Calculation of Confined Phonon Spectrum in Narrow Silicon Nanowires using the Valence Force Field Method. Abstract

Transcription

1 Calculaton of Confned Phonon Spectrum n Narrow Slcon Nanowres usng the Valence Force Feld Method Hossen Karamtaher 1,2, Neophytos Neophytou 1, Mohsen Karam Taher 3, Rahm Faez 2, and Hans Kosna 1 1 Insttute for Mcroelectroncs, TU Wen, Gußhausstraße 27-29/E360, A-1040 Wen, Austra 2 School of Electrcal Engneerng, Sharf Unversty of Technology, Tehran, Iran 3 Department of Computer Engneerng, Naragh Branch, Islamc Azad Unverstry, Naragh, Iran E-mal: {karam neophytou kosna}@ue.tuwen.ac.at Abstract We study the effect of confnement on the phonon propertes of ultra-narrow slcon nanowres of sde szes of 110nm. We use the modfed valence force feld method to compute the phononc dsperson, and extract the densty of states, the transmsson functon, the sound velocty, the ballstc thermal conductance and boundary scatternglmted dffusve thermal conductvty. We fnd that the phononc dsperson and the ballstc thermal conductance are functons of the geometrcal features of the structures,.e. the transport orentaton and confnement dmenson. The phonon group velocty and thermal conductance can vary by a factor of two dependng on the geometrcal features of the channel. The <110> nanowre has the hghest group velocty and thermal conductance, whereas the <111> the lowest. The <111> channel s thus the most sutable orentaton for thermoelectrc devces based on S nanowres snce t also has a large power factor. Our fndngs could be useful n the thermal transport desgn of slconbased devces for thermoelectrc and thermal management applcatons. Index terms: Confned phonons, slcon nanowres, lattce thermal conductance, modfed valence force feld method, Landauer formula. 1

2 I. Introducton Low dmensonal slcon nanowres (NWs), and slcon-based ultra-thn layers have attracted sgnfcant attenton as effcent thermoelectrc (TE) materals after t was demonstrated that the thermal conductvty n such materals can be drastcally reduced compared to bulk S. Values as low as 2W/mK (compared to 140W/mK for bulk S) were acheved [1, 2]. Ths resulted n ZT values close to ZT ~ 0. 5, a large mprovement compared to ZT of bulk S, ZT ~ 0. 01[1, 2]. The large reducton n the thermal conductvty was attrbuted to enhanced scatterng of phonons on the surfaces of the nanochannels. Numerous studes can be found n the lterature regardng the thermal conductvty of S NWs [3, 4, 5, 6, 7]. The effects of dfferent scatterng mechansms,.e. surface roughness scatterng, mass dopng, phonon-phonon scatterng, and phononelectron scatterng have been nvestgated by several authors [8, 9, 10, 11]. In these works, t s demonstrated that the thermal conductvty n ultra-narrow S NWs drastcally degrades once the dameter of the NW s reduced below 50nm, or when scatterng centers are ncorporated. For even smaller NW dameters,.e. below 10nm, the effect of confnement could further change the phonon spectrum sgnfcantly, and provde an addtonal mechansm n the reducton of the thermal conductvty [12]. Ths could provde addtonal benefts to the thermoelectrc fgure of mert ZT. In ths work, we employ the modfed valence force feld (MVFF) method [13] to address the effect of structural confnement on the phonon dsperson, group velocty and ballstc thermal conductance of ultra-narrow S NWs of dameters below 10nm. We consder dfferent transport orentatons, and dfferent cross sectonal szes. We fnd that although the densty of phonon modes does not show any orentaton dependence, the phonon group veloctes and sound veloctes are strongly ansotropc. Ths results n a hgher thermal transmsson n the case of the <110> NW, whch results n a hgher ballstc thermal conductance compared to dfferently orented NWs as well. On the other hand, the <111> orented NW has the lowest thermal conductance, almost 2x lower than that of the <110> NW, and could be the optmal choce for NW TE devces. Our results, therefore, ndcate bulk 2

3 that the proper choce of the NW geometry can provde a further reducton n the thermal conductvty of NW TE devces and consequently hgher ZT values. II. Approach For the calculaton of the phononc bandstructure we employ the modfed valence force fled method [13]. In ths method the nteratomc potental s modeled by the followng bond deformatons: bond stretchng, bond bendng, cross bond stretchng, cross bond bendng stretchng, and coplanar bond bendng nteractons [13]. The model accurately captures the bulk S phonon spectrum as well as the effects of confnement [14]. where In the MVFF method, the total potental energy of the system s defned as [14]: N A, U 1 2 nn, and jk jk l j jk jk jk jkl U bs Ubb Ubs-bs Ubs-bb Ubb-bb N A jnn j, knn j, k, lcop COP are the number of atoms n the system, the number of the nearest neghbors of a specfc atom '', and the coplanar atom groups for atom '', respectvely. U bs, U bb, U bs-bs, U bs-bb, and U bb-bb are the bond stretchng, bond bendng, cross bond stretchng, cross bond bendng stretchng, and coplanar bond bendng nteractons, respectvely [14]. (1) In ths formalsm we assume that the total potental energy s zero when all the atoms are located n ther equlbrum poston. Under the harmonc approxmaton, the moton of atoms can be descrbed by a dynamc matrx as [15]: D [ D 3 3] j 1 M M j Dj D l l j j (2) where dynamc matrx component between atoms '' and 'j' s gven as shown n Ref. [14] by: 3

4 D D j D D j xx j yx j zx D D D j xy j yy j zy j D xz j Dyz j D zz (3) and 2 j U elastc Dmn,, j N A and m, n [ x, y, z] j rm rn (4) s the second dervatve of the potental energy after atoms '' and 'j' are slghtly dsplaced j along the m-axs and the n-axs ( and r m r n ), respectvely. To calculate U elastc, we added only those terms of Eq. 1 that nclude atoms '' and 'j'. All other terms cancel each other out when we numercally calculate the second dervatve of potental energy. After settng up the dynamc matrx, the followng egenvalue problem s solved to calculate the phononc dsperson: 2 q. Rl qi 0 D Dl exp l (5) where D l s the dynamc matrx representng the nteracton between the unt cell and ts neghborng unt cells separated by Rl [15]. Usng the phononc dsperson, the densty of states (DOS) and the ballstc transmsson (number of modes at gven energy) are calculated as shown n Ref. [16] by: DOS DOS q q (6) and T ph h 2 M q, q q q (7) Once the transmsson s obtaned, the ballstc lattce thermal conductance s calculated wthn the framework of the Landauer theory as [17]: K l 1 h 0 T ph n T d (8) 4

5 It s well known that the thermal conductvty n S nanowres of very narrow cross sectons s domnated by boundary scatterng, whch s the reason of ther mproved thermoelectrc performance as well [2, 11, 18, 19]. The dffusve thermal conductvty can be calculated usng the phonon lfetme approxmaton n the phononc Boltzmann transport equatons as [18]: k l B, q ( q) v g, 2 ( q) ( q) kbt 2 e e ( q) / k BT ( q) / k BT where v ( q) s the group velocty of phonons of wavevector q n subband, gven by g, vg, ( q) ( q) / q, and (q) s the scatterng tme. Here we compute the boundaryscatterng-lmted thermal conductvty by evaluatng the boundary scatterng tme as: 1 1 P vg ( q), ( q) 1 P d B, where d s the effectve dameter (defned as 1 2 (9) (10) d 2W where W s the sde length of the NW cross secton) of the NWs and P s the specularty parameter that takes values form 0 to 1, determned by the detals of the surface and surface roughness [18, 20]. For a smooth surface P 1, the scatterng tme s nfnte (meanng zero scatterng rates on the surface), and boundary scatterng s fully specular. On the other hand, for a very rough surface P 0, whch results n fully dffusve boundary scatterng. Recently, studes showed that expermental results for boundary scatterng n narrow NWs are only explaned once an almost fully dffusve boundary s assumed [21, 22]. III. Results and Dscusson In ths work we examne the effects of transport orentaton and confnement length scale on the thermal propertes of S-NWs wth square cross sectons ( W H ) and sde szes up to 10 nm. We study the phononc dsperson, densty of states, ballstc transmsson, average group velocty, and lattce thermal conductvty of dfferent structures. 5

6 The phononc dspersons of S NWs of 2nm 2nm cross secton n dfferent transport orentatons, <100>, <110>, and <111>, are shown n Fg. 1. There are dfferences n the dspersons, especally n the low energy, low momentum regon, whch ndcates that the thermal propertes could be orentaton dependent as well. Fgure 2 shows the ballstc transmsson functon of a dfferent set of NWs, of sde szes W H 6nm n the same three orentatons, calculated as ndcated above n Eq. 7. The transmsson functon of the <110> NWs s the hghest n most part of the energy spectrum, whereas the transmsson of the <111> NWs s the lowest n almost the entre energy spectrum. As a result, the ballstc lattce thermal conductance of the NWs (see Fg. 3-a) calculated usng the Landauer formula as specfc n Eq. 8, shows that the <110> NWs has the hghest thermal conductance compared to the NWs orented along the <100> and <111> transport drectons. The thermal conductance s the lowest for the <111> NWs of all the sde szes we have consdered, all the way up to 10nm as shown n Fg. 3-a. The dfference between the thermal conductance of the <110> NW whch has the hghest performance, and the <111> NW whch has the lower performance, s a factor of ~2. Another observaton s that the thermal conductance ncreases as the cross secton of the NW ncreases. Ths s expected snce the larger NWs contan more transport modes. The ncrease s close to lnear. Once the conductances are normalzed by the cross secton area of the NWs, however, the resultant normalzed conductances reman almost constant. Ths s ndcated n the nset of Fg. 3-a. In ths case, agan, the <111> has clearly the lowest conductvty, almost 2 tmes lower than the <110> NW for all cross secton szes. In Fg. 3-b we show the boundary-scatterng-lmted lattce thermal conductvty for the nanowres of 2nm and 10nm cross secton sdes, versus the specularty parameter P. As P ncreases from 0 (fully dffusve surfaces) to 0.5 (semdffusve surfaces), the conductvty naturally ncreases. The scatterng s stronger for narrower NWs. For the NWs wth cross secton szes of 2nm, thermal conductvtes as low as ~2 W/mK are acheved for P=0, rasng to ~5 W/mK for P=0.5. For NWs wth cross secton szes of 10nm, the thermal conductvty s almost 5 tmes hgher, at most ~30 W/mK. These values are sgnfcantly lower than the thermal conductvty of bulk S ( 140 W/mK ), and once other scatterng mechansms are consdered (.e. phonon-phonon scatterng and mpurty scatterng) they could lower even further. Interestngly, the order 6

7 of the conductvtes for NWs of dfferent orentatons s the same as n the ballstc case of Fg. 3-a, wth the <110> NW havng the hghest and the <111> NW the lowest conductvty. The same ansotropy behavor s observed when consderng only phononphonon scatterng mechansms [23], and dfferent fully dffusve transport formalsms [18, 19], further demonstratng the mportance of the phonon bandstructure n determnng the thermal conductvty of S NWs. For thermoelectrc applcatons, therefore, where low thermal conductance s advantageous, the <111> NW would be the optmal choce as we show below. We have recently studed the role of transport orentatons and dameter on the thermoelectrc power factor of NWs usng atomstc bandstructure smulatons for electron transport [24]. Wth regards to ansotropy, n the case of n-type NWs only a small orentaton dependence was observed. In the case of p-type S NWs, however, we showed that the <111> NWs have sgnfcantly hgher power factors compared to dfferently orentated NWs. The <111> S NW channel s, therefore, the most advantageous for p-type channel thermoelectrc applcatons snce t provdes smultaneously the hghest power factor and lowest thermal conductvty compared to p- type NWs of dfferent transport orentatons. Usng the thermoelectrc power factor of Ref. [24] (Fg. 3c and Fg. 7c of Ref. [24]) and the calculated thermal conductvty, n Fg. 4a and 4b we show the ZT fgure of mert for n-type and p-type NWs, respectvely, versus the NW cross secton sze. For n-type NWs, the <111> drecton s the most advantageous, provdng ZT of ~0.5. On the other hand, a ZT value around 1 s calculated for a very narrow p-type <111> NWs as shown n Fg. 4b, almost 3 tmes hgher than that of the <110> NW. Ths s partly due to ts hgher power factor orgnatng from ts large electronc moblty [25] and partly due to ts lower lattce thermal conductvty. For the rest of the paper, we revst the ballstc results, and attempt to elucdate the reasons behnd the ansotropy of the thermal conductance and conductvty n terms of the NWs phonon bandstructure. The ballstc transmsson functon s drectly related to the product of the densty of phonon states and phonon group velocty. Below we nvestgate each of these quanttes and ther geometry dependence n order to provde 7

8 nsght nto the orentaton dependence of the thermal conductance. For ths, we consder NWs wth W H 6nm n the three dfferent orentatons under nvestgaton. Fgure 5 shows the phonon DOS, whch s nearly the same for all orentatons n the entre range of frequences. Ths can be understood by consderng the fact that the DOS s mostly related to the densty of atoms n the cross secton of the NW. The densty of atoms s mostly a bulk property and does not change drastcally n S-based structures. Each atom has three degrees of freedom ( 33 elements n ts Dynamc matrx component), and therefore provdes three egenmodes. Although for ultra-narrow structures.e. of H=W=2nm there could be some dfferences n the DOS due to fnte sze effects a 6nm 6nm structure s large enough such that the DOS s orentaton ndependent. On the other hand, as shown n Fg. 6, the sound velocty, defned by the slope of the acoustc modes near the Brlloun zone center, s not an sotropc quantty. We note here that the 2 lowest modes n phonon dsperson of NWs are flexural modes. The thrd mode s the one used to calculate the transverse sound velocty, whereas the fourth mode gves the longtudnal sound velocty. Our results, as shown n Fg. 6, are n good agreement wth prevous studes on NW sound veloctes [14, 26]. The <111> NW has the hghest longtudnal sound velocty (green-sold), whereas the <100> NW the lowest sound velocty for all NW cross secton szes. The transverse veloctes are lower, and n that case, the <100> NW has the hghest velocty. The values calculated here are dfferent, and n general lower than the bulk sound veloctes (dots n the rght sde of Fg. 6). The veloctes n NWs are lower, also n agreement wth other reports, snce confnement flattens the phonon spectrum and reduces sound veloctes [27]. At frst, followng a bulk way of analyss, t seems that accordng to the orderng n the sound velocty one would expect that the thermal conductance should be hgher n the <111> NWs, followed by the <110> NWs and then by the <100> NWs. As we show n Fg. 3, however, the order s dfferent, namely that the <110> NWs have the hghest thermal conductvty, followed by the <100>, and then by the <111> NWs. The fact s that the phonon dsperson s modfed n nanostructures so that the bulk defnton of acoustc and optcal modes s no longer the relevant quantty. The sound velocty 8

9 determnes the dsperson only n a small part of energy spectrum ( ~ 5meV of out of the total ~ 65meV). In addton, at room temperature, all the phonons n the entre energy spectrum contrbute to the thermal conducton. Therefore, the sound velocty alone cannot descrbe the NW phonon spectrum, whch s an ndcaton that the bulk treatment s neffcent for understandng thermal transport n NWs. What needs to be taken nto consderaton n order to correctly nterpret the results, therefore, s the group velocty of all phonon modes n all energes. For ths, we defne the effectve/average group velocty as follows: v g, q v g,, q q ( ( q)) ( ( q)) where the sum holds over all the modes. The velocty of a phonon s n general a functon of ts subband, ts frequency, and ts q-pont. The quantty n Eq. 11 effectvely suppresses the subband and q-pont ndces and provdes a quantty that s only frequency (or energy) dependent. Smlar effectve quanttes are also used n actual thermal conductvty calculatons wth adequate accuracy n the results [8, 28], although n our actual calculatons we utlze all the nformaton of the phonon spectrum. (11) In contrast to the sound velocty, the average weghted group velocty over dfferent modes can provde a better and more correct nsght wth regards to phonon transport. Fgure 7 shows that the average group velocty of phonons for NWs n the 110 s the hghest, or hgh enough compared to the other orentatons, n the largest part of the energy spectrum. The explanaton behnd ths les n the shape of the dspersons shown n Fg. 1. In the case of the <111> NW, the modes look overall less dspersve,.e. more flat. Ths results n the lowest average group velocty. The modes of the <110> NW, on the other hand, are more dspersve,.e. less flat, whch results n the hghest group velocty. By consderng that all NWs wth the same cross sectonal area have the same phonon DOS, the dfferences n the group velocty explan the orentaton dependence observed n the transmsson functons and n the thermal conductance shown n Fg. 2 and Fg. 3, respectvely. Eventually the transmsson 9

10 functon of the <111> NW s ~1.25x lower than that of the <100> NWs and almost 2x lower than that of the <110> NWs. IV. Conclusons In ths work we study the thermal propertes of ultra-narrow slcon nanowres usng the atomstc modfed valence force feld method for the computaton of the phonon bandstructure. We extract the thermal propertes usng the Landauer formalsm under ballstc, and dffusve boundary-scatterng-lmted transport consderatons. We address the effects of structural confnement on the phonon dsperson, the phononc densty of states, the phononc transmsson functon, the sound velocty and the effectve group velocty. Our results show that dfferently orented NWs can have up to a factor of ~2x dfference n ther effectve group velocty, transmsson functon and fnally ballstc and dffusve thermal conductance. The <110> orented NW has the hghest ballstc and dffusve thermal conductance, followed by the <100> and fnally the <111> NW. By takng nto account the thermoelectrc power factor, we show that the <111> orentaton s the optmal choce for NW TE channels, snce t provdes the lowest thermal conductance and hghest power factors for both n-type and especally p-type channels. Acknowledgements Ths work was supported by the European Communty's Seventh Framework Programme under grant agreement no. FP (NanoHTEC). 10

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14 Fgure 1: Fgure 1 capton: Phononc dspersons of S NWs of square cross sectons wth W=H=2nm for a) <100>, b) <110>, and c) <111> transport orentatons. 14

15 Fgure 2: Fgure 2 capton: Transmsson functon versus energy for NWs n dfferent transport orentatons. <110> NWs have the hghest and <111> NWs have the lowest transmsson. 15

16 Fgure 3: K l [nw/k] (a) K l /A [a.u.] W=H [nm] <100> <110> <111> W=H [nm] κl [W/Km] (b) W=H=10nm <100> <110> <111> W=H=2nm P Fgure 3 capton: (a) Ballstc lattce thermal conductance versus NW cross secton sde sze for NWs n dfferent orentatons. Inset: The thermal conductance normalzed by the cross secton area. (b) Dffusve boundary-scatterng-lmted lattce thermal conductvty for NWs wth dfferent orentatons versus the specularty parameter. Nanowres wth sde szes of 2nm and 10 nm are consdered. 16

17 Fgure 4: ZT (a) n-type <111> <100> <110> ZT (b) p-type <111> <100> <110> W=H [nm] Fgure 4 capton: The thermoelectrc fgure of mert ZT for (a) n-type and (b) p-type NWs wth dfferent orentatons versus the nanowre sde sze. For ths calculatons, we used the thermoelectrc power factors from Ref. [24]. 17

18 Fgure 5: Fgure 5 capton: The phonon DOS versus frequency for NWs n dfferent orentatons. The DOS s nearly the same for the dfferent transport orentatons. 18

19 Fgure 6: Fgure 6 capton: The sound velocty of NWs n dfferent orentatons versus the NW sde sze. Longtudnal and transverse veloctes are shown n sold and dashed lnes, respectvely. Dots: The respectve bulk S veloctes. 19

20 Fgure 7: Fgure 7 capton: Average group velocty of NWs n dfferent transport orentatons versus frequency. 20