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Ballistic thermal transport by phonons in three dimensional periodic nanostructures
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 095303 (http://iopscience.iop.org/0953-8984/27/9/095303) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 05/05/2017 at 02:54 Please note that terms and conditions apply.
You may also be interested in: The influence of flaws on phonon thermal conductance Yan Wang, Ling-Ling Wang, Li-Ming Tang et al. Acoustic phonon transport through a quantum waveguide Hai-Yan Zhang, Hong-Jian Li, Wei-Qing Huang et al. Acoustic phonon transport and thermal conductance in quantum wire Li-Ming Tang, Lingling Wang, Wei-Qing Huang et al. Acoustic phonon transport through a T-shaped quantum waveguide Wen-Xia Li, Ke-Qiu Chen, Wenhui Duan et al. Ballistic phonon transmission in a symmetric converging–diverging contact of a semiconductor nanowire Ming-Liang Zou, Wei-Qing Huang, Gui-Fang Huang et al. Phonon transport and thermal conductivity in quantum wire Wen-Xia Li, Ke-Qiu Chen, Wenhui Duan et al. Selective transmission and enhanced thermal conductance of ballistic phonon by nanocavities embedded in a narrow constriction Wei-Qing Huang, Bai-Yun Huang, Dan-Qing Yi et al. Thermal transport by phonons in zigzag graphene nanoribbons with structuraldefects Zhong-Xiang Xie, Ke-Qiu Chen and Wenhui Duan Thermal transport of isotopic-superlattice graphene nanoribbons with zigzag edge T. Ouyang, Y. P. Chen, K. K. Yang et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 095303 (6pp)
doi:10.1088/0953-8984/27/9/095303
Ballistic thermal transport by phonons in three dimensional periodic nanostructures Zhong-Xiang Xie1 , Yong Zhang1 , Xia Yu1 , Hai-Bin Wang1 , Ke-Min Li2 , Chang-Ning Pan3 and Qiao Chen4 1 Department of Mathematics and Physics, Hunan Institute of Technology, Hengyang 421002, People’s Republic of China 2 College of Physics and Eletronics, Hunan Institute of Science and Technology, Yueyang 414006, People’s Republic of China 3 School of Science, Hunan University of Technology, Zhuzhou 412000, People’s Republic of China 4 Department of Maths and Physics, Hunan Institute of Engineering, Xiangtan 411101, People’s Republic of China
E-mail: [email protected] and [email protected] Received 22 October 2014, revised 22 January 2015 Accepted for publication 28 January 2015 Published 18 February 2015 Abstract
Ballistic thermal transport properties by phonons in three dimensional (3D) periodic nanostructures is investigated. Results show that thermal transport properties in 3D periodic nanostructures can be efficiently tuned by modulating structural parameters of systems. When the incident frequency is below the first cutoff frequency, the quasi/formal-periodic oscillations of the transmission coefficient versus the periodic number/length can be observed. When the incident frequency is above the first cutoff frequency, however, these quasi/formal-periodic oscillations cannot be observed. As the periodic number is increased, the thermal conductance undergoes a prominent transition from the decrease to the constant. We also observe other intriguing physics properties such as stop-frequency gaps and quantum thermal conductance in 3D periodic nanostructures. Some similarities and differences between 2D and 3D periodic systems are identified. Keywords: thermal transport, phonon, quantum structures (Some figures may appear in colour only in the online journal)
transport [23], thermal rectification [24], negative differential thermal resistance [25], and so on, have been revealed recently. These imply that phonons can be manipulated like electrons, thus enabling controlled and managed heat transport in physically realistic nanosystems. Recently, the periodic system has been a topic of significant interest due to its potential application in thermoelectric energy conversion and thermal management. Using periodic structures (such as phononic crystals and superlattices) to control the thermal transport by manipulating coherent phonons, has been proposed theoretically and experimentally. Luckyanova et al presented a study of thermal conductance through the finite-thickness superlattices with varying numbers of periods, and found the thermal conductivity increased linearly with increasing the thickness in the temperature range from 30 to 150 K, confirming the existence of the coherent phonon transport [26]. Nika
1. Introduction
In recent years, the thermal transport in low-dimensional nanoscale systems has attracted much attention [1–6]. Different models and theories have been proposed to explore the thermal transport mechanisms. It has been demonstrated that when the feature sizes of structures become comparable to the phonon mean free path, the heat current is mainly carried by a series of discrete vibrational modes due to the boundary confinement. The thermal transport properties in a variety of nanostructures including nanowires [7–9], nanofilms [10], nanotubes [11–13], nanoribbons [14–18], have been reported. Enormous effort has been devoted to elucidate the effects of strain [19], rough surfaces [20, 21], and structural defects [22] on thermal transport properties of nanoscale structures and materials. Similar to the electronic transport, several intriguing thermal properties such as nonlinear thermal 0953-8984/15/095303+06$33.00
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Figure 1. Schematic illustration of 3D periodic nanostructures adiabatically connected to the left and right thermal reservoirs with temperatures being denoted by TL and TR , respectively. (a) represents the T-shaped periodic nanostructure with the periodic length L = T 1 + T 2, and (b) corresponds to the concavity-shaped periodic nanostructure with the period length L = C1 + C2. Here, the periodic number N = 5.
insight into the thermal conductance in 3D systems [34]. They predicted that the thermal conductance in 3D systems is larger than that in 2D systems even for the same structural parameters. In this work, using the scattering method in combination with the isotropic elastic continuum model, we study the thermal transport properties by phonons in 3D periodic nanostructures. Two different periodic manners are considered here: T-shaped and concavity-shaped periodic nanostructures.
et al [27] and Haskins et al [28] predicted that onedimensional periodic quantum-dot can be served as effective thermoelectric materials and thermal insulators due to the shape reduction of lattice thermal conductivity caused by the interface mismatching, respectively. Several recent studies have focused on the thermal conductivity in realistic periodic structures such as Si/Ge superlattices [8, 29], periodic silicon nanoporous films [30], isotopic-superlattice graphene nanoribbons [31], isotopic-superlattice structured Si nanowires [32], and so on. Among these studies, an important and long-standing discovery across these periodic systems is the existence of a minimum in the thermal conductivity for a finite periodic length. It was proved that this optimal periodic length is associated with the transition between wave-like and particle-like transport behaviors of phonons. More recently, we also investigated the transmission spectrum and thermal conductance through two-dimensional (2D) superlattice quantum-waveguides, and observed the stopband formation depending upon the periodic length [33]. In spite of these advancements in the study of thermal transport in 2D periodic films and 1D periodic nanowires, the thermal transport mechanisms in three dimensional (3D) periodic nanostructures, along with the effects of important parameters (such as the periodic number, the periodic length as well as the periodic manner) on thermal transport properties in 3D periodic nanostructures, are not thoroughly understood. It has become well known that there exist four types of acoustic modes as well as two shear modes in 3D systems, different from 2D systems where the vibrational modes are split into two independent modes: the in-plane modes and the out-of-plane modes. Recently, some researchers have attempted to provide
2. Model and formalism
We model the geometries, as illustrated in figure 1. (a) corresponds to the T-shaped periodic nanostructure with the periodic length L = T 1 + T 2, and (b) corresponds to the concavity-shaped periodic nanostructure with the periodic length L = C1 + C2. Each structure consists of two pristine semi-infinite parts adiabatically connected the left and right thermal reservoirs with the temperature being denoted by TL and TR , respectively, and the periodic part that consists of alternating blocks with different sizes. In both periodic nanostructures, the material composition of any blocks is GaAs. Note that the GaAs nanosystems have attracted much attention because of their intriguing physical properties and potential application [34]. We assume that the thermal contacts between two leads and thermal reservoirs are perfect, and only the phonon scattering occurs at the interfaces between two adjacent blocks with different sizes. Now, several methods such as the isotropic elastic continuum model [33, 34] and the lattice model [16–18], have been developed to study the thermal transport by phonons. Note that when the typical phonons have the wavelength comparable to the atomic bond 2
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length, the isotropic elastic continuum model may be incorrect quantitatively. However, it is a rather good approximation for describing the thermal transport in nanostructures at very low temperatures [6]. Considering the fact that our work mainly focuses on the acoustic phonon thermal transport in low temperature range where only several lowest modes can be excited. The wavelength of these phonon modes is generally much larger than any microscopic length including the periodic length L considered here. Thus, in this work we can employ the isotropic elastic continuum model to describe the phonon transmission. According to this model, the phonon displacement ψ(x, y, z) in the cartesian coordinate satisfies the following equation v 2 ∇ 2 ψ(x, y, z) + ω2 ψ(x, y, z) = 0,
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where ∇ = ∂ /∂x + ∂ /∂y + ∂ /∂z , v and ω represent the sound velocity and frequency of acoustic phonons, respectively. The solution to equation (1) in each block ξ (ξ is the label of each block in the periodic part) can be given by 2
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(2) ξ+ ξ− where Pmn and Pmn with +(−) being the forward (backward) propagating direction along the z axis, are constants to be ξ determined by matching the boundary condition. φmn (x, y) = ξ ξ ξ ξ φm (x)φn (y), where φm (x) and φn (y) represent the orthogonal wave function along the x and y directions, respectively. Under the stress-free boundary condition, we can obtain 2 cos( mπ x) m = 0 Xξ Xξ ξ (3a) φm (x) = 1 m=0 Xξ 2 cos( nπ y) n = 0 Yξ Yξ ξ φn (y) = . (3b) 1 n=0 Yξ
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Figure 2. (a) and (b) describe the transmission coefficient τ as a function of the incident frequency ω/ ( = πv/X1) for the T-shaped and concavity-shaped periodic nanostructures, respectively. Curves a–c correspond to N = 3, 6, and 14, respectively, and the dashed curve in (a) corresponds to the pristine case. Note that two consecutive curves are vertically offset two units for clarity. Here, X1 = 10 nm, Y 1 = 10 nm, Y 2 = 15 nm, T 1 = T 2 = 10 nm in (a), and X1 = 10 nm, Y 1 = 10 nm, Y 2 = 5 nm, C1 = C2 = 10 nm in (b).
Here, Xξ represents X1, while Yξ is Y 1 or Y 2 depending on the size of the corresponding block along the y direction. In ξ equation (2), kmn is the wave vector along the z axis, and can be expressed as ξ = ω2 /v 2 − m2 π 2 /Xξ2 − n2 π 2 /Yξ2 . (4) kmn
3. Numerical results and discussion
Figures 2(a) and (b) show the transmission coefficient τ as a function of the reduced frequency ω/ ( = π v/X1) for the T-shaped and concavity-shaped periodic nanostructures, respectively. Curves a–c in each figure correspond to the periodic number N = 3, 6, and 14, respectively, and the dashed curve in figure 2(a) corresponds to the pristine case. Clearly for the pristine case, τ exhibits a series of smooth platforms, corresponding to the available phonon channels. Note that the altitude intercept between two adjacent platforms is either 1 or 2, which is different from 2D systems. When the periodic structure is introduced, τ is sharply decreased due to the strong interface scattering. In the limit ω → 0, however, τ tends to be the ideal value, since an acoustic phonon at ω = 0 can retain the perfect transmission. This is in agreement with the case in the 2D systems [33], but different from the electron transport. As also seen in figure 2, an outstanding feature is that some stop-frequency gaps can be
ξ
For the open channel, kmn is real. Otherwise, we take ξ ξ kmn = i|kmn |. By applying the continuum condition of the displacement and stress at each interface between any adjacent blocks, and using the scattering-matrix method [34], we can calculate the transmission coefficient τmn . Then, the corresponding thermal conductance κ by phonons can be given by ∞ ω2 eβ¯hω h ¯2 1 τ (ω) dω, (5) κ= mn kB T 2 m,n 2π (eβ¯hω − 1)2 ωmn
where β = 1/(kB T ) with kB being the Boltzmann constant, h ¯ is the Planck’s constant, ωmn is the cutoff frequency of the (m, n) mode. In following numerical calculations, we will employ these values of the elastic constants and mass density of GaAs referred in [33]. 3
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Figure 3. (a1)–(a3) show the transmission coefficient τ versus the periodic length L for the incident frequency ω/ = 0.05, 0.5, and 1.2, respectively, and (b1)–(b3) show the transmission coefficient τ versus the periodic number N for the incident frequency ω/ = 0.05, 0.5, and 1.2, respectively. In each figure, the solid and dashed curves correspond to the T-shaped and concavity-shaped periodic nanostructures, respectively. Here, we take N = 8 in (a1)–(a3), and L = 20 nm in (b1)–(b3). Other parameters are the same as those in figure 2.
incremental oscillation as the periodic length L increases (figures 3(a1) and (a2)). When the incident frequency is above the first cutoff frequency (ω/ > 1), however, a similar oscillation cannot be obviously observed (figure 3(a3)). On the other hand, τ versus N exhibits the formal-periodic behavior, together with numerous peak-valley when ω/ < 1. This periodic behavior can be understood as a result of the wave interference. When the incident wave enters into the periodic region, one part can go through the periodic region, and the other part can get back due to the interface scattering. The two parts interfere with each other coherently, thus resulting in the periodic behavior. During the interfering process, when the phase difference is an even multiple of π , τ will be resonantly reinforced. Inversely, when the phase difference is an add multiple of π , τ will be strongly lowered. As a result, a periodic peak-valley behavior can be observed in the behavior of τ versus N . Examining figures 3(b1) and (b2), it is seen that the period of τ (N ) reduces as the frequency is increased. However, when the frequency is further increased (figure 3(b3)), τ versus N loses the periodic feature. In figure 3, we also note that for ω/ < 1, τ versus the structural parameters is similar in characteristic for both periodic structures, while for ω/ > 1, it is absolutely different for different periodic structures. Next, we turn to study the thermal conductance in both periodic nanostructures. Figure 4 shows that the thermal conductance reduced by the quantum value π 2 kB2 T /3h, as a function of the reduced temperature kB T /¯h . Figures 4(a) and (b) describe the thermal conductance κ0 of the zero mode for
opened at particular frequencies. Moreover, when the periodic number N is increased, the number of stop-frequency gaps increases. These stop-frequency gaps indicate that the phonon transmission can be completely forbidden in both periodic structures. The antiresonant coupling between the incident and reflected waves may be responsible for the appearance of the stop-frequency gaps. Generally, the more the periodic number N is, the stronger the coupling becomes. As a consequence, the number of the stop-frequency gaps is increased with increasing N in both periodic structures. Comparing figure 2(a) with figure 2(b), we can see that the width of stop-frequency gaps in the T-shaped periodic nanostructure is narrower than that in the concavity-shaped periodic structure. In addition, τ in the T-shaped periodic nanostructure is always higher than the corresponding value in the concavity-shaped periodic nanostructure. These differences can be attributed to the fact that the T-shaped periodic nanostructure can provide more channels to facilitate the phonon transmission as compared to the concavity-shaped periodic nanostructure, although the number of modes excited in the incident lead is the same for both periodic nanostructures. In order to further reveal the phonon transmission properties, we describe the transmission coefficient as a function of the structural parameters such as the periodic length L and the periodic number N , as shown in figure 3. We find that τ versus L displays different behaviors for different incident frequencies. When the incident frequency is below the first cutoff frequency (ω/ < 1), τ presents the quasi-periodic 4
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Figure 4. The thermal conductance as a function of the reduced temperature kB T /¯h . (a) and (b) denote κ0 of the zero mode for the T-shaped and concavity-shaped periodic nanostructures, respectively, while (c) and (d) correspond to the total thermal conductance κt for them, respectively. The solid, dashed, and dot–dashed curves in (a)–(d) correspond to N = 3, 6, and 14, respectively, and the dotted curves in (a) and (c) correspond to the pristine case. Other parameters are the same as those in figure 2.
the T-shaped and concavity-shaped periodic nanostructures, respectively, while figures 4(c) and (d) correspond to the total thermal conductance κt for them, respectively. For the pristine case, the quantum platform of κ0 is clearly observed, while for both periodic nanostructures, this quantum platform cannot be observed. In the zero temperature limit, however, κ0 approaches the quantum value π 2 kB2 T /3h regardless of the variation of the periodic number N. This is consistent with the isotopic-superlattice graphene nanoribbons [31] and 2D superlattice quantum-waveguides [33]. Comparing figure 4(a) with figure 4(b), some distinct differences of κ0 between both periodic nanostructures can be identified as follows: (i) κ0 for the T-shaped periodic nanostructure firstly decreases to a minimum value, then increases gently, while that for the concavity-shaped periodic nanostructure monotonously decreases as the temperature is increased; (ii) at high temperature, κ0 for the T-shaped periodic nanostructure is more sensitive to the variation of N , compared to the concavity-shaped periodic nanostructure; (iii) κ0 for the Tshaped periodic nanostructure is higher than the corresponding value for the concavity-shaped periodic nanostructure. These differences indicate that the thermal conductance in 3D periodic nanostructures sensitively depends upon the periodic manner. It is the decrease of κ0 that results in the decrease of κt in the low temperature region, where only the zero mode can be excited (figures 4(c) and (d)). This behavior seems to be in agreement with the experimental result qualitatively [3].
With increasing the temperature (kB T /¯h > 0.1), all the κt for both periodic nanostructures gradually increase, since higher order modes are successively excited to conduct heat. Note that the thermal conductance of higher order modes monotonously increases from the zero value due to their nonzero cutoff frequencies (not shown), different from the decreasing behavior of the zero mode presented here. Finally, we show the total thermal conductance κt as a function of the periodic number N for three given temperatures in figure 5. It is seen that the curves in κt versus N for both periodic nanostructures are similar in characteristics. The value of κt firstly decreases from the maximum value, then becomes nearly independent of N . In other words, when N is gradually increased, κt undergoes a noticeable transformation from the decrease to the constant. This transformation can be attributed to structural features of 3D periodic nanostructures. When N is relatively small, increasing the N will lead to the augment of the number of scattering interfaces, thus resulting in the reduction of κt . However, when N is increased to a certain extent, the centrally periodic part can be regarded as an integrated whole. So the transformation can occur in the behavior of κt versus N . A similar transformation was also recently observed in 2D superlattice wave-guides [33]. Comparing figures 5(a) with (b), it is found that the constant of κt for the T-shaped periodic nanostructure is larger than the corresponding value for the concavity-shaped periodic nanostructure, in agreement with 5
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Acknowledgments
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This work is supported supported by the National Natural Science Foundation of China under Grant (Nos. 11404110 and 11204074), the Nature Science Foundation of Hunan Province under Grant (Nos. 14JJ3139 and 2015JJ2050), the Outstanding Young Program of Hunan Provincial education department of China under Grant (No. 14B046), and the Key Research Program of Hunan provincial education department, China (Grant No. 12A059).
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References [1] Cahill D G et al 2014 Appl. Phys. Rev. 1 011305 [2] Li N B, Ren J, Wang L, Zhang G, Hangg P and Li B 2012 Rev. Mod. Phys. 84 1045 [3] Dubi Y and Ventra M D 2011 Rev. Mod. Phys. 83 131 [4] Marconnet A M, Panzer M A and Goodson K E 2013 Rev. Mod. Phys. 85 1295 [5] Balandin A A 2011 Nat. Mater. 10 569 [6] Wang J S, Wang J and Lu J T 2008 Eur. Phys. J. B 62 381 [7] Chen J, Zhang G and Li B 2012 Nano Lett. 12 2826 [8] Hu M and Poulikakos D 2012 Nano Lett. 12 5487 [9] Chen J, Zhang G and Li B 2011 J. Chem. Phys. 135 104508 [10] Hua Y C and Cao B Y 2014 Int. J. Heat Mass Transfer 78 755 [11] Chen X B, Xu Y, Zou X L, Gu B L and Duan W H 2013 Phys. Rev. B 87 155438 [12] Chen J, Zhang G and Li B 2010 Nano Lett. 10 3978 [13] Wang J, Li L and Wang J S 2011 Appl. Phys. Lett. 99 091905 [14] Huang H Q, Xu Y, Zou X L, Wu J and Duan W H 2013 Phys. Rev. B 87 205415 [15] Yu C X and Zhang G 2013 J. Appl. Phys. 113 044306 [16] Xie Z X, Chen K Q and Duan W H 2011 J. Phys.: Condens. Matter 23 315302 [17] Xie Z X, Tang L M, Pan C N, Li K M, Chen K Q and Duan W H 2012 Appl. Phys. Lett. 100 073105 [18] Xu Y, Chen X B, Wang J S, Gu B L and Duan W H 2010 Phys. Rev. B 81 195425 [19] Li X B, Maute K, Dunn M L and Yang R 2010 Phys. Rev. B 81 245318 [20] Gotsmann B and Lantz M A 2013 Nat. Mater. 12 59 [21] Lim J, Hippalgaonkar K, Andrews S C, Majumdar A and Yang P 2012 Nano Lett. 12 2475 [22] Tan S H, Tang L M, Xie Z X, Pan C N and Chen K Q 2013 Carbon 65 181 [23] Sanchez D and Lopez R 2013 Phys. Rev. Lett. 110 026804 [24] Chang C W, Okawa D, Majumdar A and Zettl A 2006 Science 314 1121 [25] Shao Z G, Ai B Q and Zhong W R 2014 Appl. Phys. Lett. 104 013106 [26] Luckyanova M N et al 2012 Science 338 936 [27] Nika D L, Pokatilov E P, Balandin A A, Formin V M, Rastelli A and Schmidt O G 2011 Phys. Rev. B 86 165415 [28] Haskins J B, Kinaci A and Cagin T 2011 Nanotechnology 22 155701 [29] Tian Z, Esfarjani K and Chen G 2014 Phys. Rev. B 89 235307 [30] Jain A, Yu Y J and McGaughey A J H 2013 Phys. Rev. B 87 195301 [31] Ouyang T, Chen Y P, Yang K K and Zhong J X 2009 Europhys. Lett. 88 28002 [32] Yang N, Zhang G and Li B 2008 Nano Lett. 8 276 [33] Xie Z X, Zhang Y, Yu X, Li K M and Chen Q 2014 J. Appl. Phys. 115 104309 [34] Peng X F, Chen K Q, Zou B S and Zhang Y 2007 Appl. Phys. Lett. 90 193502
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Figure 5. The total thermal conductance κt as a function of the periodic number N for three certain temperatures. (a) and (b) correspond to the T-shaped and concavity-shaped periodic nanostructures, respectively. The solid, dashed, and dot–dashed curves correspond to T = 0.5, 1, and 3 K, respectively. Here, X1 = 10 nm, Y 1 = 10 nm, Y 2 = 15 nm, T 1 = T 2 = 10 nm in (a), and X1 = 10 nm, Y 1 = 10 nm, Y 2 = 5 nm, C1 = C2 = 10 nm in (b).
that discussed above. Note that the constant is not larger for higher temperatures in 3D periodic structures. This is different from 2D superlattice structures, where the constant linearly increases with increasing the temperature [33]. 4. Summary
In summary, using the scattering-matrix method in combination with the isotropic elastic continuum model, we investigated ballistic thermal transport properties by phonons in 3D periodic nanostructures. We found that the thermal transport properties in 3D periodic nanostructures can be efficiently tuned by modulating structural parameters of systems. When the incident frequency is below the first cutoff frequency, the transmission coefficient versus the periodic number/length presents quasi/formal-periodic oscillations regardless of the periodic manners. When the incident frequency is above the first cutoff frequency, these quasi/formal-periodic oscillations cannot be observed, and the transmission coefficient versus the periodic number/length strongly depends on the periodic manners. At particular frequencies, some stop-frequency can be observed in these systems. In the limit T → 0, the thermal conductance in 3D periodic nanostructures approaches the quantum value π 2 kB2 T /3h regardless of the periodic number and manner. As the periodic number is increased, a transition of the thermal conductance from the decrease to the constant can be clearly observed. Some similarities and differences between 2D and 3D periodic systems are identified. Our results will not only enrich thermal transport properties in periodic systems, but also provide a valuable reference for modulating the thermal transport by phonons in nanostructures. 6
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Ballistic thermal transport by phonons in three dimensional periodic nanostructures
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 095303 (http://iopscience.iop.org/0953-8984/27/9/095303) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 05/05/2017 at 02:54 Please note that terms and conditions apply.
You may also be interested in: The influence of flaws on phonon thermal conductance Yan Wang, Ling-Ling Wang, Li-Ming Tang et al. Acoustic phonon transport through a quantum waveguide Hai-Yan Zhang, Hong-Jian Li, Wei-Qing Huang et al. Acoustic phonon transport and thermal conductance in quantum wire Li-Ming Tang, Lingling Wang, Wei-Qing Huang et al. Acoustic phonon transport through a T-shaped quantum waveguide Wen-Xia Li, Ke-Qiu Chen, Wenhui Duan et al. Ballistic phonon transmission in a symmetric converging–diverging contact of a semiconductor nanowire Ming-Liang Zou, Wei-Qing Huang, Gui-Fang Huang et al. Phonon transport and thermal conductivity in quantum wire Wen-Xia Li, Ke-Qiu Chen, Wenhui Duan et al. Selective transmission and enhanced thermal conductance of ballistic phonon by nanocavities embedded in a narrow constriction Wei-Qing Huang, Bai-Yun Huang, Dan-Qing Yi et al. Thermal transport by phonons in zigzag graphene nanoribbons with structuraldefects Zhong-Xiang Xie, Ke-Qiu Chen and Wenhui Duan Thermal transport of isotopic-superlattice graphene nanoribbons with zigzag edge T. Ouyang, Y. P. Chen, K. K. Yang et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 095303 (6pp)
doi:10.1088/0953-8984/27/9/095303
Ballistic thermal transport by phonons in three dimensional periodic nanostructures Zhong-Xiang Xie1 , Yong Zhang1 , Xia Yu1 , Hai-Bin Wang1 , Ke-Min Li2 , Chang-Ning Pan3 and Qiao Chen4 1 Department of Mathematics and Physics, Hunan Institute of Technology, Hengyang 421002, People’s Republic of China 2 College of Physics and Eletronics, Hunan Institute of Science and Technology, Yueyang 414006, People’s Republic of China 3 School of Science, Hunan University of Technology, Zhuzhou 412000, People’s Republic of China 4 Department of Maths and Physics, Hunan Institute of Engineering, Xiangtan 411101, People’s Republic of China
E-mail: [email protected] and [email protected] Received 22 October 2014, revised 22 January 2015 Accepted for publication 28 January 2015 Published 18 February 2015 Abstract
Ballistic thermal transport properties by phonons in three dimensional (3D) periodic nanostructures is investigated. Results show that thermal transport properties in 3D periodic nanostructures can be efficiently tuned by modulating structural parameters of systems. When the incident frequency is below the first cutoff frequency, the quasi/formal-periodic oscillations of the transmission coefficient versus the periodic number/length can be observed. When the incident frequency is above the first cutoff frequency, however, these quasi/formal-periodic oscillations cannot be observed. As the periodic number is increased, the thermal conductance undergoes a prominent transition from the decrease to the constant. We also observe other intriguing physics properties such as stop-frequency gaps and quantum thermal conductance in 3D periodic nanostructures. Some similarities and differences between 2D and 3D periodic systems are identified. Keywords: thermal transport, phonon, quantum structures (Some figures may appear in colour only in the online journal)
transport [23], thermal rectification [24], negative differential thermal resistance [25], and so on, have been revealed recently. These imply that phonons can be manipulated like electrons, thus enabling controlled and managed heat transport in physically realistic nanosystems. Recently, the periodic system has been a topic of significant interest due to its potential application in thermoelectric energy conversion and thermal management. Using periodic structures (such as phononic crystals and superlattices) to control the thermal transport by manipulating coherent phonons, has been proposed theoretically and experimentally. Luckyanova et al presented a study of thermal conductance through the finite-thickness superlattices with varying numbers of periods, and found the thermal conductivity increased linearly with increasing the thickness in the temperature range from 30 to 150 K, confirming the existence of the coherent phonon transport [26]. Nika
1. Introduction
In recent years, the thermal transport in low-dimensional nanoscale systems has attracted much attention [1–6]. Different models and theories have been proposed to explore the thermal transport mechanisms. It has been demonstrated that when the feature sizes of structures become comparable to the phonon mean free path, the heat current is mainly carried by a series of discrete vibrational modes due to the boundary confinement. The thermal transport properties in a variety of nanostructures including nanowires [7–9], nanofilms [10], nanotubes [11–13], nanoribbons [14–18], have been reported. Enormous effort has been devoted to elucidate the effects of strain [19], rough surfaces [20, 21], and structural defects [22] on thermal transport properties of nanoscale structures and materials. Similar to the electronic transport, several intriguing thermal properties such as nonlinear thermal 0953-8984/15/095303+06$33.00
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insight into the thermal conductance in 3D systems [34]. They predicted that the thermal conductance in 3D systems is larger than that in 2D systems even for the same structural parameters. In this work, using the scattering method in combination with the isotropic elastic continuum model, we study the thermal transport properties by phonons in 3D periodic nanostructures. Two different periodic manners are considered here: T-shaped and concavity-shaped periodic nanostructures.
et al [27] and Haskins et al [28] predicted that onedimensional periodic quantum-dot can be served as effective thermoelectric materials and thermal insulators due to the shape reduction of lattice thermal conductivity caused by the interface mismatching, respectively. Several recent studies have focused on the thermal conductivity in realistic periodic structures such as Si/Ge superlattices [8, 29], periodic silicon nanoporous films [30], isotopic-superlattice graphene nanoribbons [31], isotopic-superlattice structured Si nanowires [32], and so on. Among these studies, an important and long-standing discovery across these periodic systems is the existence of a minimum in the thermal conductivity for a finite periodic length. It was proved that this optimal periodic length is associated with the transition between wave-like and particle-like transport behaviors of phonons. More recently, we also investigated the transmission spectrum and thermal conductance through two-dimensional (2D) superlattice quantum-waveguides, and observed the stopband formation depending upon the periodic length [33]. In spite of these advancements in the study of thermal transport in 2D periodic films and 1D periodic nanowires, the thermal transport mechanisms in three dimensional (3D) periodic nanostructures, along with the effects of important parameters (such as the periodic number, the periodic length as well as the periodic manner) on thermal transport properties in 3D periodic nanostructures, are not thoroughly understood. It has become well known that there exist four types of acoustic modes as well as two shear modes in 3D systems, different from 2D systems where the vibrational modes are split into two independent modes: the in-plane modes and the out-of-plane modes. Recently, some researchers have attempted to provide
2. Model and formalism
We model the geometries, as illustrated in figure 1. (a) corresponds to the T-shaped periodic nanostructure with the periodic length L = T 1 + T 2, and (b) corresponds to the concavity-shaped periodic nanostructure with the periodic length L = C1 + C2. Each structure consists of two pristine semi-infinite parts adiabatically connected the left and right thermal reservoirs with the temperature being denoted by TL and TR , respectively, and the periodic part that consists of alternating blocks with different sizes. In both periodic nanostructures, the material composition of any blocks is GaAs. Note that the GaAs nanosystems have attracted much attention because of their intriguing physical properties and potential application [34]. We assume that the thermal contacts between two leads and thermal reservoirs are perfect, and only the phonon scattering occurs at the interfaces between two adjacent blocks with different sizes. Now, several methods such as the isotropic elastic continuum model [33, 34] and the lattice model [16–18], have been developed to study the thermal transport by phonons. Note that when the typical phonons have the wavelength comparable to the atomic bond 2
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length, the isotropic elastic continuum model may be incorrect quantitatively. However, it is a rather good approximation for describing the thermal transport in nanostructures at very low temperatures [6]. Considering the fact that our work mainly focuses on the acoustic phonon thermal transport in low temperature range where only several lowest modes can be excited. The wavelength of these phonon modes is generally much larger than any microscopic length including the periodic length L considered here. Thus, in this work we can employ the isotropic elastic continuum model to describe the phonon transmission. According to this model, the phonon displacement ψ(x, y, z) in the cartesian coordinate satisfies the following equation v 2 ∇ 2 ψ(x, y, z) + ω2 ψ(x, y, z) = 0,
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(2) ξ+ ξ− where Pmn and Pmn with +(−) being the forward (backward) propagating direction along the z axis, are constants to be ξ determined by matching the boundary condition. φmn (x, y) = ξ ξ ξ ξ φm (x)φn (y), where φm (x) and φn (y) represent the orthogonal wave function along the x and y directions, respectively. Under the stress-free boundary condition, we can obtain 2 cos( mπ x) m = 0 Xξ Xξ ξ (3a) φm (x) = 1 m=0 Xξ 2 cos( nπ y) n = 0 Yξ Yξ ξ φn (y) = . (3b) 1 n=0 Yξ
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Figure 2. (a) and (b) describe the transmission coefficient τ as a function of the incident frequency ω/ ( = πv/X1) for the T-shaped and concavity-shaped periodic nanostructures, respectively. Curves a–c correspond to N = 3, 6, and 14, respectively, and the dashed curve in (a) corresponds to the pristine case. Note that two consecutive curves are vertically offset two units for clarity. Here, X1 = 10 nm, Y 1 = 10 nm, Y 2 = 15 nm, T 1 = T 2 = 10 nm in (a), and X1 = 10 nm, Y 1 = 10 nm, Y 2 = 5 nm, C1 = C2 = 10 nm in (b).
Here, Xξ represents X1, while Yξ is Y 1 or Y 2 depending on the size of the corresponding block along the y direction. In ξ equation (2), kmn is the wave vector along the z axis, and can be expressed as ξ = ω2 /v 2 − m2 π 2 /Xξ2 − n2 π 2 /Yξ2 . (4) kmn
3. Numerical results and discussion
Figures 2(a) and (b) show the transmission coefficient τ as a function of the reduced frequency ω/ ( = π v/X1) for the T-shaped and concavity-shaped periodic nanostructures, respectively. Curves a–c in each figure correspond to the periodic number N = 3, 6, and 14, respectively, and the dashed curve in figure 2(a) corresponds to the pristine case. Clearly for the pristine case, τ exhibits a series of smooth platforms, corresponding to the available phonon channels. Note that the altitude intercept between two adjacent platforms is either 1 or 2, which is different from 2D systems. When the periodic structure is introduced, τ is sharply decreased due to the strong interface scattering. In the limit ω → 0, however, τ tends to be the ideal value, since an acoustic phonon at ω = 0 can retain the perfect transmission. This is in agreement with the case in the 2D systems [33], but different from the electron transport. As also seen in figure 2, an outstanding feature is that some stop-frequency gaps can be
ξ
For the open channel, kmn is real. Otherwise, we take ξ ξ kmn = i|kmn |. By applying the continuum condition of the displacement and stress at each interface between any adjacent blocks, and using the scattering-matrix method [34], we can calculate the transmission coefficient τmn . Then, the corresponding thermal conductance κ by phonons can be given by ∞ ω2 eβ¯hω h ¯2 1 τ (ω) dω, (5) κ= mn kB T 2 m,n 2π (eβ¯hω − 1)2 ωmn
where β = 1/(kB T ) with kB being the Boltzmann constant, h ¯ is the Planck’s constant, ωmn is the cutoff frequency of the (m, n) mode. In following numerical calculations, we will employ these values of the elastic constants and mass density of GaAs referred in [33]. 3
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Figure 3. (a1)–(a3) show the transmission coefficient τ versus the periodic length L for the incident frequency ω/ = 0.05, 0.5, and 1.2, respectively, and (b1)–(b3) show the transmission coefficient τ versus the periodic number N for the incident frequency ω/ = 0.05, 0.5, and 1.2, respectively. In each figure, the solid and dashed curves correspond to the T-shaped and concavity-shaped periodic nanostructures, respectively. Here, we take N = 8 in (a1)–(a3), and L = 20 nm in (b1)–(b3). Other parameters are the same as those in figure 2.
incremental oscillation as the periodic length L increases (figures 3(a1) and (a2)). When the incident frequency is above the first cutoff frequency (ω/ > 1), however, a similar oscillation cannot be obviously observed (figure 3(a3)). On the other hand, τ versus N exhibits the formal-periodic behavior, together with numerous peak-valley when ω/ < 1. This periodic behavior can be understood as a result of the wave interference. When the incident wave enters into the periodic region, one part can go through the periodic region, and the other part can get back due to the interface scattering. The two parts interfere with each other coherently, thus resulting in the periodic behavior. During the interfering process, when the phase difference is an even multiple of π , τ will be resonantly reinforced. Inversely, when the phase difference is an add multiple of π , τ will be strongly lowered. As a result, a periodic peak-valley behavior can be observed in the behavior of τ versus N . Examining figures 3(b1) and (b2), it is seen that the period of τ (N ) reduces as the frequency is increased. However, when the frequency is further increased (figure 3(b3)), τ versus N loses the periodic feature. In figure 3, we also note that for ω/ < 1, τ versus the structural parameters is similar in characteristic for both periodic structures, while for ω/ > 1, it is absolutely different for different periodic structures. Next, we turn to study the thermal conductance in both periodic nanostructures. Figure 4 shows that the thermal conductance reduced by the quantum value π 2 kB2 T /3h, as a function of the reduced temperature kB T /¯h . Figures 4(a) and (b) describe the thermal conductance κ0 of the zero mode for
opened at particular frequencies. Moreover, when the periodic number N is increased, the number of stop-frequency gaps increases. These stop-frequency gaps indicate that the phonon transmission can be completely forbidden in both periodic structures. The antiresonant coupling between the incident and reflected waves may be responsible for the appearance of the stop-frequency gaps. Generally, the more the periodic number N is, the stronger the coupling becomes. As a consequence, the number of the stop-frequency gaps is increased with increasing N in both periodic structures. Comparing figure 2(a) with figure 2(b), we can see that the width of stop-frequency gaps in the T-shaped periodic nanostructure is narrower than that in the concavity-shaped periodic structure. In addition, τ in the T-shaped periodic nanostructure is always higher than the corresponding value in the concavity-shaped periodic nanostructure. These differences can be attributed to the fact that the T-shaped periodic nanostructure can provide more channels to facilitate the phonon transmission as compared to the concavity-shaped periodic nanostructure, although the number of modes excited in the incident lead is the same for both periodic nanostructures. In order to further reveal the phonon transmission properties, we describe the transmission coefficient as a function of the structural parameters such as the periodic length L and the periodic number N , as shown in figure 3. We find that τ versus L displays different behaviors for different incident frequencies. When the incident frequency is below the first cutoff frequency (ω/ < 1), τ presents the quasi-periodic 4
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the T-shaped and concavity-shaped periodic nanostructures, respectively, while figures 4(c) and (d) correspond to the total thermal conductance κt for them, respectively. For the pristine case, the quantum platform of κ0 is clearly observed, while for both periodic nanostructures, this quantum platform cannot be observed. In the zero temperature limit, however, κ0 approaches the quantum value π 2 kB2 T /3h regardless of the variation of the periodic number N. This is consistent with the isotopic-superlattice graphene nanoribbons [31] and 2D superlattice quantum-waveguides [33]. Comparing figure 4(a) with figure 4(b), some distinct differences of κ0 between both periodic nanostructures can be identified as follows: (i) κ0 for the T-shaped periodic nanostructure firstly decreases to a minimum value, then increases gently, while that for the concavity-shaped periodic nanostructure monotonously decreases as the temperature is increased; (ii) at high temperature, κ0 for the T-shaped periodic nanostructure is more sensitive to the variation of N , compared to the concavity-shaped periodic nanostructure; (iii) κ0 for the Tshaped periodic nanostructure is higher than the corresponding value for the concavity-shaped periodic nanostructure. These differences indicate that the thermal conductance in 3D periodic nanostructures sensitively depends upon the periodic manner. It is the decrease of κ0 that results in the decrease of κt in the low temperature region, where only the zero mode can be excited (figures 4(c) and (d)). This behavior seems to be in agreement with the experimental result qualitatively [3].
With increasing the temperature (kB T /¯h > 0.1), all the κt for both periodic nanostructures gradually increase, since higher order modes are successively excited to conduct heat. Note that the thermal conductance of higher order modes monotonously increases from the zero value due to their nonzero cutoff frequencies (not shown), different from the decreasing behavior of the zero mode presented here. Finally, we show the total thermal conductance κt as a function of the periodic number N for three given temperatures in figure 5. It is seen that the curves in κt versus N for both periodic nanostructures are similar in characteristics. The value of κt firstly decreases from the maximum value, then becomes nearly independent of N . In other words, when N is gradually increased, κt undergoes a noticeable transformation from the decrease to the constant. This transformation can be attributed to structural features of 3D periodic nanostructures. When N is relatively small, increasing the N will lead to the augment of the number of scattering interfaces, thus resulting in the reduction of κt . However, when N is increased to a certain extent, the centrally periodic part can be regarded as an integrated whole. So the transformation can occur in the behavior of κt versus N . A similar transformation was also recently observed in 2D superlattice wave-guides [33]. Comparing figures 5(a) with (b), it is found that the constant of κt for the T-shaped periodic nanostructure is larger than the corresponding value for the concavity-shaped periodic nanostructure, in agreement with 5
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Acknowledgments
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This work is supported supported by the National Natural Science Foundation of China under Grant (Nos. 11404110 and 11204074), the Nature Science Foundation of Hunan Province under Grant (Nos. 14JJ3139 and 2015JJ2050), the Outstanding Young Program of Hunan Provincial education department of China under Grant (No. 14B046), and the Key Research Program of Hunan provincial education department, China (Grant No. 12A059).
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References [1] Cahill D G et al 2014 Appl. Phys. Rev. 1 011305 [2] Li N B, Ren J, Wang L, Zhang G, Hangg P and Li B 2012 Rev. Mod. Phys. 84 1045 [3] Dubi Y and Ventra M D 2011 Rev. Mod. Phys. 83 131 [4] Marconnet A M, Panzer M A and Goodson K E 2013 Rev. Mod. Phys. 85 1295 [5] Balandin A A 2011 Nat. Mater. 10 569 [6] Wang J S, Wang J and Lu J T 2008 Eur. Phys. J. B 62 381 [7] Chen J, Zhang G and Li B 2012 Nano Lett. 12 2826 [8] Hu M and Poulikakos D 2012 Nano Lett. 12 5487 [9] Chen J, Zhang G and Li B 2011 J. Chem. Phys. 135 104508 [10] Hua Y C and Cao B Y 2014 Int. J. Heat Mass Transfer 78 755 [11] Chen X B, Xu Y, Zou X L, Gu B L and Duan W H 2013 Phys. Rev. B 87 155438 [12] Chen J, Zhang G and Li B 2010 Nano Lett. 10 3978 [13] Wang J, Li L and Wang J S 2011 Appl. Phys. Lett. 99 091905 [14] Huang H Q, Xu Y, Zou X L, Wu J and Duan W H 2013 Phys. Rev. B 87 205415 [15] Yu C X and Zhang G 2013 J. Appl. Phys. 113 044306 [16] Xie Z X, Chen K Q and Duan W H 2011 J. Phys.: Condens. Matter 23 315302 [17] Xie Z X, Tang L M, Pan C N, Li K M, Chen K Q and Duan W H 2012 Appl. Phys. Lett. 100 073105 [18] Xu Y, Chen X B, Wang J S, Gu B L and Duan W H 2010 Phys. Rev. B 81 195425 [19] Li X B, Maute K, Dunn M L and Yang R 2010 Phys. Rev. B 81 245318 [20] Gotsmann B and Lantz M A 2013 Nat. Mater. 12 59 [21] Lim J, Hippalgaonkar K, Andrews S C, Majumdar A and Yang P 2012 Nano Lett. 12 2475 [22] Tan S H, Tang L M, Xie Z X, Pan C N and Chen K Q 2013 Carbon 65 181 [23] Sanchez D and Lopez R 2013 Phys. Rev. Lett. 110 026804 [24] Chang C W, Okawa D, Majumdar A and Zettl A 2006 Science 314 1121 [25] Shao Z G, Ai B Q and Zhong W R 2014 Appl. Phys. Lett. 104 013106 [26] Luckyanova M N et al 2012 Science 338 936 [27] Nika D L, Pokatilov E P, Balandin A A, Formin V M, Rastelli A and Schmidt O G 2011 Phys. Rev. B 86 165415 [28] Haskins J B, Kinaci A and Cagin T 2011 Nanotechnology 22 155701 [29] Tian Z, Esfarjani K and Chen G 2014 Phys. Rev. B 89 235307 [30] Jain A, Yu Y J and McGaughey A J H 2013 Phys. Rev. B 87 195301 [31] Ouyang T, Chen Y P, Yang K K and Zhong J X 2009 Europhys. Lett. 88 28002 [32] Yang N, Zhang G and Li B 2008 Nano Lett. 8 276 [33] Xie Z X, Zhang Y, Yu X, Li K M and Chen Q 2014 J. Appl. Phys. 115 104309 [34] Peng X F, Chen K Q, Zou B S and Zhang Y 2007 Appl. Phys. Lett. 90 193502
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Figure 5. The total thermal conductance κt as a function of the periodic number N for three certain temperatures. (a) and (b) correspond to the T-shaped and concavity-shaped periodic nanostructures, respectively. The solid, dashed, and dot–dashed curves correspond to T = 0.5, 1, and 3 K, respectively. Here, X1 = 10 nm, Y 1 = 10 nm, Y 2 = 15 nm, T 1 = T 2 = 10 nm in (a), and X1 = 10 nm, Y 1 = 10 nm, Y 2 = 5 nm, C1 = C2 = 10 nm in (b).
that discussed above. Note that the constant is not larger for higher temperatures in 3D periodic structures. This is different from 2D superlattice structures, where the constant linearly increases with increasing the temperature [33]. 4. Summary
In summary, using the scattering-matrix method in combination with the isotropic elastic continuum model, we investigated ballistic thermal transport properties by phonons in 3D periodic nanostructures. We found that the thermal transport properties in 3D periodic nanostructures can be efficiently tuned by modulating structural parameters of systems. When the incident frequency is below the first cutoff frequency, the transmission coefficient versus the periodic number/length presents quasi/formal-periodic oscillations regardless of the periodic manners. When the incident frequency is above the first cutoff frequency, these quasi/formal-periodic oscillations cannot be observed, and the transmission coefficient versus the periodic number/length strongly depends on the periodic manners. At particular frequencies, some stop-frequency can be observed in these systems. In the limit T → 0, the thermal conductance in 3D periodic nanostructures approaches the quantum value π 2 kB2 T /3h regardless of the periodic number and manner. As the periodic number is increased, a transition of the thermal conductance from the decrease to the constant can be clearly observed. Some similarities and differences between 2D and 3D periodic systems are identified. Our results will not only enrich thermal transport properties in periodic systems, but also provide a valuable reference for modulating the thermal transport by phonons in nanostructures. 6
