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Scaling behavior of the thermal conductivity of width-modulated nanowires and nanofilms for heat transfer control at the nanoscale
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Nanotechnology Nanotechnology 25 (2014) 465402 (9pp)
doi:10.1088/0957-4484/25/46/465402
Scaling behavior of the thermal conductivity of width-modulated nanowires and nanofilms for heat transfer control at the nanoscale Xanthippi Zianni1,2, Valentin Jean3, Konstantinos Termentzidis3 and David Lacroix3 1
Department of Aircraft Technology, Technological Educational Institution of Sterea Ellada, 34400 Psachna, Greece 2 Department of Microelectronics, INN, NCSR’Demokritos’, 153 10 Athens, Greece 3 Université de Lorraine, LEMTA, CNRS, UMR 7563, Faculté des Sciences et Technologies, BP 70239, F-54506, Vandoeuvre les Nancy cedex, France E-mail: [email protected] and [email protected] Received 15 July 2014, revised 21 September 2014 Accepted for publication 6 October 2014 Published 31 October 2014 Abstract
We report on scaling behavior of the thermal conductivity of width-modulated nanowires and nanofilms that have been studied with the phonon Monte Carlo technique. It has been found that the reduction of the thermal conductivity scales with the nanostructure transmissivity, a property entirely determined by the modulation geometry, irrespectively of the material choice. Tuning of the thermal conductivity is possible by the nanostructure width-modulation without strict limitations for the modulation profile. In addition, a very significant constriction thermal resistance due to width-discontinuity has been identified, in analogy to the contact thermal resistance between two dissimilar materials. The constriction thermal resistance also scales with the modulated nanostructure transmissivity. Our conclusions are generic indicating that a wide range of materials can be used for the modulated nanostructures. Direct heat flow control can be provided by designing the nanostructure width-modulation. Keywords: nanoscale heat transfer, phonon Monte Carlo, modulated nanowires, nanofilms, boundary scattering, transmissivity, porosity (Some figures may appear in colour only in the online journal) 1. Introduction
dimensions in the quantum confinement regime. Much research is also devoted to complex materials where phonon scattering and the thermal conductivity could be controlled by the material composition [3–5]. Porous materials and lowdimensional structures with characteristic dimensions above the quantum confinement regime, show reduced thermal conductivity due to enhanced phonon boundary scattering [6– 10]. Here, we propose an alternative strategy for tuning the thermal conductivity: the use of width-modulated nanowires and nanofilms that are relatively thick and operate in the boundary scattering regime. We report on the possibility for heat flow control by designing the nanostructure width-
Control of the heat transfer at the nanoscale is currently a major technological research issue. It is highly required in order to increase the efficiency of nanoelectronic, optoelectronic, and photovoltaic devices. In thermoelectrics, heat transfer control is particularly demanding because of the requirement for limiting the thermal conductivity without suppressing the electrical conductivity. Phonon bandstructure engineering is one of the strategies proposed for heat manipulation at the nanoscale [1, 2]. Limitations are posed by strict requirements for perfectly periodic structures with 0957-4484/14/465402+09$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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modulation without strict limitations for the modulation profile. This is enabled by the scaling behavior that we found simulating the modulated nanostructures with the Monte Carlo (MC) technique: the reduction of the thermal conductivity scales with the nanostructure transmissivity, a property entirely determined by the modulation geometry, irrespectively of the material choice. We point out that the width-modulation acts like an ‘external porosity’ that could be easier to handle and control by the fabrication technology compared to the ‘internal porosity’. We also report on a novel effect: a very significant constriction thermal resistance due to the width discontinuity has been identified, in analogy to the contact thermal resistance between two dissimilar materials. The constriction thermal resistance scales with the transmissivity. Our conclusions are generic. They indicate that a wide range of materials can be used to develop width-modulated nanowires and nanofilms and directly control the heat flow by designing the width-modulation. Width-modulated nanowires have been previously proposed as efficient thermoelectric materials [11]. It has been theoretically predicted that the transport properties of electrons and phonons are modified resulting in enhanced thermal to electrical energy conversion efficiency. In the quantum confinement regime, the electron transmission coefficient shows resonances, minibands and sharp propagation thresholds and thereby Seebeck coefficient enhancement [11, 12]. Observed thermoelectric power factor enhancement has been attributed to resonance states [13]. In modulated nanowires, electrons occupy energy minibands [14] favoring the thermoelectric efficiency [14–16]. In the quantum confinement regime, the phonon transmission coefficient and phonon energy states are also modified so that the thermal conductivity decreases [17–19]. Above the quantum confinement regime, the thermal conductivity of modulated nanowires has been previously studied within the kinetic theory for phonons and using an approximate model for boundary scattering [20]. A big reduction of the thermal conductivity was found due to enhanced boundary scattering and the decrease of the phonon transmissivity by the nanowire width-modulation. Here, we report on MC simulations of the phonon transport properties of width-modulated nanowires and nanofilms that are free from the approximations used in the kinetic theory model [20]. The exhibited behavior has been interpreted using physics arguments and modeling. General conclusions have been extracted applicable to a wide range of materials. We have chosen Si as reference material, because of its technological importance and because contrary to common strategies that work in the quantum confinement regime and require Si nanostructures with very small dimensions, width-modulated nanostructures that are proposed here are free from this requirement.
resolution of the Boltzmann transport equation (BTE) in the frame of the relaxation time approximation (RTA). Silicon dispersion properties have been used for the sampling of phonon frequency and group velocity, as well as for the calculation of the scattering relaxation times following the model proposed by Holland [23]. 2.1. BTE for phonons
The BTE for phonons is related to the variations of the distribution function f (T , r, p , K) which depends on time t, location r, polarization p and wave vector K. f (T , r, p , K) can also be defined as the mean particle number at time t in the d3r volume around r with K wave vector and d3K accuracy for a given polarization p. In the absence of external force, the BTE expression is [24] ∂f ∂f + K ω ⋅ r f = ∂t ∂t
,
(1)
collisions
with the phonon group velocity vg = K ω and f 0 (T , p , K) the equilibrium thermodynamic phonons population for polarization p and wave vector K, which is described by the Bose–Einstein distribution function f 0 (T , p , K) =
1 e
( ℏω / k B T )
.
(2)
−1
In the frame of the RTA, the right hand side term of equation (1) can be written as follows ∂f ∂t
= collisions
f 0 (T , p , K) − f (T , r , p , K) , τ ( T , p , K, L c )
(3)
where τ is the phonon scattering relaxation time, which depends on the phonon dispersion properties, on the temperature and possibly on a characteristic length Lc. In the frame of MC modeling, the former equations are solved in two steps. Firstly, the advective part of the BTE is addressed (left hand side term of equation (1)) assuming that there is no scattering. Thus, a statistically representative population of phonons is allowed to drift with respect to: (i) its group velocity within the nanostructure, (ii) the internal reflections at pore surfaces and borders. Secondly, the phonon scattering step is achieved considering equation (3). The characteristic relaxation time is given according to the ‘Matthiessen’ summation rule. It takes into account: normal (N), umklapp (U) and impurity (I) scattering rates according to Holland [23] formalism. As mentioned previously, the simulation is frequency dependent. This means that dispersion properties of materials shall be taken into account. Here, we used data (dispersion relation and relaxation time parameters) extensively detailed in previous works for silicon [8, 23]. They are not recalled here.
2. Methodology
2.2. MC procedure
The flowchart of the MC procedure mainly involves successive stages which are reproduced for each time steps of the simulation until the convergence (temperature and flux steady
We have performed MC simulations for the phonon transport using a previously developed simulation tools. The numerical method is extensively described in [8, 21, 22]. It lies on the 2
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nanowire axis is calculated with N
φz =
state) is reached within the nanostructure. First of all, the nanostructure geometry is defined and meshed with rectangular cells of volume V. In some of them tiny volumes are removed in order to obtain constrictions, modulations or porosities (see figures 1 and 5). Once the geometry is known, the temperature gradient at the edges of the nanostructure (first and last cells) is set and the other cells are initially at the lower temperature. On the basis of these prescribed temperatures the energy within each cell can be determined from the Bose–Einstein distribution and the density of states. Then a phonon distribution with different frequencies ω can be randomly sampled in each cell using a cumulative distribution function (CDF) based on the energy [8, 22] until the reference energy at a given temperature is reached
3. Results and discussion It is well known that the thermal conductivity of nanowires and nanofilms is significantly smaller than the thermal conductivity of the corresponding bulk materials when the characteristic dimensions are small. Measurements of the thermal conductivity of Si nanowires and nanofilms with widths ranging from ∼20 nm to ∼100 nm and at temperatures up to 350 K have been reported in [25, 26]. Lower thermal conductivities than in bulk Si have been measured in all nanostructures. These experimental data have been interpreted by several theoretical studies and the measured decrease of the thermal conductivity has been attributed to the predominant boundary scattering [25, 27]. Boundary scattering dominates because the phonon bulk mean free path (∼270 nm at 300 K) is greater than the distance between the nanostructure boundaries. In addition, phonon confinement effects and especially dispersion properties modifications can be neglected at these dimensions in Si. Phonons in Si nanostructures with characteristic dimensions down to ∼20 nm, can be considered as bulk-like and their transport is determined by the bulk material scattering mechanisms and additional boundary scattering. We have performed MC simulations in this transport regime. We report on simulations of the thermal conductivity of nanowires and nanofilms with boundaries separations in the range 20–100 nm. Diffusive boundary scattering has been assumed. At room temperature, the phonon’s wavelength is about 1 nm, depending on the way it is evaluated. Thus, the specularity parameter ( p = exp − 16π 3δ 2 /λ 2 , where δ is the roughness parameter and λ is the wavelength [28]) is close to 0, meaning that every single reflection is diffuse. Nevertheless, Chen et al [29] showed that the reflection is frequency dependent. This dependence is particularly important at low temperatures, when the average phonon’s wavelength can be much larger than the average roughness height (about 2 nm in their work, including the disordered phase). This is not the case here, since all calculations were done at 300 K. Moreover, at room temperature, phonons with large wavelength (low frequency)
N
∑ℏ ω i .
(5)
where V is the volume of the cell and vgzi the group velocity in the z direction (nanowire axis) of the phonon with frequency ωi. The thermal conductivity is thus determined assuming the Fourier formalism. The calculations are achieved on a cluster with a parallel architecture. In order to have an uncertainty in the range 0.5–5% on the thermal conductivity, up two 12 nodes are used and several thousand of phonons are sampled according to the nanostructure size. The time step of the simulation is 1 ps and the steady state is typically reached between 20 and 100 ns according to the nanowire length and the magnitude of the modulations.
Figure 1. Schematics of the reference modulated nanostructures: (a) a periodic sequence of constrictions of width b, and (b) a nonperiodic sequence of constrictions of average width 〈b〉.
th E cell (V , T ) =
1 ∑ℏωi × vgzi , V i=1
(4)
i=1
For each sampled phonon the frequency and the polarization are randomly obtained using the CDF. The group velocity is derived from the dispersion properties knowing ω and p. The initial propagation direction is isotropically sampled. This concludes the initialization procedure. During the drift stage, all the sampled phonons are allowed to move according their propagation direction on a distance equal to vg × Δt , unless they reach a boundary and thus they are diffusely reflected in the case of a nanowire. At the end of the drift procedure, according to the phonon locations the energy in each cell and the corresponding temperature are calculated. This results in a shift from the equilibrium distribution function which is corrected applying the collision procedure. In the latter one, a collision probability is calculated for each phonon on the basis of the relaxation time evaluation at the corresponding values of ω, p and T [8, 21]. A set of random numbers are drawn to determine whether the phonon scatters and if it is the case the nature of the scattering process (U, N, I). In the case of a N process the phonon propagation direction remains the same, if it is U or I process the later one is resampled. Besides, the phonon frequency is resampled in the case of a three phonons process (N or U). The drift and collision stages are repeated until the convergence of the simulation. Once steady state is reached, the phonon heat flux along the
(
3
)
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as
weakly contribute to heat transport, as their density of states is small compared to the transverse acoustic ones close to the edge of the Brillouin zone. Thus, the assumption of complete diffuse scattering at the boundaries remains appropriate as long as the temperature is high enough and the roughness is moderate. The simulation parameters have been optimized to interpret the experimental data on the thermal conductivity of Si nanowires and nanofilms [25, 26]. We consider nanowires and nanofilms with variable widths. In order to investigate the effect of the width-modulation on the thermal conductivity, we consider as reference a non-modulated nanowire/nanofilm of length L with constant width a and depth d. For the nanowires d ∼ a ≪ L; for the nanofilms d ≫ a and d ∼ L. There is no restriction on the choice of the profile shape of the width-modulation. Here, we use as reference the modulation profiles shown in figure 1. The model width-modulation that we use consists in a sequence of constrictions along the direction of heat flow. The cross sections of the wide and narrow segments are assumed orthogonal for the sake of simplicity. Hereafter, we refer to the wide segments as ‘openings’ and to the narrow segments as ‘constrictions’.
transmissivity Tr ≡
constriction cross section . opening cross section
(6)
For the geometry of figure 1(a), this reduces to the ratio of the two characteristic dimensions of the width modulation Tr =
b . a
(7)
Moreover, a simple scaling behavior has been identified by the analysis of our simulation data: ‘the ratio of the thermal conductivity of the modulated nanowire over that of the reference nanowire scales with the transmissivity’. This behavior is shown in figure 2(b) and is quantitatively expressed by the relation k modulated wire ≈ Tr. k wire
(8)
The nanowire transmissivity, as defined in equation (6), expresses the percentage of the nanowire’s geometry modulation. This definition also applies for the nanofilms (figure 1(a)). The simulations data for the nanofilms are shown together with those of the nanowires in figure 3. It can be seen that the same scaling behavior holds for the nanofilms as for the nanowires. In what follows, we present the simulation results for nanowires with d = a. Nevertheless, the obtained evidence and interpretation also hold for the nanofilms with d ≫ a. Equation (8) can be interestingly interpreted as that: the percentage of the width modulation is equal to the percentage of the thermal conductivity reduction, i.e. to the heat flow transmissivity. It thereby becomes evident that ‘direct control on the heat flow by the nanowire/nanofilm width-modulation takes place’. Τhe same scaling behavior holds for the nanofilms as for the nanowires when the heat flow is normal to the widthmodulations. In the case of nanofilms, if the wide and narrow sections are collinear to the thermal gradient, the in-plane modification of the thermal conductivity should not be the same as in the present case where modulated sections are perpendicular to the heat flux. Considering a simple electrical analogy, those two cases correspond to parallel and serial association of thermal resistances respectively. Cross-plane thermal conductivity assessment for modulated thin films would require different treatment because hot and cold boundaries are not the same for the whole film. The above scaling behavior holds for ‘adequate’ modulation density for which the heat flow density is quite uniform along the nanowire. The effect of the modulation density on thermal conductivity is shown in figure 4 and it can be modeled by introducing an offset C in equation (8)
3.1. Modulated nanowires and nanofilms
We start our discussion with a periodic sequence of modulating constrictions, as shown in figure 1(a). Each constriction has width b, length c and depth d. The length of the constriction c is assumed much smaller than the constriction width. The data shown here are for c = 10 nm. The MC simulations uncertainty is in the range of 0.5–5% for the thermal conductivity. In figure 2 are shown the simulated thermal conductivities of nanowires with representative dimensions: a ‘thick’ nanowire of width 100 nm and a ‘thin’ nanowire of width 60 nm and for d = a. A weak dependence on the constriction length c was found for c varying in the range 10–20 nm and for constant dimensions of the openings. The trend is that the thermal conductivity decreases linearly with decreasing b. Our data show that the decrease of the thermal conductivity can be very significant. In figure 2(a), a decrease of ∼10 times is shown for the smallest values of b considered. The thermal conductivity decrease is more pronounced for the thicker nanowire. The thermal conductivity of the thick nanowire is lower than that of the thin nanowire for same b. This behavior was also found in a previous theoretical study of the thermal conductivity of diameter modulated nanowires within the kinetic theory model where geometry-dependent boundary scattering was used [20]. Two main assumptions were used in [20] for boundary scattering: (i) the variable nanowire width was taken into account by a geometrical averaging of the boundary scattering length, and (ii) the effect of the non-uniform phonon propagation along the nanowire was taken into account by the transmissivity. The qualitative agreement with the present MC simulation, that is free from the above two assumptions, validates the assumptions of [20]. Interestingly, both models indicate that the size dependence of the thermal conductivity can be interpreted in terms of the nanowire transmissivity Tr, defined
k modulated wire = (1 − C ) Tr + C . k wire
(9)
For a decreasing modulation density, the thermal conductivity increases. The offset C increases correspondingly. The width modulation becomes, then, less efficient in decreasing the thermal conductivity. A more detailed discussion is devoted 4
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Figure 2. Scaling behavior of the thermal conductivity. (a) The thermal conductivity of two periodically modulated nanowires with a = d, and a = 100 nm (red squares) and a = 60 nm (blue dots) versus the constriction width b. (b) The ratio of the thermal conductivity of the periodically modulated nanowire over the thermal conductivity of the non-modulated nanowire versus the transmissivity. (c) and (d) are as (a) and (b) respectively with additionally shown the corresponding data for non-periodically modulated nanowires with average width b (stars). The length of the constriction is c = 10 nm.
Figure 4. The effect of the modulation density. The ratio of the
Figure 3. The scaling behavior of the thermal conductivity for
thermal conductivity of the modulated nanowire over that of the reference nanowire versus the transmissivity for the constriction density (squares) in figure 2(a), for the double (triangles) and the half density (dots). The length of the wires is kept constant.
modulated nanowires and nanofilms. For nanowires (solid symbols): d = a. For nanofilms (stars) d = 1 μm.
to this dependence in the following, when we investigate the thermal conductivity of a nanowire modulated by a single constriction. We have, so far, assumed a periodic modulation. In this way, we have been able to identify a single ratio of the dimensions of the periodic modulation that explains the
reduction of the thermal conductivity. We would expect that this behavior should not be bound to the periodic modulation profile. We are in the particle-like regime, where phonon states are not related to the presence of periodicity in the boundary conditions. Indeed, our simulations showed that ‘it 5
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Figure 5. Schematics of a width-modulated nanowire and the
corresponding porous nanowire that are discussed in the main text.
is still possible to relate the reduction of the thermal conductivity to the reduction of the transmissivity of nanowires/ nanofilms that are modulated non-periodically’. This becomes evident in figures 2(c) and (d). By dots and squares are shown the data for periodic modulations and by stars the data for non-periodic modulations (figure 2(d)). In the nonperiodic modulation simulations, the width of the openings and the density of the modulation were kept constant. The widths of the constrictions were varied randomly. The thermal conductivity is plotted at the average constriction width. Now, the scaling behavior although blurred, is still shown as a clear trend: ‘the ratio of the thermal conductivity of the modulated nanowire/nanofilm over that of the reference nanostructure follows the linear reduction of the transmissivity’. In this case, a nanowire/nanofilm transmissivity can be determined by the average constriction cross-section transmissivity Tr ≡
constriction cross section . opening cross section
Figure 6. Single constriction modulation. The thermal conductivity
of a nanowire modulated by a single constriction versus the nanowire length for various values of the constriction width b (squares). The length of the constriction is c = 10 nm. Periodically modulated nanowires with the same parameters a and b (dashed lines).
physical quantity as the transmissivity of the width-modulated nanostructures, introduced here in equation (6). It could be also useful to point out that the nanowire width-modulation can be seen as an ‘external porosity’ since the ‘missing’ volumes of the cavities are located at the surface of the modulated nanowires. The use of porous nanostructures is commonly proposed to achieve thermal conductivity reduction. ‘The same thermal conductivity reduction should be achieved using width-modulated nanowires with ‘external porosity’ equivalent to the ‘internal porosity’ of porous nanowires’. This conclusion could be useful because widthmodulated nanowires/nanofilms (‘external porosity’) should be much easier to handle and control by the nano-fabrication technologies than the corresponding porous nanostructures (‘internal porosity’).
(10)
The above evidence points out that by designing the average width modulation, the thermal conductivity can be tuned and the heat flow can be controlled. ‘Very importantly, it is pointed out that there are no posed strict requirements for the geometry modulation profile: it would be sufficient to use a profile with average width corresponding to the transmissivity required for a desired thermal conductivity reduction. Moreover, our simulations indicate that more efficient structures should emerge when thick nanowires are modulated’. These results provide technology guidelines for heat flow control at the nanoscale. We have compared the thermal conductivities of the width-modulated nanowires and of porous nanowires with pores of the same shape and dimensions with the ‘missing’ volumes at the surface of the modulated wires (figure 5). It should be noticed that the change in the volume of the modulated nanowires by the width-modulations at their surface is the same as that of the corresponding porous nanowires where the pores are located inside their volume. We found quantitative agreement for the thermal conductivities of the two types of nanowires. The calculated thermal conductivities for the porous nanowires agree well with the data shown in figure 2 within the simulations accuracy. The porous nanowires also showed the same scaling behavior of the thermal conductivity with the transmissivity. It is well known that in porous materials, the reduction of the thermal conductivity can be directly related to the porosity [8, 10] defined as the ratio of the involved volumes: (Vnanowire − Vcavities)/ Vnanowire. The porosity expresses the percentage of the volume modulation of the material and should be conceived as similar
3.2. Nanowires with a single constriction
We now discuss the case of a single width modulation, namely a ‘constriction’. The thermal conductivity decreases with the decrease being more important with decreasing constriction width b (figure 6). The thermal conductivity shows a weak dependence on the constriction length c for the considered range of values. The thermal conductivity of the modulated nanowire depends on the wire length as is shown in figure 6. This dependence can be interpreted by two effects: (i) the decrease of the thermal conductivity with decreasing nanowire length, and (ii) the additional decrease of the thermal conductivity by the constriction thermal resistance. Concerning the effect (i). The thermal conductivity of a nanowire increases when its length increases and saturates to a certain value when fully diffusive transport takes place. This is well known behavior of nanowires that is also found here for the modulated nanowires. The length dependence is shown in figure 6 with black squares for the non-modulated 6
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nanowires, and with colored squares for the modulated nanowires. The thermal conductivity of a modulated nanowire is lower than that of the corresponding non-modulated nanowire of the same length. It decreases with decreasing b, as expected. When the constriction deformation is shallow, the decrease of the thermal conductivity is small and its length dependence is very similar to that of the non-modulated nanowire. A steeper length dependence of the thermal conductivity is shown for bigger values of b. This effect is characteristic for a transition regime between two kinds of geometries. First, if we look at the nanowire without constriction (black squares), the expected decrease of the thermal conductivity with increasing wire length L is shown. For a long wire and shallow modulation (as for a = 100 nm and b = 90 nm shown with red squares), the constriction has a weak effect on the nanowire thermal conductivity as it is a local effect on phonon boundary scattering. The dominant characteristic size of the nanowire is the length ‘a’. For nanowires with a deep modulation (as for a = 100 nm and b = 20 nm shown with pink squares), there is a ‘blocking’ effect on the phonon transport which is more significant for small length L for the reasons discussed above. In this case, the dominant characteristic size is ‘b’. A transition between the above two regimes is shown at intermediate dimensions as e.g. for b = 60 nm, a = 100 nm shown with blue squares. The constriction has a weak effect on the heat flow for long nanowires where c/L is small, whereas it has a greater impact for short modulated nanowires where c/L is larger. Furthermore, analyzing our data we have concluded that this behavior as well as the actual values of the thermal conductivity for each value of b, can be interpreted in terms of the constriction thermal resistance (effect (ii)). Concerning the effect (ii). A thermal resistance can be attributed to the constriction originating from the width discontinuity. The constriction thermal resistance increases with decreasing constriction cross section discontinuity and it is nearly independent of the wire length. This is evident in the temperature profiles along the wires shown in figure 7 where modulated nanowires of three lengths are depicted. It can be thereby concluded that ‘the identified thermal resistance is a property of the constriction itself’. For the largest value of b (shallow deformation), the temperature drop is nearly linear implying a uniform nanowire thermal conductivity. With increasing b, the linear drop starts to deform and eventually shows a discontinuity for small b (deep deformation). The temperature gradient on both sides of the discontinuity shows a small length-dependence. For short nanowires, the temperature jump close the discontinuity is larger than for long nanowires because of the short distance from the hot and cold contact boundaries of the structure. This is no longer true when the nanowire length increases. For a sufficiently long nanowire the temperature jump remains constant, solely depending on the constriction cross-section discontinuity. The discontinuity implies a non-uniform thermal conductivity. The opening is the signature of ballistic transport through the constriction and resembles to the Kapitza resistance at the interface of two dissimilar materials [30, 32]. ‘It can be very importantly concluded that the width
Figure 7. The evolution of the temperature profile with increasing modulation depth. The temperature profile along the nanowire for three values of b and for three nanowire lengths. Temperature is plotted at the center of each cell. All cells have the same length.
discontinuity acts like a contact thermal resistance’. The origin of the thermal resistance is the heat flux transmission discontinuity due to the width (cross-section) discontinuity. This is in analogy to the Kapitza resistance at the interface of two dissimilar materials that is due to the heat flux discontinuity due to the density of states discontinuity [30, 31]. The magnitude of the contact thermal resistance and its effect on the overall effective conductivity of the nanowire are estimated below. The constriction thermal conductance has been calculated directly by the simulation data and is shown in figure 8(a) by solid symbols versus the width b. The data can be interpreted by the approximate relation th R mw = R cth + R wth ,
(11)
th where Rmw , Rcth , Rwth are the thermal resistances of the modulated nanowire, of the constriction and of the nonmodulated nanowires respectively. The above relation is 7
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Figure 8. Scaling behavior of the constriction thermal conductance. The constriction thermal conductance (squares) versus (a) the constriction width b, and (b) the transmissivity. The inset shows the linear relation between the constriction thermal conductance and (1 − Tr)−1 as predicted by equation (13) of the main text. In (a), the data using equation (13) of the main text are also shown with stars.
written as
Deviations are found for very shallow deformation when the constriction thermal resistance is very small. The steeper length dependence of the thermal conductivity of the modulated nanowires by a single constriction compared to that of the non-modulated nanowire is explained by the additional contribution of the constriction thermal resistance (equations (11) and (12)) that becomes more significant when the nanowire length decreases. In the limiting case where the length of the nanowire is comparable to the constriction modulation unit, the overall conductance is nearly equal to the constriction thermal conductance. Kc explains the minimum values of the thermal conductivity for the various values of b in figure 6. For adequate modulation density, the heat flow density is uniform along the wire. A linear drop of the temperature along the wire has been found in the simulations. In this regime, the reduction of the thermal conductivity of the width-modulated nanowires is interpreted by the transmissivity. For a single constriction, the reduction of the thermal conductivity of the modulated nanowire is interpreted by the constriction thermal conductance. Kc decreases rapidly when the modulation becomes deeper (figure 8(a)) and causes the more rapid decrease of the overall thermal conductivity shown in figure 6. The temperature profile shown in figure 7 indicates that the heat flow density is non-constant in this case. The single constriction is less efficient in ‘guiding’ the heat flow and reducing the thermal conductivity compared to a sequence of constrictions. This is shown in figure 6 where the data for the thermal conductivity of the periodically modulated nanowires (dashed lines) are plotted together with the data for the single constriction modulation for comparison. The thermal conductivity of the nanowires modulated by a single constriction becomes smaller than that of periodically
⎛ L left L mw L wright ⎞ 1 ⎟⎟ , (12) = + ⎜⎜ w left + A w k mw A w k wright ⎠ ( A w − A c ) Kc ⎝ A w k w
where k mw , k w , are the thermal conductivities of the modulated wire and of the non-modulated wire respectively. Kc is the constriction thermal conductance. Aw and Ac respectively denote the cross sections of the wire and of the constriction. Lmw, L wleft(right) , are the lengths of the modulated wire and of the left (right) segment of the wire with constant thickness a. For long nanowires, it can be assumed that L ≡ L mw ≈ L w = L wleft + L wright . Then, equation (12) gives the following expression for the constriction thermal conductance
(
Kc =
)
1 ⎡ 1 ⎛ k mw k w ⎞ ⎤ ⎢ ⎜ ⎟ ⎥, 1 − Tr ⎣ L ⎝ k w − k mw ⎠ ⎦
(13)
where Tr = Ac /Aw (also see equation (6)). We have used equation (13) to estimate the constriction thermal conductance. The results are shown in figure 8(a) by stars. It can be seen that they agree well with the results of the MC (squares). This agreement validates relations (11) and (12). Furthermore, equation (13) indicates that ‘the constriction thermal conductance scales with the transmissivity Tr’. In particular, it turns out that Kc ∝
1 . 1 − Tr
(14)
This dependence is predicted by equation (8) and is confirmed by our MC simulations shown in figure 8(b). The inset in figure 8(b) shows that equation (9) is followed well. 8
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References
modulated nanowires only for short nanowires because the constriction thermal conductance has, in this case, a bigger weight in the resistances network (equations (11) and (12)).
[1] Cahill D G et al 2014 Appl. Phys. Rev. 1 011305 [2] Han H, Potyomina L G, Darinskii A A, Volz S and Kosevich Y 2014 Phys. Rev. B 89 180301 [3] Zebarjadi M, Esfarjani K, Dresselhaus M S, Ren Z F and Chen G 2012 Energy Environ. Sci. 5 5147 [4] Pernot G, Stoffel M, Savic I, Pezzoli F, Chen P, Savelli G, Jacquot A, Schumann J, Denker U and Moench I 2010 Nature 9 491 [5] Mingo N, Hauser D, Kobayashi N P, Plissonnier M and Shakouri A 2010 Nano Lett. 10 2288 [6] Heron J S, Fournier T, Mingo N and Bourgeois O 2010 Nano Lett. 10 2288 [7] Heremans J P, Dresselhaus M S, Bell L E and Morelli D T 2013 Nat. Nanotechnology 8 471 [8] Jean V, Fumeron S, Termentzidis K, Tutashkonko S and Lacroix D 2014 J. Appl. Phys. 115 024304 [9] Jain A, Yu Y-J and McGaughey J H 2009 Phys. Rev. B 87 195301 [10] Liu L-C and Huang M-J 2010 Int. J. Therm. Sci. 49 1547 [11] Zianni X 2010 Appl. Phys. Lett. 97 233106 [12] Zianni X 2011 Nanoscale Res. Lett. 6 286 [13] Wu P M, Gooth J, Zianni X, Fahlvik Svensson S, Thelander K D, Thelander C, Nielsch K and Linke H 2013 Nano Lett. 13 4080 [14] Zianni X 2014 J. Electron. Mater. 43 3753 [15] Balandin A A and Lazarenkova O L 2003 Appl. Phys. Lett. 82 415 [16] Bahk J-H, Sadeghian R B, Bian Z and Shakouri A 2012 J. Electron. Mater. 41 1498 [17] Zianni X 2012 J. Solid State Chem. 193 53 [18] Nika D L, Cocemasov A I, Isacova C I, Balandin A A, Fomin V M and Schmidt O G 2012 Phys. Rev. B 85 205439 [19] Termentzidis K, Barreteau T, Ni Y, Merabia S, Zianni X, Chalopin Y, Chantrenne P and Volz S 2013 Phys. Rev. B 87 125410 [20] Zianni X and Chantrenne P 2013 J. Electron. Mater. 42 1509 [21] Lacroix D, Joulain K and Lemonnier D 2005 Phys. Rev. B 72 064305 [22] Peraud J-P M and Hadjiconstantinou N G 2011 Phys. Rev. B 84 205331 [23] Holland M G 1963 Phys. Rev. 132 2461 [24] Neil W A and David N M 1976 Solid State Physics (Philadelphia, PA: Saunders) [25] Li D, Wu Y, Kim P, Shi L, Yang P and Majumdar A 2003 Appl. Phys. Lett. 83 2934 [26] Asheghi M, Toulzelbaev M N, Goodson K, Leung Y K and Wong S S 1998 ASME J. Heat Transfer 120 30 [27] Lacroix D, Joulain K, Terris D and Lemonnier D 2006 Appl. Phys. Lett. 89 103104 [28] Moore A L, Saha S K, Prasher P S and Shi L 2008 Appl. Phys. Lett. 93 083112 [29] Chen R, Hochbaum A I, Murphy P, Moore J, Yang P and Majumdarin A 2008 Phys. Rev. Lett. 101 105501 [30] Kapitza P L 1941 Zh Eksp. Teor. Fiz. 11 1 [ English transl.: J. Phys. USSR. 4, 181 (1941)]. [31] Pollack G L 1969 Rev. Mod. Phys. 41 48 [32] Prasher R 2005 Nano Lett. 5 2155
4. Conclusions We have performed MC simulations on the heat flow in width modulated nanowires and nanofilms and we provided physics interpretation and modeling of the simulation results. Our work indicates that width-modulated nanowires/nanofilms that are relatively thick and operate in the boundary scattering regime could be used to tune the thermal conductivity. We have made evident the possibility for heat flow control by designing the width-modulation without strict limitations for the modulation profile. This is enabled by the identified scaling behavior: the reduction of the thermal conductivity scales with the transmissivity that is entirely determined by the modulation geometry irrespectively of the material choice. More efficient structures should emerge when thick nanowires/nanofilms are modulated. We have additionally pointed out that the width modulation acts like an ‘external porosity’ that could be easier to handle and control by the fabrication technology. We have also identified a novel effect: a very significant constriction thermal resistance, in analogy to the Kapitza contact resistance. The origin of the constriction thermal resistance is the width discontinuity and it scales with the transmissivity. Our conclusions are generic. We have used here Si as reference material. Our numerical method can be applied to several materials for which phonons are the main heat carriers, mainly insulating materials and semi-conductors like Ge, GaAs, InSb, C, etc, as long as dispersion properties and relaxation times are known. The obtained scaling behavior of the thermal conductivity with the modulation geometry should also hold for these materials. The sizelimitations on characteristic dimensions of the modulation geometry may vary in different materials. Our work indicates that a wide range of materials can be suitable for efficient nanoscale heat management by designing the nanostructure width modulation.
Acknowledgments XZ acknowledges invited professorship of the University of Lorraine (LEMTA), the project ‘SiMoNE’, PEPS Mirabelle 2013 financed by CNRS and the University of Loraine, and the European Social Fund (ESF)-European Unions and National Resources within the framework of the Grant of Excellence ‘ARISTEIA’.
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Scaling behavior of the thermal conductivity of width-modulated nanowires and nanofilms for heat transfer control at the nanoscale
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Nanotechnology Nanotechnology 25 (2014) 465402 (9pp)
doi:10.1088/0957-4484/25/46/465402
Scaling behavior of the thermal conductivity of width-modulated nanowires and nanofilms for heat transfer control at the nanoscale Xanthippi Zianni1,2, Valentin Jean3, Konstantinos Termentzidis3 and David Lacroix3 1
Department of Aircraft Technology, Technological Educational Institution of Sterea Ellada, 34400 Psachna, Greece 2 Department of Microelectronics, INN, NCSR’Demokritos’, 153 10 Athens, Greece 3 Université de Lorraine, LEMTA, CNRS, UMR 7563, Faculté des Sciences et Technologies, BP 70239, F-54506, Vandoeuvre les Nancy cedex, France E-mail: [email protected] and [email protected] Received 15 July 2014, revised 21 September 2014 Accepted for publication 6 October 2014 Published 31 October 2014 Abstract
We report on scaling behavior of the thermal conductivity of width-modulated nanowires and nanofilms that have been studied with the phonon Monte Carlo technique. It has been found that the reduction of the thermal conductivity scales with the nanostructure transmissivity, a property entirely determined by the modulation geometry, irrespectively of the material choice. Tuning of the thermal conductivity is possible by the nanostructure width-modulation without strict limitations for the modulation profile. In addition, a very significant constriction thermal resistance due to width-discontinuity has been identified, in analogy to the contact thermal resistance between two dissimilar materials. The constriction thermal resistance also scales with the modulated nanostructure transmissivity. Our conclusions are generic indicating that a wide range of materials can be used for the modulated nanostructures. Direct heat flow control can be provided by designing the nanostructure width-modulation. Keywords: nanoscale heat transfer, phonon Monte Carlo, modulated nanowires, nanofilms, boundary scattering, transmissivity, porosity (Some figures may appear in colour only in the online journal) 1. Introduction
dimensions in the quantum confinement regime. Much research is also devoted to complex materials where phonon scattering and the thermal conductivity could be controlled by the material composition [3–5]. Porous materials and lowdimensional structures with characteristic dimensions above the quantum confinement regime, show reduced thermal conductivity due to enhanced phonon boundary scattering [6– 10]. Here, we propose an alternative strategy for tuning the thermal conductivity: the use of width-modulated nanowires and nanofilms that are relatively thick and operate in the boundary scattering regime. We report on the possibility for heat flow control by designing the nanostructure width-
Control of the heat transfer at the nanoscale is currently a major technological research issue. It is highly required in order to increase the efficiency of nanoelectronic, optoelectronic, and photovoltaic devices. In thermoelectrics, heat transfer control is particularly demanding because of the requirement for limiting the thermal conductivity without suppressing the electrical conductivity. Phonon bandstructure engineering is one of the strategies proposed for heat manipulation at the nanoscale [1, 2]. Limitations are posed by strict requirements for perfectly periodic structures with 0957-4484/14/465402+09$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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modulation without strict limitations for the modulation profile. This is enabled by the scaling behavior that we found simulating the modulated nanostructures with the Monte Carlo (MC) technique: the reduction of the thermal conductivity scales with the nanostructure transmissivity, a property entirely determined by the modulation geometry, irrespectively of the material choice. We point out that the width-modulation acts like an ‘external porosity’ that could be easier to handle and control by the fabrication technology compared to the ‘internal porosity’. We also report on a novel effect: a very significant constriction thermal resistance due to the width discontinuity has been identified, in analogy to the contact thermal resistance between two dissimilar materials. The constriction thermal resistance scales with the transmissivity. Our conclusions are generic. They indicate that a wide range of materials can be used to develop width-modulated nanowires and nanofilms and directly control the heat flow by designing the width-modulation. Width-modulated nanowires have been previously proposed as efficient thermoelectric materials [11]. It has been theoretically predicted that the transport properties of electrons and phonons are modified resulting in enhanced thermal to electrical energy conversion efficiency. In the quantum confinement regime, the electron transmission coefficient shows resonances, minibands and sharp propagation thresholds and thereby Seebeck coefficient enhancement [11, 12]. Observed thermoelectric power factor enhancement has been attributed to resonance states [13]. In modulated nanowires, electrons occupy energy minibands [14] favoring the thermoelectric efficiency [14–16]. In the quantum confinement regime, the phonon transmission coefficient and phonon energy states are also modified so that the thermal conductivity decreases [17–19]. Above the quantum confinement regime, the thermal conductivity of modulated nanowires has been previously studied within the kinetic theory for phonons and using an approximate model for boundary scattering [20]. A big reduction of the thermal conductivity was found due to enhanced boundary scattering and the decrease of the phonon transmissivity by the nanowire width-modulation. Here, we report on MC simulations of the phonon transport properties of width-modulated nanowires and nanofilms that are free from the approximations used in the kinetic theory model [20]. The exhibited behavior has been interpreted using physics arguments and modeling. General conclusions have been extracted applicable to a wide range of materials. We have chosen Si as reference material, because of its technological importance and because contrary to common strategies that work in the quantum confinement regime and require Si nanostructures with very small dimensions, width-modulated nanostructures that are proposed here are free from this requirement.
resolution of the Boltzmann transport equation (BTE) in the frame of the relaxation time approximation (RTA). Silicon dispersion properties have been used for the sampling of phonon frequency and group velocity, as well as for the calculation of the scattering relaxation times following the model proposed by Holland [23]. 2.1. BTE for phonons
The BTE for phonons is related to the variations of the distribution function f (T , r, p , K) which depends on time t, location r, polarization p and wave vector K. f (T , r, p , K) can also be defined as the mean particle number at time t in the d3r volume around r with K wave vector and d3K accuracy for a given polarization p. In the absence of external force, the BTE expression is [24] ∂f ∂f + K ω ⋅ r f = ∂t ∂t
,
(1)
collisions
with the phonon group velocity vg = K ω and f 0 (T , p , K) the equilibrium thermodynamic phonons population for polarization p and wave vector K, which is described by the Bose–Einstein distribution function f 0 (T , p , K) =
1 e
( ℏω / k B T )
.
(2)
−1
In the frame of the RTA, the right hand side term of equation (1) can be written as follows ∂f ∂t
= collisions
f 0 (T , p , K) − f (T , r , p , K) , τ ( T , p , K, L c )
(3)
where τ is the phonon scattering relaxation time, which depends on the phonon dispersion properties, on the temperature and possibly on a characteristic length Lc. In the frame of MC modeling, the former equations are solved in two steps. Firstly, the advective part of the BTE is addressed (left hand side term of equation (1)) assuming that there is no scattering. Thus, a statistically representative population of phonons is allowed to drift with respect to: (i) its group velocity within the nanostructure, (ii) the internal reflections at pore surfaces and borders. Secondly, the phonon scattering step is achieved considering equation (3). The characteristic relaxation time is given according to the ‘Matthiessen’ summation rule. It takes into account: normal (N), umklapp (U) and impurity (I) scattering rates according to Holland [23] formalism. As mentioned previously, the simulation is frequency dependent. This means that dispersion properties of materials shall be taken into account. Here, we used data (dispersion relation and relaxation time parameters) extensively detailed in previous works for silicon [8, 23]. They are not recalled here.
2. Methodology
2.2. MC procedure
The flowchart of the MC procedure mainly involves successive stages which are reproduced for each time steps of the simulation until the convergence (temperature and flux steady
We have performed MC simulations for the phonon transport using a previously developed simulation tools. The numerical method is extensively described in [8, 21, 22]. It lies on the 2
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nanowire axis is calculated with N
φz =
state) is reached within the nanostructure. First of all, the nanostructure geometry is defined and meshed with rectangular cells of volume V. In some of them tiny volumes are removed in order to obtain constrictions, modulations or porosities (see figures 1 and 5). Once the geometry is known, the temperature gradient at the edges of the nanostructure (first and last cells) is set and the other cells are initially at the lower temperature. On the basis of these prescribed temperatures the energy within each cell can be determined from the Bose–Einstein distribution and the density of states. Then a phonon distribution with different frequencies ω can be randomly sampled in each cell using a cumulative distribution function (CDF) based on the energy [8, 22] until the reference energy at a given temperature is reached
3. Results and discussion It is well known that the thermal conductivity of nanowires and nanofilms is significantly smaller than the thermal conductivity of the corresponding bulk materials when the characteristic dimensions are small. Measurements of the thermal conductivity of Si nanowires and nanofilms with widths ranging from ∼20 nm to ∼100 nm and at temperatures up to 350 K have been reported in [25, 26]. Lower thermal conductivities than in bulk Si have been measured in all nanostructures. These experimental data have been interpreted by several theoretical studies and the measured decrease of the thermal conductivity has been attributed to the predominant boundary scattering [25, 27]. Boundary scattering dominates because the phonon bulk mean free path (∼270 nm at 300 K) is greater than the distance between the nanostructure boundaries. In addition, phonon confinement effects and especially dispersion properties modifications can be neglected at these dimensions in Si. Phonons in Si nanostructures with characteristic dimensions down to ∼20 nm, can be considered as bulk-like and their transport is determined by the bulk material scattering mechanisms and additional boundary scattering. We have performed MC simulations in this transport regime. We report on simulations of the thermal conductivity of nanowires and nanofilms with boundaries separations in the range 20–100 nm. Diffusive boundary scattering has been assumed. At room temperature, the phonon’s wavelength is about 1 nm, depending on the way it is evaluated. Thus, the specularity parameter ( p = exp − 16π 3δ 2 /λ 2 , where δ is the roughness parameter and λ is the wavelength [28]) is close to 0, meaning that every single reflection is diffuse. Nevertheless, Chen et al [29] showed that the reflection is frequency dependent. This dependence is particularly important at low temperatures, when the average phonon’s wavelength can be much larger than the average roughness height (about 2 nm in their work, including the disordered phase). This is not the case here, since all calculations were done at 300 K. Moreover, at room temperature, phonons with large wavelength (low frequency)
N
∑ℏ ω i .
(5)
where V is the volume of the cell and vgzi the group velocity in the z direction (nanowire axis) of the phonon with frequency ωi. The thermal conductivity is thus determined assuming the Fourier formalism. The calculations are achieved on a cluster with a parallel architecture. In order to have an uncertainty in the range 0.5–5% on the thermal conductivity, up two 12 nodes are used and several thousand of phonons are sampled according to the nanostructure size. The time step of the simulation is 1 ps and the steady state is typically reached between 20 and 100 ns according to the nanowire length and the magnitude of the modulations.
Figure 1. Schematics of the reference modulated nanostructures: (a) a periodic sequence of constrictions of width b, and (b) a nonperiodic sequence of constrictions of average width 〈b〉.
th E cell (V , T ) =
1 ∑ℏωi × vgzi , V i=1
(4)
i=1
For each sampled phonon the frequency and the polarization are randomly obtained using the CDF. The group velocity is derived from the dispersion properties knowing ω and p. The initial propagation direction is isotropically sampled. This concludes the initialization procedure. During the drift stage, all the sampled phonons are allowed to move according their propagation direction on a distance equal to vg × Δt , unless they reach a boundary and thus they are diffusely reflected in the case of a nanowire. At the end of the drift procedure, according to the phonon locations the energy in each cell and the corresponding temperature are calculated. This results in a shift from the equilibrium distribution function which is corrected applying the collision procedure. In the latter one, a collision probability is calculated for each phonon on the basis of the relaxation time evaluation at the corresponding values of ω, p and T [8, 21]. A set of random numbers are drawn to determine whether the phonon scatters and if it is the case the nature of the scattering process (U, N, I). In the case of a N process the phonon propagation direction remains the same, if it is U or I process the later one is resampled. Besides, the phonon frequency is resampled in the case of a three phonons process (N or U). The drift and collision stages are repeated until the convergence of the simulation. Once steady state is reached, the phonon heat flux along the
(
3
)
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as
weakly contribute to heat transport, as their density of states is small compared to the transverse acoustic ones close to the edge of the Brillouin zone. Thus, the assumption of complete diffuse scattering at the boundaries remains appropriate as long as the temperature is high enough and the roughness is moderate. The simulation parameters have been optimized to interpret the experimental data on the thermal conductivity of Si nanowires and nanofilms [25, 26]. We consider nanowires and nanofilms with variable widths. In order to investigate the effect of the width-modulation on the thermal conductivity, we consider as reference a non-modulated nanowire/nanofilm of length L with constant width a and depth d. For the nanowires d ∼ a ≪ L; for the nanofilms d ≫ a and d ∼ L. There is no restriction on the choice of the profile shape of the width-modulation. Here, we use as reference the modulation profiles shown in figure 1. The model width-modulation that we use consists in a sequence of constrictions along the direction of heat flow. The cross sections of the wide and narrow segments are assumed orthogonal for the sake of simplicity. Hereafter, we refer to the wide segments as ‘openings’ and to the narrow segments as ‘constrictions’.
transmissivity Tr ≡
constriction cross section . opening cross section
(6)
For the geometry of figure 1(a), this reduces to the ratio of the two characteristic dimensions of the width modulation Tr =
b . a
(7)
Moreover, a simple scaling behavior has been identified by the analysis of our simulation data: ‘the ratio of the thermal conductivity of the modulated nanowire over that of the reference nanowire scales with the transmissivity’. This behavior is shown in figure 2(b) and is quantitatively expressed by the relation k modulated wire ≈ Tr. k wire
(8)
The nanowire transmissivity, as defined in equation (6), expresses the percentage of the nanowire’s geometry modulation. This definition also applies for the nanofilms (figure 1(a)). The simulations data for the nanofilms are shown together with those of the nanowires in figure 3. It can be seen that the same scaling behavior holds for the nanofilms as for the nanowires. In what follows, we present the simulation results for nanowires with d = a. Nevertheless, the obtained evidence and interpretation also hold for the nanofilms with d ≫ a. Equation (8) can be interestingly interpreted as that: the percentage of the width modulation is equal to the percentage of the thermal conductivity reduction, i.e. to the heat flow transmissivity. It thereby becomes evident that ‘direct control on the heat flow by the nanowire/nanofilm width-modulation takes place’. Τhe same scaling behavior holds for the nanofilms as for the nanowires when the heat flow is normal to the widthmodulations. In the case of nanofilms, if the wide and narrow sections are collinear to the thermal gradient, the in-plane modification of the thermal conductivity should not be the same as in the present case where modulated sections are perpendicular to the heat flux. Considering a simple electrical analogy, those two cases correspond to parallel and serial association of thermal resistances respectively. Cross-plane thermal conductivity assessment for modulated thin films would require different treatment because hot and cold boundaries are not the same for the whole film. The above scaling behavior holds for ‘adequate’ modulation density for which the heat flow density is quite uniform along the nanowire. The effect of the modulation density on thermal conductivity is shown in figure 4 and it can be modeled by introducing an offset C in equation (8)
3.1. Modulated nanowires and nanofilms
We start our discussion with a periodic sequence of modulating constrictions, as shown in figure 1(a). Each constriction has width b, length c and depth d. The length of the constriction c is assumed much smaller than the constriction width. The data shown here are for c = 10 nm. The MC simulations uncertainty is in the range of 0.5–5% for the thermal conductivity. In figure 2 are shown the simulated thermal conductivities of nanowires with representative dimensions: a ‘thick’ nanowire of width 100 nm and a ‘thin’ nanowire of width 60 nm and for d = a. A weak dependence on the constriction length c was found for c varying in the range 10–20 nm and for constant dimensions of the openings. The trend is that the thermal conductivity decreases linearly with decreasing b. Our data show that the decrease of the thermal conductivity can be very significant. In figure 2(a), a decrease of ∼10 times is shown for the smallest values of b considered. The thermal conductivity decrease is more pronounced for the thicker nanowire. The thermal conductivity of the thick nanowire is lower than that of the thin nanowire for same b. This behavior was also found in a previous theoretical study of the thermal conductivity of diameter modulated nanowires within the kinetic theory model where geometry-dependent boundary scattering was used [20]. Two main assumptions were used in [20] for boundary scattering: (i) the variable nanowire width was taken into account by a geometrical averaging of the boundary scattering length, and (ii) the effect of the non-uniform phonon propagation along the nanowire was taken into account by the transmissivity. The qualitative agreement with the present MC simulation, that is free from the above two assumptions, validates the assumptions of [20]. Interestingly, both models indicate that the size dependence of the thermal conductivity can be interpreted in terms of the nanowire transmissivity Tr, defined
k modulated wire = (1 − C ) Tr + C . k wire
(9)
For a decreasing modulation density, the thermal conductivity increases. The offset C increases correspondingly. The width modulation becomes, then, less efficient in decreasing the thermal conductivity. A more detailed discussion is devoted 4
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Figure 2. Scaling behavior of the thermal conductivity. (a) The thermal conductivity of two periodically modulated nanowires with a = d, and a = 100 nm (red squares) and a = 60 nm (blue dots) versus the constriction width b. (b) The ratio of the thermal conductivity of the periodically modulated nanowire over the thermal conductivity of the non-modulated nanowire versus the transmissivity. (c) and (d) are as (a) and (b) respectively with additionally shown the corresponding data for non-periodically modulated nanowires with average width b (stars). The length of the constriction is c = 10 nm.
Figure 4. The effect of the modulation density. The ratio of the
Figure 3. The scaling behavior of the thermal conductivity for
thermal conductivity of the modulated nanowire over that of the reference nanowire versus the transmissivity for the constriction density (squares) in figure 2(a), for the double (triangles) and the half density (dots). The length of the wires is kept constant.
modulated nanowires and nanofilms. For nanowires (solid symbols): d = a. For nanofilms (stars) d = 1 μm.
to this dependence in the following, when we investigate the thermal conductivity of a nanowire modulated by a single constriction. We have, so far, assumed a periodic modulation. In this way, we have been able to identify a single ratio of the dimensions of the periodic modulation that explains the
reduction of the thermal conductivity. We would expect that this behavior should not be bound to the periodic modulation profile. We are in the particle-like regime, where phonon states are not related to the presence of periodicity in the boundary conditions. Indeed, our simulations showed that ‘it 5
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Figure 5. Schematics of a width-modulated nanowire and the
corresponding porous nanowire that are discussed in the main text.
is still possible to relate the reduction of the thermal conductivity to the reduction of the transmissivity of nanowires/ nanofilms that are modulated non-periodically’. This becomes evident in figures 2(c) and (d). By dots and squares are shown the data for periodic modulations and by stars the data for non-periodic modulations (figure 2(d)). In the nonperiodic modulation simulations, the width of the openings and the density of the modulation were kept constant. The widths of the constrictions were varied randomly. The thermal conductivity is plotted at the average constriction width. Now, the scaling behavior although blurred, is still shown as a clear trend: ‘the ratio of the thermal conductivity of the modulated nanowire/nanofilm over that of the reference nanostructure follows the linear reduction of the transmissivity’. In this case, a nanowire/nanofilm transmissivity can be determined by the average constriction cross-section transmissivity Tr ≡
constriction cross section . opening cross section
Figure 6. Single constriction modulation. The thermal conductivity
of a nanowire modulated by a single constriction versus the nanowire length for various values of the constriction width b (squares). The length of the constriction is c = 10 nm. Periodically modulated nanowires with the same parameters a and b (dashed lines).
physical quantity as the transmissivity of the width-modulated nanostructures, introduced here in equation (6). It could be also useful to point out that the nanowire width-modulation can be seen as an ‘external porosity’ since the ‘missing’ volumes of the cavities are located at the surface of the modulated nanowires. The use of porous nanostructures is commonly proposed to achieve thermal conductivity reduction. ‘The same thermal conductivity reduction should be achieved using width-modulated nanowires with ‘external porosity’ equivalent to the ‘internal porosity’ of porous nanowires’. This conclusion could be useful because widthmodulated nanowires/nanofilms (‘external porosity’) should be much easier to handle and control by the nano-fabrication technologies than the corresponding porous nanostructures (‘internal porosity’).
(10)
The above evidence points out that by designing the average width modulation, the thermal conductivity can be tuned and the heat flow can be controlled. ‘Very importantly, it is pointed out that there are no posed strict requirements for the geometry modulation profile: it would be sufficient to use a profile with average width corresponding to the transmissivity required for a desired thermal conductivity reduction. Moreover, our simulations indicate that more efficient structures should emerge when thick nanowires are modulated’. These results provide technology guidelines for heat flow control at the nanoscale. We have compared the thermal conductivities of the width-modulated nanowires and of porous nanowires with pores of the same shape and dimensions with the ‘missing’ volumes at the surface of the modulated wires (figure 5). It should be noticed that the change in the volume of the modulated nanowires by the width-modulations at their surface is the same as that of the corresponding porous nanowires where the pores are located inside their volume. We found quantitative agreement for the thermal conductivities of the two types of nanowires. The calculated thermal conductivities for the porous nanowires agree well with the data shown in figure 2 within the simulations accuracy. The porous nanowires also showed the same scaling behavior of the thermal conductivity with the transmissivity. It is well known that in porous materials, the reduction of the thermal conductivity can be directly related to the porosity [8, 10] defined as the ratio of the involved volumes: (Vnanowire − Vcavities)/ Vnanowire. The porosity expresses the percentage of the volume modulation of the material and should be conceived as similar
3.2. Nanowires with a single constriction
We now discuss the case of a single width modulation, namely a ‘constriction’. The thermal conductivity decreases with the decrease being more important with decreasing constriction width b (figure 6). The thermal conductivity shows a weak dependence on the constriction length c for the considered range of values. The thermal conductivity of the modulated nanowire depends on the wire length as is shown in figure 6. This dependence can be interpreted by two effects: (i) the decrease of the thermal conductivity with decreasing nanowire length, and (ii) the additional decrease of the thermal conductivity by the constriction thermal resistance. Concerning the effect (i). The thermal conductivity of a nanowire increases when its length increases and saturates to a certain value when fully diffusive transport takes place. This is well known behavior of nanowires that is also found here for the modulated nanowires. The length dependence is shown in figure 6 with black squares for the non-modulated 6
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nanowires, and with colored squares for the modulated nanowires. The thermal conductivity of a modulated nanowire is lower than that of the corresponding non-modulated nanowire of the same length. It decreases with decreasing b, as expected. When the constriction deformation is shallow, the decrease of the thermal conductivity is small and its length dependence is very similar to that of the non-modulated nanowire. A steeper length dependence of the thermal conductivity is shown for bigger values of b. This effect is characteristic for a transition regime between two kinds of geometries. First, if we look at the nanowire without constriction (black squares), the expected decrease of the thermal conductivity with increasing wire length L is shown. For a long wire and shallow modulation (as for a = 100 nm and b = 90 nm shown with red squares), the constriction has a weak effect on the nanowire thermal conductivity as it is a local effect on phonon boundary scattering. The dominant characteristic size of the nanowire is the length ‘a’. For nanowires with a deep modulation (as for a = 100 nm and b = 20 nm shown with pink squares), there is a ‘blocking’ effect on the phonon transport which is more significant for small length L for the reasons discussed above. In this case, the dominant characteristic size is ‘b’. A transition between the above two regimes is shown at intermediate dimensions as e.g. for b = 60 nm, a = 100 nm shown with blue squares. The constriction has a weak effect on the heat flow for long nanowires where c/L is small, whereas it has a greater impact for short modulated nanowires where c/L is larger. Furthermore, analyzing our data we have concluded that this behavior as well as the actual values of the thermal conductivity for each value of b, can be interpreted in terms of the constriction thermal resistance (effect (ii)). Concerning the effect (ii). A thermal resistance can be attributed to the constriction originating from the width discontinuity. The constriction thermal resistance increases with decreasing constriction cross section discontinuity and it is nearly independent of the wire length. This is evident in the temperature profiles along the wires shown in figure 7 where modulated nanowires of three lengths are depicted. It can be thereby concluded that ‘the identified thermal resistance is a property of the constriction itself’. For the largest value of b (shallow deformation), the temperature drop is nearly linear implying a uniform nanowire thermal conductivity. With increasing b, the linear drop starts to deform and eventually shows a discontinuity for small b (deep deformation). The temperature gradient on both sides of the discontinuity shows a small length-dependence. For short nanowires, the temperature jump close the discontinuity is larger than for long nanowires because of the short distance from the hot and cold contact boundaries of the structure. This is no longer true when the nanowire length increases. For a sufficiently long nanowire the temperature jump remains constant, solely depending on the constriction cross-section discontinuity. The discontinuity implies a non-uniform thermal conductivity. The opening is the signature of ballistic transport through the constriction and resembles to the Kapitza resistance at the interface of two dissimilar materials [30, 32]. ‘It can be very importantly concluded that the width
Figure 7. The evolution of the temperature profile with increasing modulation depth. The temperature profile along the nanowire for three values of b and for three nanowire lengths. Temperature is plotted at the center of each cell. All cells have the same length.
discontinuity acts like a contact thermal resistance’. The origin of the thermal resistance is the heat flux transmission discontinuity due to the width (cross-section) discontinuity. This is in analogy to the Kapitza resistance at the interface of two dissimilar materials that is due to the heat flux discontinuity due to the density of states discontinuity [30, 31]. The magnitude of the contact thermal resistance and its effect on the overall effective conductivity of the nanowire are estimated below. The constriction thermal conductance has been calculated directly by the simulation data and is shown in figure 8(a) by solid symbols versus the width b. The data can be interpreted by the approximate relation th R mw = R cth + R wth ,
(11)
th where Rmw , Rcth , Rwth are the thermal resistances of the modulated nanowire, of the constriction and of the nonmodulated nanowires respectively. The above relation is 7
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Figure 8. Scaling behavior of the constriction thermal conductance. The constriction thermal conductance (squares) versus (a) the constriction width b, and (b) the transmissivity. The inset shows the linear relation between the constriction thermal conductance and (1 − Tr)−1 as predicted by equation (13) of the main text. In (a), the data using equation (13) of the main text are also shown with stars.
written as
Deviations are found for very shallow deformation when the constriction thermal resistance is very small. The steeper length dependence of the thermal conductivity of the modulated nanowires by a single constriction compared to that of the non-modulated nanowire is explained by the additional contribution of the constriction thermal resistance (equations (11) and (12)) that becomes more significant when the nanowire length decreases. In the limiting case where the length of the nanowire is comparable to the constriction modulation unit, the overall conductance is nearly equal to the constriction thermal conductance. Kc explains the minimum values of the thermal conductivity for the various values of b in figure 6. For adequate modulation density, the heat flow density is uniform along the wire. A linear drop of the temperature along the wire has been found in the simulations. In this regime, the reduction of the thermal conductivity of the width-modulated nanowires is interpreted by the transmissivity. For a single constriction, the reduction of the thermal conductivity of the modulated nanowire is interpreted by the constriction thermal conductance. Kc decreases rapidly when the modulation becomes deeper (figure 8(a)) and causes the more rapid decrease of the overall thermal conductivity shown in figure 6. The temperature profile shown in figure 7 indicates that the heat flow density is non-constant in this case. The single constriction is less efficient in ‘guiding’ the heat flow and reducing the thermal conductivity compared to a sequence of constrictions. This is shown in figure 6 where the data for the thermal conductivity of the periodically modulated nanowires (dashed lines) are plotted together with the data for the single constriction modulation for comparison. The thermal conductivity of the nanowires modulated by a single constriction becomes smaller than that of periodically
⎛ L left L mw L wright ⎞ 1 ⎟⎟ , (12) = + ⎜⎜ w left + A w k mw A w k wright ⎠ ( A w − A c ) Kc ⎝ A w k w
where k mw , k w , are the thermal conductivities of the modulated wire and of the non-modulated wire respectively. Kc is the constriction thermal conductance. Aw and Ac respectively denote the cross sections of the wire and of the constriction. Lmw, L wleft(right) , are the lengths of the modulated wire and of the left (right) segment of the wire with constant thickness a. For long nanowires, it can be assumed that L ≡ L mw ≈ L w = L wleft + L wright . Then, equation (12) gives the following expression for the constriction thermal conductance
(
Kc =
)
1 ⎡ 1 ⎛ k mw k w ⎞ ⎤ ⎢ ⎜ ⎟ ⎥, 1 − Tr ⎣ L ⎝ k w − k mw ⎠ ⎦
(13)
where Tr = Ac /Aw (also see equation (6)). We have used equation (13) to estimate the constriction thermal conductance. The results are shown in figure 8(a) by stars. It can be seen that they agree well with the results of the MC (squares). This agreement validates relations (11) and (12). Furthermore, equation (13) indicates that ‘the constriction thermal conductance scales with the transmissivity Tr’. In particular, it turns out that Kc ∝
1 . 1 − Tr
(14)
This dependence is predicted by equation (8) and is confirmed by our MC simulations shown in figure 8(b). The inset in figure 8(b) shows that equation (9) is followed well. 8
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References
modulated nanowires only for short nanowires because the constriction thermal conductance has, in this case, a bigger weight in the resistances network (equations (11) and (12)).
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4. Conclusions We have performed MC simulations on the heat flow in width modulated nanowires and nanofilms and we provided physics interpretation and modeling of the simulation results. Our work indicates that width-modulated nanowires/nanofilms that are relatively thick and operate in the boundary scattering regime could be used to tune the thermal conductivity. We have made evident the possibility for heat flow control by designing the width-modulation without strict limitations for the modulation profile. This is enabled by the identified scaling behavior: the reduction of the thermal conductivity scales with the transmissivity that is entirely determined by the modulation geometry irrespectively of the material choice. More efficient structures should emerge when thick nanowires/nanofilms are modulated. We have additionally pointed out that the width modulation acts like an ‘external porosity’ that could be easier to handle and control by the fabrication technology. We have also identified a novel effect: a very significant constriction thermal resistance, in analogy to the Kapitza contact resistance. The origin of the constriction thermal resistance is the width discontinuity and it scales with the transmissivity. Our conclusions are generic. We have used here Si as reference material. Our numerical method can be applied to several materials for which phonons are the main heat carriers, mainly insulating materials and semi-conductors like Ge, GaAs, InSb, C, etc, as long as dispersion properties and relaxation times are known. The obtained scaling behavior of the thermal conductivity with the modulation geometry should also hold for these materials. The sizelimitations on characteristic dimensions of the modulation geometry may vary in different materials. Our work indicates that a wide range of materials can be suitable for efficient nanoscale heat management by designing the nanostructure width modulation.
Acknowledgments XZ acknowledges invited professorship of the University of Lorraine (LEMTA), the project ‘SiMoNE’, PEPS Mirabelle 2013 financed by CNRS and the University of Loraine, and the European Social Fund (ESF)-European Unions and National Resources within the framework of the Grant of Excellence ‘ARISTEIA’.
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![On the scaling behavior of electric conductivity in [C4mim][NTf2].](/assets/img/hold.png)
