. This deficit ofevidence is not an indication of the lack of rectification asmuch as itis an underscore on the uncertainty in experimental measure-ments, the uncertainty in the theoretical mechanisms and imma-turity of the field. Researchers are only now starting to think aboutwhat devices can be fabricated, what quantities can be measuredand whether the theoretical systems can be realized in the lab. Infact, experimental study of thermal rectification is said to beongoing in quite a few research labs.

3. Summary of thermal rectification measurementsand predictions

In this section we summarize the results from all the reports ofeach of the mechanisms discussed previously and use them topropose the future direction of thermal rectification research thatshould get us to improved levels of rectification and a possiblefuture with available rectifying materials. Tables 1 and 2 show thelevels of rectification using equation (4) reported from bothexperimental and theoretical studies, respectively.

Based on the previous sections and Tables 1 and 2 it is clear thatthermal rectification can occur and the difference in transport inthe two directions can be quite different. One important aspectwith many of the applications for thermal rectifiers would be thatthey exhibit good conducting properties in the greater transportdirection. Unfortunately in many of these reports the mechanismthat causes thermal rectification does so by reducing the transport(in one direction more than the other), which can make it difficultto maintain high levels of conduction. In some of these studies,specifically the work with the asymmetric CNTs and graphene[5,42,44,46e49,55,56], the transport appears to be increased in onedirection and decreased in the opposite which makes these idealmaterials for thermal rectification applications.

4. Conclusions

This paper explored the past, current and future of thermalrectification in solid materials. Thermal rectification exists at themacro and micro scales and a better understanding is beingdeveloped in both, which will eventually lead to efficient thermalrectifiers. Currently the many mechanisms that are known to existin solid systems have shown only small levels of rectification andare not predictable, but theoretical and analytical models havepredicted large rectification possible and are becoming more andmore accurate. Experimental validation and realization of some of

N.A. Roberts, D.G. Walker / International Journal of Thermal Sciences 50 (2011) 648e662 661

these mechanisms is crucial to the future applications involvingthermal rectifiers even if only a fraction of the predicted levels ofrectification are achieved.

Nomenclature

A area (m2)gC electron DOSq heat rate (W)C heat capacity (J/K)h convect. coeff. (W/m2 K)R therm. res. (K/W)Do deformation pot. (J)Z Planck’s constantRa Rayleigh numberE radiation (W/m2)N occupation numberr radius (m)F Fermi energy (J)Nu Nusselt numberT temperature (K)G rate of energy transfer (W)n power law exponentg gravitational accel. (m/s2)Pr Prandtl number

Greek lettersa thermal diffusivity (m2/s)3 level of rectification emissivityf work function (J)b thermal exp. coeff. (m3/K)r therm. res. ratio mass density (kg/m3)G load/mech. properties coeff.k thermal cond. (W/m-K)g effective rectificationl wavelength (m)s scattering rate (1/s)D normalized temp. diff.n fluid viscosity (m2/s)u frequency (Hz)

SubscriptsA,B,1,2 material typee electron emittedo originb black bodyp phononc coldeep electronephononpt phonon trans. contacth hotr reflected rightE energyl left

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