Non rectification of heat in graded Si-Ge alloys
Sandra Carillo, Maria Grazia Naso, Elena Vuk, Federico Zullo
NNon rectification of heat in graded Si-Ge alloys
S. Carillo , M.G. Naso , E. Vuk , and F. Zullo Dipartimento di Scienze di Base e Applicate per l’Ingegneria,Universit`a di Roma “
La Sapienza ”, Via Antonio Scarpa 16, 00161 Rome, Italy,&I.N.F.N. - Sezione Roma1, Gr. IV - M.M.N.L.P., Rome, Italy [email protected] , DICATAM, Universit`a degli Studi di Brescia, Via Valotti 9, Brescia, Italy [email protected] , [email protected] , [email protected] Abstract.
We investigate the possibility to obtain a thermal diode withfunctionally graded Si-Ge alloys. A wire with variable section is consid-ered. After the introduction of a formula giving the thermal conductivityof the wire as a function of the species content and of the diameter of thewire, numerical and analytical results are presented supporting the im-practicability to get a thermal diode with the characteristics here consid-ered.However, the present study opens the way to further generalisationsamenable to give applicative promising results.
Keywords:
Thermal rectification, graded materials, Fourier law,silicon germanium alloys
The thermal phenomenon that allows the heat to be transferred in a suitable di-rection in a given material, while the flow is impeded in the opposite direction, iscalled thermal rectification [1,2]. This is the analogue of the current rectificationof the electronic diodes and for this reason any device showing some thermalrectifying feature is called thermal diode. An homogeneous material possessinga constant thermal conductivity is known to not possess any rectifying prop-erty [3]: the heat flows, under the same thermal gradient applied, equally in allthe directions. It follows that, if a material possesses a rectifying effect, thenthe thermal conductivity λ is a non-homogeneous function of the temperature.With non-homogeneous we mean that the thermal conductivity also depends onthe space variable x . Despite to be necessary, this condition is however far frombeing sufficient: indeed, it has been shown [3] that, if the thermal conductiv-ity is a separable function, i.e. if there exist two functions f and g such that λ ( T, x ) = f ( T ) g ( x ), then no rectifying effect can be observed in the material.A practicable process to obtain non-homogeneous values of macroscopic prop-erties, such as the thermal conductivity, is the manufacturing of functionallygraded materials, i.e. materials with a specific gradation in the composition in a r X i v : . [ c ond - m a t . o t h e r] N ov S. Carillo, M.G. Naso, E. Vuk, and F. Zullo order to achieve particular performances or functions [4,5]. One of the most com-mon example of graded materials are materials made of binary alloys A c B − c ,where A and B are two different atomic species and c is the content of thespecie A (so that c ∈ (0 , – the device is a wire of variable diameter D , hence the problem can be con-sidered one-dimensional; – the Fourier law describes with enough accuracy the distribution of temper-ature along the device; – the anharmonic, alloy and boundary scattering of the phonons all give con-tributions to the value of the thermal conductivity; – the concentration c and the diameter D are unknown variables and must bedetermined in order to optimize the rectifying effect.The paper is organized as follows: in Section 2 we introduce the main equa-tions and the rectification coefficient R . Also, for the sake of clearness and com-pleteness we briefly describe the approach given in [1] to maximize this coeffi-cient. In Section 3 we apply the formulae to a Si-Ge graded wire with variablediameter. An analytical formula for the thermal conductivity, mainly based on[6] (see also [7,8,9]), is presented and discusses. In Section 4 we apply the re-sults given in Section 2 to the formula for the thermal conductivity obtained.Finally, in the conclusions, the main aspects of this work are emphasized, undera constructive point of view. Indeed, even if it is true that our analysis seems topreclude the possibility to get significant values of the rectification coefficient,different perspectives which may indicate the methodology to achieve applicableresults. This section is mostly based on a work of Peyrard [2] and on a previous workof some of the co-authors [1] and introduces the main findings described in Sec-tion 3. It has been shown in [2] that it is possible to get a rectifying effect froma device composed of two materials with at least one of them with a tempera-ture dependent thermal conductivity. When the temperature range consideredis large, almost every metallic or semiconductor material has a temperature de-pendent thermal conductivity. Furthermore, for graded materials, the thermalconductivity may be also a function of the gradation in composition: for exam-ple, for binary alloys A c B − c , λ is a function of T and of the species content c (see e.g. [10]). If the species content c is variable inside the material, i.e. c = c ( x ),then the thermal conductivity becomes dependent on T and x , λ = λ ( T, c ( x )).In [1] a systematic way to choose the spatial distribution of the composition on rectification of heat in graded Si-Ge alloys 3 c ( x ) and the geometry of the device presenting the more interesting rectifica-tion performances has been introduced. Here we will recall the main findings forcompleteness and easy of readability of Section 3. By assuming that the Fourierlaw holds, the steady state distribution of the temperature is described by thesolution of the following equation: dTdx = − qλ ( T, c ( x )) . (1)In a steady state situation, like the one we are considering, the heat flux q acrossthe device is a constant (since it solves ∇ · q = 0). Then, the implicit solution ofequation (1) is given by T ( x ) = T (0) − q (cid:90) x λ ( T ( y ) , ν ( y )) dy, (2)giving q = − T ( L ) − T (0) (cid:90) L λ ( T ( y ) , ν ( y )) dy . (3)The efficiency of a thermal rectifier can be evaluated through the rectificationcoefficient, defined by the ratio of the heat flux in two opposite configurations,the “direct” and the “inverse”. To fix ideas, we take the two boundaries of thedevice at x = 0 and x = L at the temperatures T H and T L , where T H > T L .In the direct configuration the end x = 0 is at the temperature T L and the end x = L at the higher temperature T H . In the reverse configuration the boundary x = 0 is at the higher temperature T H whereas the end at x = L is at the lowertemperature T L . From equation (2), if q d and q r are respectively the direct andreverse heat fluxes, the rectification coefficient R is defined as: R = | q d || q r | = (cid:90) L λ ( τ r ( y ) , c ( y )) dy (cid:90) L λ ( τ d ( y ) , c ( y )) dy , (4)where τ r and τ d are the solutions of the steady Fourier equation (1) in thereverse and direct configurations respectively [1]. From (4) it is clear that if λ was a constant, the rectification constant would be equal to 1, i.e. the heatfluxes in the direct and inverse configurations are equal and no rectifying effect isobserved. Notice that, if λ is constant, then the distribution of the temperaturesin the two configurations is represented by the distributions of the temperatures T d and T r in the direct and reverse configurations, given, respectively, by T d = T l + ( T h − T l ) xL , T r = T h − ( T h − T l ) xL . (5)If we look at the plane ( x, T ), these temperatures lie exactly on the diagonalsof the rectangle with vertices in (0 , T L ), (0 , T H ), ( L, T L ) and ( L, T H ). According S. Carillo, M.G. Naso, E. Vuk, and F. Zullo to [1], if λ is a regular continuous function of T and x , at steady state thetemperature profiles roughly will follow the same diagonals. Then, if the valuethat the thermal conductivity assumes on one of these diagonals is much greaterthen the value it assumes on the other diagonal, a considerable rectifying effectshould be observed. Based on this reasoning, in [1] the following methodologyhas been proposed to maximize the rectification coefficient (4):1. Among the various possible geometries and distributions c ( x ), look for theparticular geometry and distribution c ( x ) giving a saddle point to the func-tion λ ( T, x ) in the middle of the domain of interest. In the plane ( x, T ), thesaddle must present a maximum on one diagonal and a minimum on theother diagonal.2. Among the various possible geometries and distributions c ( x ), look for theparticular geometry and the particular distribution c ( x ) which maximizesthe difference between the values that λ ( T, x ) assumes on the vertices ofthe rectangle whose diagonals are given by (5). More precisely, if T A ( x )denotes the values on one of the two diagonals and T B ( x ) the values on theother, maximize the differences λ ( T A (0) , − λ ( T B (0) ,
0) and λ ( T A ( L ) , L ) − λ ( T B ( L ) , L ).In the next section, in order to apply the above line of reasoning to graded Si-Ge devices, we will describe the dependence of the thermal conductivity on thetemperature, gradation and dimension of the section of the wire. The thermal conductivity of silicon germanium alloys is in general lower thanthe corresponding thermal conductivities of the bulk materials. From a micro-scopic point of view, this is due to an additional scattering mechanism of thephonons inside the material, the so-called alloy scattering. The geometry of thematerial may play also a role in the determination of the thermal conductivity.Indeed, if the dimensions are small enough, the scattering of the phonons withthe boundaries of the material may become strong enough and contribute to afurther lowering of the thermal conductivity. The properties of the Si-Ge alloyscan be computed by using first-principles approaches, like the density-functionalperturbation theory (see e.g. [11] for bulk materials with diffusive boundary con-ditions or [12] for disordered silicon-germanium alloys). As for the case of poroussilicon, we need an analytical formulation of the thermal conductivity accurateenough and simple enough to be manipulated. In [6] the authors presented a the-oretical, phenomenological formulation of the thermal conductivity of Si − c Ge c nanowire alloys, with the section of the wire explicitly taken into account. Thisformulation has been compared with experimental results: between a tempera-ture range of (100 , − c Ge c nanowire is on rectification of heat in graded Si-Ge alloys 5 then given by [6] λ = k B T π v (cid:126) (cid:90) (cid:126) w c /k B T y e y ( e y − τ − dy, (6)where k B and (cid:126) are the Boltzmann’s and reduced Planck’s constants, the inversescattering rate τ − is given, by the Mathiessen’s rule, by the sum of three terms,i.e. τ − = τ − u + τ − a + τ − b , where τ − u , τ − a and τ − b are respectively the contributions due to anharmonic,alloy and boundary scattering. The anharmonic contribution is described by theweighted average between the Si and Ge terms, τ − u = (1 − c ) τ − Si + cτ − Ge , with τ − Si proportional to w T e − C Si /T and τ − Ge proportional to w T e − C Ge /T ( C Si and C Ge are constants, w is the frequency of the phonons). The alloy scattering termis approximated by a quadratic function of c , that must be zero for c = 0 and c = 1, giving τ − a ∼ c (1 − c ) w (see also [9] for the proportionality to the w term). The boundary term is taken in [6] to be equal to τ − b = v/D , where D is the diameter of the wire and v is the average speed of sound, given by v − = (1 − c ) v − Si + cv − Ge ( v Si and v Ge are the average speeds of sound in siliconand germanium). The overall cutoff frequency w c is given by w c = w cut v/v Si ,where w cut ∼ . λ = ˆ λ T β (cid:90) Θ c /βT y e y ( e y − − (1 − c ) y T e − C Si /T + q cy T e − C Ge /T + q c (1 − c ) T y + q / ( βD ) dy, (7)where ˆ λ = 2 . − K − , β = √ c , Θ c = 1 . · K, C Si = 139 . C Ge = 69 . q i are given by q = 1 . q = 3 . − , q = 96 . m.In Figure (1) the plots of the value of the thermal conductivity for T = 300K andfor different values of D are reported. In the next subsection we apply our methodologyto the thermal conductivity (7): as we will see, in this case we find that it is not possibleto obtain high values of the rectification coefficient (4) following our approach. In the approach explained in Section 2 to obtain sufficiently large values of the recti-fication coefficient it is crucial, for both the steps 1) and 2), to have large values forthe differences λ ( T A (0) , − λ ( T B (0) ,
0) and λ ( T A ( L ) , L ) − λ ( T B ( L ) , L ), where T A ( x )defines the values of the temperatures on one of the two diagonals and T B the valueson the other. It is clear however that, in order to satisfy these conditions, the function λ ( T, x = 0), as a function of T in the domain ( T l , T h ), is required to be increasing atleast in some interval of the domain, whereas the function λ ( T, x = L ) is required to be S. Carillo, M.G. Naso, E. Vuk, and F. Zullo Fig. 1:
The values of the thermal conductivity obtained from formula (7) with T =300K for D = ∞ (corresponding to a bulk alloy), D = 1000nm, D = 400nm and D = 150nm. A log scale has been used on the vertical axis. Fig. 2:
The values of the thermal conductivity for c ∈ (0 . , .
95) for different valuesof the temperature.on rectification of heat in graded Si-Ge alloys 7
Fig. 3:
The values of the thermal conductivity as a function of the temperature in thetwo dilute zones given by c ∈ (0 , .
05) and c ∈ (0 . ,
1) for some values of c . decreasing at least in some interval of the same domain. Since in formula (7) we assumethat both the concentration c and the diameter D are function of x , the values x = 0and x = L pick out some values c (0), c ( L ), D (0) and D ( L ). If the function λ ( T, c, D )would be increasing, as a function of T , in some region of the variables c and D , anddecreasing in some other regions, then it would be possible, by tuning the dependenceof c and D on x , to give a thermal conductivity presenting an interesting rectificationcoefficient. But, in the range of temperatures (100 , λ ( T, c, D ) doesnot satisfy the requested properties. Indeed, in the region c = (0 . , . λ is almostflat for all values of T and for all the reasonable values of D , i.e. from 10nm to 2 · nm.In this range of c , the values of λ are bounded in the interval (1 − − K − , asshown in (2), whereas in the full range of variation of c (i.e. c ∈ (0 , λ can reach values as large as 500Wm − K − . This suggests to look at values of c givingalmost pure silicon ( c ∈ (0 , . c ∈ (0 . , decreasing function of the temper-ature, independently on the value the concentration assumes, apart very flat maximafor fixed and small values of D (see the plots (3) for the dilute zones). This seems topreclude the possibility to tune the dependence of c and D on x in such a way to getan interesting value of the rectification coefficient. The above result could explain thesmall values of R found in literature for Si − c Ge c devices (see, e.g., [13]). In this work we systematically analyzed the possibility to obtain a thermal diode forfunctionally graded Si-Ge alloys. We tried to get the particular spatial distributionof the species content c and the geometry of the wire giving reasonably high valuesof the rectification coefficient. This same methodology is applied to porous siliconmaterials in [1] showing the possibility to obtain a rectification coefficient equal to 3.15.The same approach applied to Si-Ge materials shows that a thermal diode with thecharacteristics here described has instead few chances to become a good heat rectifier. S. Carillo, M.G. Naso, E. Vuk, and F. ZulloClearly, this negative result only implies the impracticability to get a thermal diodewith the above characteristics. The model here presented, although to be physicalconsistent and accurate, can be regarded as a first approximation to more complexapproaches and if new variables or different heat transfer laws are embedded in themodel here considered, more satisfactory results could be achieved. Work performed under the auspices of the Gruppo Nazionale per la Fisica Matematica(GNFM-INdAM).M.G.N., E.V., and F.Z. would like to acknowledge the financial support of the Univer-sit`a degli Studi di Brescia.S.C. acknowledges the partial financial support of Universit`a di Roma “
La Sapienza ”.S.C. and F.Z. would like to acknowledge the financial support of INFN.
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