Non-Equilibrium Phonon Distributions in Sub-100 nm Silicon Transistors

Transcription

1 S. Sinha Thermocience Diviion, Mechanical Engineering Department, Stanford Univerity, California E. Pop R. W. Dutton Electrical Engineering Department, Stanford Univerity, California K. E. Goodon Thermocience Diviion, Mechanical Engineering Department, Stanford Univerity, California Non-Equilibrium Phonon Ditribution in Sub-00 nm Silicon Tranitor Intene electron-phonon cattering near the peak electric field in a emiconductor device reult in nanometer-cale phonon hotpot. Pat tudie have argued that ballitic phonon tranport near uch hotpot erve to retrict heat conduction. We reexamine thi aertion by developing a new phonon tranport model. In a departure from previou tudie, we treat iotropic diperion in all phonon branche and include a phonon emiion pectrum from independent Monte Carlo imulation of electron-phonon cattering. We cat the model in term of a non-equilibrium phonon ditribution function and compare prediction from thi model with data for ballitic tranport in ilicon. The olution to the teady-tate tranport equation for bulk ilicon tranitor how that energy tagnation at the hotpot reult in an exce equivalent temperature rie of about 3% in a 90 nm gate-length device. Longitudinal optical phonon with non-zero group velocitie dominate tranport. We find that the reitance aociated with ballitic tranport doe not overwhelm that from the package unle the peak power denity approache 50 W/ m 3. A tranient calculation how negligible phonon accumulation and retardation between ucceive logic tate. Thi work highlight and reduce the knowledge gap in the electro-thermal imulation of tranitor. DOI: 0.5/ Keyword: device, heat tranfer, modeling, nanocale, thermophyical Introduction The quetion of an anomalou temperature rie due to ballitic phonon tranport near the heat ource in a tranitor remain unreolved depite much recent attention 4. It i well known 5 that localized phonon emiion from hot electron near the drain of a emiconductor device reult in a heat ource with dimenion on the order of 0 nm. Figure how a typical heat ource in a bulk ilicon metal-oxide-emiconductor field-effect tranitor MOSFET. The heat generation contour were obtained from a hydrodynamic imulation of electron tranport in a 90 nm gatelength device at a upply voltage, V dd, of.2 V with the gate and the drain terminal biaed at.2 V, and the ource terminal grounded. The peak power denity i nearly 5 W/ m 3 at aturation condition in the device. The patial localization of thi hotpot near the drain i a conequence of the harp peak in the electric field and the hort mean free path for electron-phonon cattering. A olution to the heat diffuion equation for thi ource would predict the temperature field to vary on a length cale comparable to the phonon mean free path in ilicon, which i clearly unphyical. Several ub-continuum treatment of thi problem are available in the literature a mentioned above. The common concluion i that there i a ubtantial difference between the temperature rie predicted by olving the heat conduction equation and the equivalent temperature rie predicted by olving implified verion of the phonon Boltzmann tranport equation BTE. However, pat work ha failed to reach a conenu on the cale of thi difference and what it implie for tranitor deep in the ub- 00 nm regime. Thi i further compounded by the difficulty in experimentally verifying any of the claimed effect. We expect the phonon ditribution function to be nonlocal 6 near the ource, reulting in le heat flow than predicted by the Contributed by the Heat Tranfer Diviion of ASME for publication in the JOUR- NAL OF HEAT TRANSFER. Manucript received June 7, 2005; final manucript received December 2, Review conducted by C. P. Grigoropoulo. Paper preented at the 2004 ASME International Mechanical Engineering Congre IMECE2004, November 3 9, 2004, Anaheim, California, USA. theory of heat diffuion. However, the magnitude of thi reduction varie widely depending on the aumption about the phonon diperion relationhip and the phonon frequencie dominating the heat production. The pioneering work of Lai and Majumdar on concurrent electron and phonon tranport employed the hydrodynamic model for electron and a gray-body two-tep energy conervation model for optical and acoutic phonon. A temperature rie of 7 K wa calculated for the acoutic phonon in a 0.24 m gate-length ilicon MOSFET diipating mw per unit micrometer width of the device. Heat generation wa aumed to be only through optical phonon emiion, and the group velocitie of optical phonon mode were aumed to be zero. To the bet of our knowledge, thi aumption ha been retained in all ubequent model. We note that no major ub-continuum effect wa oberved in thi tudy. However, Chen 7 howed that ballitic heat conduction would decreae thermal conductivity locally in the vicinity of a heated nano-particle when the ize of the particle wa comparable to the phonon mean free path. Expanding upon thi reult, Sverdrup et al. 2 conidered ballitic tranport in a 0.4 m ilicon-on-inulator device. Their model howed the lattice temperature rie to be 60% higher than that obtained uing the heat diffuion equation. Thi work aumed a two-fluid diperion for phonon. More recent effort 3 extended the energymoment formulation to account for phonon polarization in the acoutic mode and include frequency-dependent relaxation rate. While the above reult were baed on energy-moment formulation of the phonon BTE, a recent tudy 4 ued the balliticdiffuive equation BDE to alo how thi extra temperature rie at the ource. Interetingly, none of the ilicon MOSFET with gate length le than 00 nm reported in the literature ee, for example, Ref. 8 demontrate any anomalou thermal behavior to the bet of our knowledge. While elf-heating i a tandard problem in the ilicon-on-inulator device, there are no indication of any anomalou heat ource ize effect a far a their current-voltage characteritic are concerned. Thi bring up everal important quetion: Doe the preence of a ub-continuum ource affect the device characteritic at all? Alternately, do device ource really 638 / Vol. 28, JULY 2006 Copyright 2006 by ASME Tranaction of the ASME

2 v k, N k, = N k, N k, where k, refer to the phonon mode with wave vector k and polarization, v i the group velocity, N i the phonon occupation number, N i the equilibrium Boe-Eintein ditribution function, i the net relaxation time from all cattering event. A common approach to olve the BTE i to write the ditribution function a a departure from equilibrium function,, added to the Boe- Eintein equilibrium ditribution function Fig. Contour of heat generation in a 90 nm gate-length bulk ilicon n-mosfet are calculated uing the hydrodynamic model for electron tranport. The peak power denity at the center i nearly 5 W/ m 3. correpond to the ub-continuum ource that microcale heat tranfer reearch ha targeted in the pat? Are the aumption ued in predicting ub-continuum phonon tranport omehow reponible for the reported temperature jump at the ource? Finally, are any of thee prediction realitic when it come to practical device? In thi paper we attempt to anwer the above quetion by reformulating the problem and relaxing the aumption of a graybody heat ource. Going beyond thi implification require detail of electron-phonon cattering, which we obtain through Monte Carlo imulation. We olve the phonon BTE for the phonon departure from equilibrium taking into account the imulated electron-phonon cattering term a well a phonon diperion in the acoutic and optical branche. In our formulation, we plit the phonon departure from equilibrium function into two component: one that trace the evolution of the emitted phonon before they thermalize through cattering, and another that trace the diffuion of the thermalized phonon. The former i obtained by olving the ballitic BTE in a patial region of the order of a mean free path. The latter correpond to the olution of the BTE in the limit of diffuive tranport. We compare the model prediction with exiting data on hotpot in a ilicon thin film. The olution i extended to predict the temperature field in a bulk ilicon device. We make another departure from previou tudie by uing boundary condition that take into account the thermal reitance due to the enveloping package. We how that the preence of a hotpot impede heat conduction locally at the drain in agreement with earlier tudie. However, the magnitude i much lower than earlier prediction. Baed on thi reult, we conclude that the emiion pectrum in combination with the phonon diperion ultimately determine the magnitude of the reduced thermal conductance near the hotpot. We argue that the approach of olving either the energy moment of the phonon BTE or the BDE with no conideration of electron-phonon cattering, lead to erroneou concluion. Finally, we olve the tranient problem to invetigate phonon accumulation and retardation during witching. Thi work aim to highlight and cloe key knowledge gap in the electrothermal modeling of ub-00 nm gate-length device. 2 BTE Formulation for a Phonon Source We begin the dicuion on our model by firt reviewing the tandard form of the phonon Boltzmann tranport equation. Thi help u to clarify the reaoning behind our propoed model. Under the relaxation time approximation, the phonon Boltzmann tranport equation at teady tate i N = N T + k, 2 Subtitution in the BTE yield an explicit olution for k, under the aumption that term of econd order in the temperature gradient and the term involving the gradient in the departure function are much maller than the other term, and may be neglected. The departure from equilibrium function thu obtained i proportional to the temperature gradient and conform to the Fourier heat flux law N k, = k, v k, T 3 T The above firt order approximation work well provided the temperature gradient i mall enough that the change in temperature over the relaxation length i much maller than the abolute value of the temperature 9. That i T v T x 4 a condition that i uually valid. However, when the econd order derivative of T varie on a length cale comparable to the mean free path, the heat flux i nonlocal in the phonon ditribution function. Claro and Mahan 6,0 howed that thi lead to much higher temperature gradient than thoe predicted by the Fourier law of heat conduction. We expect a confined high denity ource of phonon uch a a device hot pot to alo how non-local effect. In thi cae the nonlocality arie not due to higher order temperature derivative but due to the gradient in the departure function itelf. The departure from equilibrium function will be eentially dictated to firt order by the ditribution of emitted phonon in the real and reciprocal pace of the crytal. Thu, large patial gradient in the ource function that arie when the hotpot i mall compared to the phonon mean free path, tranlate to large gradient in the departure function a well, that i * k, x * ṅ * k, k, 5 to firt order where the ṅ k, i the net phonon emiion from electron-phonon cattering. The aterik in the upercript indicate a normalized quantity, with the mean free path and the equilibrium ditribution function choen a the appropriate caling factor. The term on the right i large provided the electron-phonon relaxation rate i greater than the phonon-phonon relaxation rate. The diparity in the electron and phonon energy relaxation rate i etimated to be about two order of magnitude in a ubmicrometer MOSFET operating at room temperature. The relaxation rate i about 0 3 for electron and about 0 for optical phonon,2 in ilicon. Thu, the gradient in the departure from equilibrium function cannot be neglected in the cae of electro-thermal tranport in tranitor a it i in the formulation for thermal conductivity. We modify the formulation by firt including a ource term in the phonon BTE that provide the net phonon emiion rate due to electron cattering. The actual ource function i obtained through detailed Monte Carlo imulation a decribed later. Other phonon cattering event are modeled through the relaxation time approxi- Journal of Heat Tranfer JULY 2006, Vol. 28 / 639

3 mation and a reciprocal um of the relaxation time i taken to repreent the overall relaxation rate of a mode. The expreion for the relaxation time can be obtained from firt order perturbation theory and empirical fit have been developed by Holland 3. Thu, the teady-tate evolution can be written a v k, N k, = N k, N + ṅ k, 6 k, where ṅ i the phonon emiion. Proceeding a in the formulation for thermal conductivity, we write N a a mall perturbation,, over the equilibrium ditribution function a given in Eq. 2. The BTE, written in term of k,,i N v k, + T = T k, + ṅ k, 7 k, To obtain a olution to the non-homogeneou BTE, we firt ubtract the near-equilibrium departure function a given by Eq. 3 from k,. The remainder i the far-from-equilibrium departure function that i due to the phonon emiion ource term. Thu, N k, = k, v k, T T + n k, 8 where n k, i the far-from-equilibrium departure function. The idea behind Eq. 8 i that the phonon flux at any point in pace i due to a near-equilibrium part that obey the Fourier law uperpoed on a econd contribution due to the emiion pectrum that doe not obey the Fourier law. A we move farther away from the hotpot, we expect the econd contribution to diminih trongly a the emitted phonon thermalize, and the Fourier law contribution to increae proportionately to maintain energy continuity. The BTE thu become N v k, n k, k, v k, T = T n k, + ṅ k, 9 k, Additionally, macrocopic energy continuity mut be atified and thi i written at teady tate a J = Q 0 where J i the heat flux vector and Q i the heat generation rate. Thee are expreed in term of the non-equilibrium ditribution function a follow J = Q = 8 3 ṅ k, k, dk 3 v k, n k, k, dk v k, k, v k, T k,dk N 8 T 2 where the integration i taken over the firt Brillouin zone. The econd term in the heat flux vector can be written in term of the thermal conductivity tenor, K, o that Eq. 2 reduce to J = 3 v k, n k, k, dk K T 3 8 With the above expreion for the heat flux vector, Eq. 0 can be integrated to give 8 3 v k, n k, k, dk K T = Q dr 4 Equation 9 and 4 form a cloed ytem with the unknown being n k, and T. However, thi ytem i till difficult to olve without further implification. We note that the econd term on the left in Eq. 9 Fig. 2 A cro ection of the experimental tructure ued to probe ballitic conduction near a doped reitor in ilicon 4 i hown at the top. The reitor acted a a hotpot inide the ilicon membrane. The ymmetry in the problem i ued to olve the BTE in the domain hown below. i a econd order term in the temperature gradient and can be neglected. Thi approximation caue the temperature field to be independent of the emitted phonon until they thermalize through cattering. The field doe influence the far-from-equilibrium ditribution indirectly through the temperature dependence of the phonon cattering rate. With the above approximation, the BTE i of the form v k, n k, = n k, + ṅ k, 5 k, To olve the ytem we need to pecify the boundary condition. We take thi up in the following ection when we develop analytical olution for pecific cae. Knowing the ditribution function n k, the continuity of energy mut be olved for the temperature field. Proceeding a above, we can alo derive time dependent equation. Auming that the temporal variation in temperature i much lower than that in the departure function, we can drop the tranient temperature term in the BTE. The tranient form i thu n k, t + v k, n k, = n k, k, + ṅ k, t The time dependent energy 3 conervation equation i n k, k, dk 8 C T t + t = Q where C i the lattice heat capacity v k, n k, k, dk K T 7 3 Comparion With Data for Ballitic Tranport In thi ection we ue the above model to predict the thermal reitance aociated with ballitic tranport near a hotpot in ilicon. Sverdrup et al. 4 ued heating in a doped reitor thermometer in ilicon to create a micrometer cale phonon ource in a membrane tructure, hown chematically in Fig. 2. In thi experiment, a 3- m-wide region in a 5- m-thick n-type ilicon membrane wa p doped to a depth of about 0.3 m. A bia applied acro the terminal of the reitor forced a current through the doped region. By revere biaing the p-n junction between the reitor and the ubtrate, the current wa confined within the doped region. Joule heating in the doped reitor induced a temperature rie in the membrane tructure. The thermal reitance of 640 / Vol. 28, JULY 2006 Tranaction of the ASME

4 the ilicon membrane wa determined from the temperature rie detected by the metallic enor, placed parallel to the doped reitor, and by the reitor itelf. The experiment wa conducted in the ambient temperature range of about K. The thermal reitance howed a large deviation from the prediction of the Fourier law at low temperature indicating non-diffuive behavior cloe to the reitor. We now calculate the thermal reitance meaured in the experiment uing our model. The problem i much implified by conidering the ymmetry of the ource and the boundary condition. We expect cattering at the membrane boundary to be largely diffue in the temperature range of the experiment ince the phonon wavelength i comparable to the urface roughne. Thu, the BTE of Eq. 5 may be olved in the domain hown at the bottom of Fig. 2 with homogeneou boundary condition at all four boundarie. The two-dimenional BTE in rectangular coordinate i v x n k, x + v n k, y y = n k, + ṅ k, k, 8 We olve the above wave equation by the method of characteritic and get the olution in velocity pace a follow We define the equivalent temperature, T EQ, imilar to previou work,7,6 on non-equilibrium phonon tranport 3 N T EQ dk = 3 N T dk n dk 2 We note that the firt term on the right in Eq. 8, which i proportional to the temperature gradient, doe not contribute to the energy integral. With the above definition we can repreent the nonequilibrium among the mode in term of the equivalent temperature. In order to obtain R BTE we rewrite Eq. 2 in term of the lattice heat capacity, C, and etimate the difference in the peak lattice and equivalent temperature. The integration over phonon frequencie i implified by auming phonon near the ource to have a mean free path and a mean relaxation time. Uing the velocity pace olution given above C T EQ T ref = C T T ref + 4 ṅ g d 4 W H 0 + H W 0 d d 2 2 e d d 2 2 e h w or, T EQ T x=0 = C ṅ g d f w,h, where the function F and G are defined a F x,y 0 G x,y 0 y ṅ x,y v y x ṅ x,y v y e y y v y dy e x x v x dx 9 Since the reitor wa p doped, phonon emiion in the experiment wa from hole-phonon cattering. Unfortunately, the emiion pectrum for hole at the electric field employed in the experiment i not available in the literature. In the abence of any pectral information about the emitted phonon, we proceed by integrating the departure function over frequency to get the net power denity which i known from the experiment. The net thermal reitance at the ource i the um of the diffuion and BTE reitance. Thu R = R diffuion + R BTE with R diffuion = K M W 2H + ln H 2h + w 20 where w and h are the cro-ectional width and height, repectively, of the doped reitor, H i the thickne of the membrane, W i half the width of the membrane, K M i the thermal conductivity in the membrane. Detail regarding derivation of the twodimenional thermal reitance from diffuion theory may be found in Hahne and Grigull 5. The BTE reitance of Eq. 20 i a hypothetical reitance and i a meaure of the difference between the diffuion temperature and the equivalent temperature calculated from the BTE model. Q max = f w,h,w,h, 22 C where, are the direction coine along x and y, repectively, i the phonon mean free path, and Q max i the peak heat generation rate, at x 0. The heat generated in the diode i fairly uniform acro it cro ection and we may ue the average power intead of Q max. The effective cattering time i given by the ratio of the ummation of ṅ over all mode to the peak generation rate and i a fitting parameter in thi calculation ince the function ṅ i unknown. The thermal reitance utaining the peak difference between T EQ and T i R BTE = T EQ T x=0 q Q max = f w,h,w,h, 23 qc where q i the total heat current flowing to the ink. The geometry factor f i unity at room temperature, where w,h, and decreae lightly with temperature a the mean free path increae in relation to the ource dimenion. The dominant effect of the decreaing temperature i to increae R BTE. The net increae i, in turn, due to the decreae in the heat capacity and the increae in the effective cattering time of emitted phonon. Thi i hown in Fig. 3 which compare the thermal reitance meaured by the doped reitor at different bae temperature with prediction baed on Eq. 20 and 23. We note that in the abence of information about the emiion pectrum, we have fit the model uing a contant value for the effective cattering time for emitted phonon,, which i on the order of 0 n. Thi repreent a weighted average of the cattering time of all emitted phonon, the weight being the relative excitation number of different mode. A better fit could be obtained in principle if we include a temperature dependence of the mean cattering time. However, thi i quite complicated in practice becaue it Journal of Heat Tranfer JULY 2006, Vol. 28 / 64

5 Fig. 3 A comparion of the thermal reitance meaured at different temperature. Prediction baed on the propoed model are given by the continuou line. The prediction uffer from lack of information on temperature dependent cattering rate of phonon emitted by hole in ilicon. involve the temperature dependence of not jut the phonon relaxation time but alo that of the weight. The latter would require detailed Monte Carlo imulation of hole-phonon cattering at each bae temperature. Hence, we omit any temperature dependence of the mean cattering time in the abence of a clear intuition about the temperature dependence of the weight. The dahed line how the thermal reitance of the membrane tructure calculated from diffuion theory uing the bulk thermal conductivity of ilicon. Phonon boundary cattering reduce the thermal conductivity of the membrane o that the bulk prediction are ignificantly lower than the data. The dotted line i the thermal reitance calculated from diffuion theory Eq. 20 but uing the thermal conductivity of the membrane meaured in itu. The meaurement increaingly differ from prediction of diffuion theory a the temperature decreae. We attribute thi to ballitic phonon tranport near the doped reitor. Phonon emitted at the heat ource increaingly undergo boundary cattering a the ambient temperature decreae. Thi lead to a temperature lip at the heat ource and caue the temperature rie to deviate from heat diffuion theory. The olid line repreent the um total of the BTE and diffuion reitance. The prediction from Eq. 20 agree well with the data at 00 and 40 K, where the departure from diffuion theory i ubtantial. The model i, however, unable to match the data at 90 K. Additionally, the trend curve for the data appear to have an inflection point between 40 and 90 K. The ub-continuum contribution appear to aymptote at low temperature, wherea the model predict ever-increaing contribution due to increaing mean free path. We peculate that thi deviation from the model may be due to the mean free path in the thin film becoming contrained by boundary cattering at low temperature. We note that a mean free path baed on the thin-film thermal conductivity and calculated from the imple kinetic theory expreion of =3 K/Cv would not reproduce thi trend. Thu, although the qualitative behavior of the trend in the data i undertandable, we are unable to obtain a good quantitative match, primarily due to lack of information about the temperature dependence of the mean cattering time for reaon dicued above. 4 Application to a Bulk Silicon MOSFET In thi ection we ue our model to compute the phonon ditribution function in a 90 nm gate-length bulk ilicon n-type MOS- FET NMOS. We pecifically chooe an NMOS device becaue it higher power denity would increae the magnitude of any ub-continuum effect. A hown in Fig., the hot pot predicted Fig. 4 The boundary condition ued in the device calculation are hown above. Heat i aumed to flow out through the metallic contact at the top which act a fin. The heat tranfer coefficient i relatively high due to thi preading effect. Mot of the heat flow i toward the heat ink through the bulk ilicon at the bottom. by a hydrodynamic calculation i formed in the channel under the gate toward the drain ide. We note that device Monte Carlo imulation, however, predict the location to be hifted more toward the drain. The precie location i unimportant in thi work ince the boundarie for the thermal calculation are much larger in extent when compared to thi hift in the location of the hot pot. We are more intereted in the hape of the hot pot, which i predicted reaonably well by hydrodynamic calculation. The hape i approximately emicylindrical in a bulk MOSFET, with the axi aligned along the width of the device. In order to proceed toward a emi-analytical olution to our model, we approximate the location of the ource to be midway between the ource and the drain. A noted above, thi hift in location i much maller than the lateral extent of the device. Hence, we do not expect a major difference in our reult due to thi approximation. Figure 4 how the device with the approximated ource at the origin. The top boundary i aumed to be adiabatic due to the inulating gate oxide. In reality, there i a mall heat flux acro thi interface which i ignored in our calculation to implify the BTE olution. The ide boundarie are both adiabatic due to thick iolation oxide. Some heat i lot through the metallic contact a hown. We obtain the heat tranfer coefficient by treating the metallic interconnect a fin 7. The heat produced in the tranitor motly flow out at the bottom toward the heat ink, after paing through the bulk ilicon. We aume the power denity at the ource to be 5W/ m 3, caled down by an activity factor of 0. to account for average circuit activity, and the radiu to be 20 nm which cloely approximate realitic device hotpot. The device hot pot i further aumed to be a tep function a hown in Fig. 5. The phonon pectrum in the heat production region i obtained through a Monte Carlo imulation a decribed below. 4. Electron-Phonon Scattering Uing the Monte Carlo Method. The purpoe behind formulating our BTE model in term of the non-equilibrium phonon ditribution function, a oppoed to ome fictitiou temperature, i to olve it with a frequency-dependent ource term without invoking quetionable argument about equivalent temperature. Before we can olve the implified phonon BTE, however, we need the ource term, ṅ k,, appearing in Eq. 5. Ideally, the ource term hould couple the conervation of energy equation for electron and phonon. In thi work, we avoid that coniderable complication by decoupling electron and phonon tranport. In effect, we aume electronphonon cattering to proceed between non-equilibrium electron and equilibrium phonon. We employ Monte Carlo imulation of electron-phonon cattering to extract phonon emiion rate in 642 / Vol. 28, JULY 2006 Tranaction of the ASME

6 Fig. 5 The cylindrical geometry reulting from the aumption that the hotpot i located at the center of the channel erve to reduce the complexity. We further aume a tep profile with the heat ource confined to a radiu a, a hown chematically. ilicon at a contant electric field. Detail of thi work are provided eparately in Ref. 8,9 and are beyond the cope of thi paper. We include a brief overview here for the reader convenience. Our Monte Carlo approach employ analytic approximation for the electron energy band and the phonon diperion. Thi band approximation for electron i reaonable for device voltage near or below the ilicon band gap. ev, uch a thoe of future nano-technologie, and it repreent an efficient implementation of the otherwie time-conuming Monte Carlo approach. The imulation treat all phonon cattering event a inelatic. Electron exchange energy with the lattice a determined by the phonon diperion and cattering election rule. Scattering with intravalley longitudinal acoutic LA and tranvere acoutic TA phonon, a well a with intervalley longitudinal optical LO and tranvere optical TO phonon, i conidered individually. The phonon diperion relationhip i ued to compute the final electronic tate in a manner that conerve both momentum and energy. During the imulation all phonon aborbed and emitted are tallied and net phonon emiion tatitic can be computed. Figure 6 how the analytic quadratic phonon diperion ued in thi work and the computed phonon emiion pectrum at a field of 4 MV/m. To facilitate comparion the vertical axe are drawn with the ame energy unit. The peak in the phonon generation rate occur due to election rule for the electron-phonon interaction. The relative magnitude of the peak depend on the choice of cattering deformation potential, which are calibrated acro a wide temperature range 8. In term of energy, LO emiion make up about half, LA emiion about a third and TO emiion about a tenth of the heat generation rate in bulk ilicon at typical electric field. In the next ection, we ue the emiion rate to compute teady-tate phonon occupation for individual phonon frequencie. 4.2 Analytical Solution to the Phonon BTE. The twodimenional BTE for the device problem i a given in Eq. 8. By approximating the ource to be a emicylinder located at the origin, we can take advantage of the reulting radial ymmetry. We convert Eq. 8 to radial coordinate a follow v r n k, r + v 2 n k, r v r v v r n k, + n k, = ṅ k, r v k, 24 where v r and v are the radial and tangential velocitie a depicted in Fig. 5. Due to axial ymmetry, there i no dependence on the azimuthal angle,. To obtain an analytic olution we plit the olution into two domain, ubcripted with indice and 2 a hown chematically in Fig. 5. For r a, where a i the radiu within which the ource i confined, the BTE i nonhomogeneou, wherea for r a the equation i homogeneou 2 n v r r + v n r 2 n 2 v r r + v n 2 r v r v v r r v r v v r r n + n v = ṅ n 2 + n 2 v =0 25 The olution to Eq. 25 can be obtained by reducing the ytem to a form that correpond to the BTE for electron tranport in a metallic wire 20 for which the general olution i known. We find the general olution to Eq. 25 to be rv r n = ṅ exp v 2 r + v 2 rv,v r f 2 + v 2 rv r n 2 = ṅ exp v 2 r + v 2 2 rv,v r f 2 + v 2 26 where f and f 2 are arbitrary function to be determined from boundary condition. The boundary condition for the problem are n 2 r = l, v r,v =0 n r =0, v r,v = n r =0, v r,v Fig. 6 The phonon diperion in ilicon along 00 i obtained from a fit to neutron cattering data from Dolling 23. The above diperion i aumed to hold along all direction in our calculation. At an electric field of 4 MV/m, the ource term in the BTE, ṅ k,, ha the frequency dependence hown on the right. n r = a, ± v r,v = n 2 r = a, ± v r,v Applying the boundary condition, the final olution i n = ṅ exp rv r + a 2 v 2 r + v 2 r 2 2 v n 2 v r 0 =2ṅ exp rv r v r 2 + v 2 v r 2 + v 2 27 inh a 2 v 2 r + v 2 r 2 2 v v 2 r + v 2 n 2 v r 0 =0 28 In order to evaluate the above expreion, we need the phonon relaxation time. For the relaxation time of the acoutic mode, we ue the expreion developed by Holland 3 and fit to thermal conductivity data for bulk ilicon. We ue a ingle value of 0 p for all optical mode which i on the order of the room temperature lifetime of zone-center optical mode in ilicon meaured by Raman pectrocopy 2. We now compare the model prediction with a numerical olution to the heat diffuion equa- Journal of Heat Tranfer JULY 2006, Vol. 28 / 643

7 Fig. 7 The phonon number denity for different branche i hown cloe to the hotpot, which i 20 nm in extent. The LO contribution dominate the number denity and conequently, the energy denity. tion in a 90 nm gate length device. It i intereting to compare the non-equilibrium contribution of the different phonon branche to the number denity and the non- Fourier heat flux. Due to trong emiion in the LO mode, the LO contribution i more than 50% of the total non-equilibrium number denity. Figure 7 how the contribution from all four branche near the hotpot. The energy denity i alo proportionately higher for the LO branche ince LO frequencie are higher compared to other branche. The LO and the LA branche are dominant contributor to heat conduction. Thi i evident from Fig. 8 where the contribution of individual branche to the heat flux are plotted. We note that the flux i zero for all branche at r=0 due to the impoed ymmetry condition. The contribution of TA and TO are inignificant. The non-fourier heat flux diminihe rapidly outide the hotpot, a expected from the ballitic nature of the equation. At the point where it peak, the non-fourier heat flux i till only 5% of the total heat flux. Therefore, heat conduction i dominated by the Fourier contribution according to thi Fig. 8 The non-fourier heat flux due to different phonon branche i hown. Ballitic heat conduction i predominantly through LO phonon. The cumulative flux i only about 5% of the total, the ret being the flux due to thermalized phonon that obey the Fourier law. Fig. 9 A comparion of the temperature field obtained from continuum heat diffuion and phonon heat conduction i hown above. Ballitic conduction augment the overall thermal reitance between the tranitor and the ambient by about 3%. model. The impact of the non-local departure from equilibrium i, however, quite trong near the hotpot in term of the local energy denity, which i characterized by the equivalent temperature of Eq. 2. Figure 9 compare the temperature field in the device obtained from the heat diffuion equation with that from the propoed model. The BTE model predict an exce temperature rie of 5 K, above the peak temperature rie of about 38 K from heat diffuion. The LO branch i hotter than the acoutic branche, with the branch temperature being 358 K. We note that thee magnitude are very enitive to boundary condition. Baed on thee number, the peak equivalent lattice temperature rie i nearly 3% more than the peak temperature rie computed auming heat diffuion for a 90 nm device. Mot of the exce energy i reident in longitudinal optical phonon. The equivalent temperature rie for LO mode alone i 37% higher than that for the entire phonon enemble, and i a meaure of the nonequilibrium among phonon. We now examine the implication of the above reult in the context of the quetion poed at the beginning. We emphaize that the major departure in the preent treatment come from handling of the heat ource term in the phonon BTE and from olving directly for phonon ditribution function intead of hypothetical equivalent temperature. The econd apect i critical becaue it altogether avoid attaching hypothetical equivalent temperature to phonon branche a well a defining energy exchange between branche. We feel that the latter approach i arbitrary becaue energy mixing between phonon mode doe not proceed trictly on the bai of polarization. Polarization merely decide the ymmetry for election rule in mot cae. The firt thing to note in the above reult i the preponderance of LO phonon, not jut in term of the energy denity but alo in term of the heat flux. Earlier tudie have repeatedly aumed optical phonon to poe zero group velocitie baed on the uual flatne of their diperion relation. We note that in reality the g-lo phonon, that i mot likely to catter with electron, poee a group velocity of about 500 m/. Thi i reflected in our reult in term of the dominating LO contribution to the heat flux. Further, by including boundary condition that take into account thermal reitance all the way up to the package, we are able to gauge the importance of 644 / Vol. 28, JULY 2006 Tranaction of the ASME

8 Fig. 0 The ditribution of free path in room temperature ilicon a a function of phonon frequency and polarization i hown. The mean free path of phonon emitted by hot electron in a device are alo given for comparion. The ue of a gray-body approximation for the heat ource would lead to ignificant error in predictiing the ballitic nature of tranport near hotpot. ub-continuum reitance in relation to that of the continuum junction to ambient reitance. While ballitic tranport erve to create hotpot more intene than predicted otherwie by diffuion theory, the chip package remain the dominant thermal reitance. Thu, our reult do not ugget the ource-ize effect to lead to device reliability iue in bulk device at current electric field level. If, however, the peak electric field and the current denity increae in future device o a to augment the peak volumetric power denity by an order to magnitude to about 50 W/ m 3, then the reultant phonon denity at the hotpot would lead to reliability concern in the drain. In thi cae, we etimate that ub-continuum effect would increae the thermal reitance by about 30 40%. An important apect not conidered thu far i the implication of ub-continuum effect on leakage current. The drain to ubtrate junction i an important ource of leakage current. High phonon denitie in the drain could promote leakage current by increaing the thermal energy of the electron. To the bet of our knowledge, thi conideration i not available in the literature. We caution that the ue of equivalent temperature obtained from BTE calculation in the uual empirical relation between leakage current and temperature ee 2, for example would lead to unreaonable prediction ince uch relation are derived on the aumption of thermodynamic equilibrium. Finally, we examine the often-ued concept of a phonon mean free path in the context of tranport in ilicon tranitor. Generally, a phonon mean free path on the order of 00 nm i ued for gray-body calculation. However, from our conideration of electron-phonon cattering and ubequent phonon tranport, we find that the above figure derived from conideration of thermal conductivity of ilicon i mileading, when it come to tranport near hotpot in tranitor. A pointed out above, tranport in the vicinity of the hotpot i far from equilibrium and ha little to do with thermal conductivity of the medium. The mean free path of phonon emitted at the hot pot depend trongly on the phonon emiion pectrum. Figure 0 how the free path of phonon in ilicon a a function of frequency and polarization at 300 K baed on the Holland model 3 for thermal conductivity. Alo hown are the mean free path of phonon in the emiion pectrum for different branche. A evident from the figure, the phonon dominating the emiion pectrum the LO phonon have a mean free path of about 5 nm, much maller than the commonly ued figure of 00 nm. We explain the origin of the large deviation from heat diffuion theory reported in previou tudie a follow. Eentially, none of thee tudie decribe a pure ource-ize effect. The Fig. A tep-like phonon ource ymmetric about x=0 with a uniform power denity of 5 W/ m 3 i conidered for a ample tranient calculation. The extent of the ource, a, i taken to be 20 nm, conitent with device hotpot. A ink at 300 K i aumed to be preent at x=±l where l=300 nm. choice of a large mean free path baed on a thermal phonon ditribution and the election of a thin film geometry ilicon-oninulator device lead to the termination of the hot phonon free path at the film boundary. Thi give rie to a large temperature lip at the heat ource in the ame manner a in the low temperature experiment dicued previouly. The ize of the ource i not the primary factor but rather the thickne of the film in relation to the mean free path ued in the tudy. The choice of a zero group velocity for optical phonon erve to increae the magnitude of the lip. Baed on our calculation above, the phonon ditribution at the heat ource correpond to a maller mean free path. Additionally, the optical contribution correpond to a non-zero group velocity. Thu, we do not expect the heat ource in real device operating at room temperature to demontrate a temperature lip unle the device i built on ultrathin film ilicon with thickne down to 5 nm. Thi etimate of the thickne i baed on the aumption that the peak electric field remain imilar to that conidered here. However, for a lip to occur phonon mut be emitted in the direction of the thin-film boundarie. Since quantum confinement of electron in an ultrathin film ilicon channel lead to the heat ource migrating away from the gate oxide interface to the middle of the channel, it i difficult to predict if the condition required for the temperature lip would be exactly atified in thi geometry. 5 Tranient Phonon Ditribution During Switching Although the above analyi conidered a teady-tate phonon ource, in reality, the ource i time dependent if the tranitor i operating in a circuit. A tranitor in a digital circuit typically witche on a time cale of about 00 p. Ignoring leakage power, a complementary metal-oxide-emiconductor device diipate power over only a fraction of thi period, referred to a the duty cycle, which i typically le than 0.3. Thi witching time i, however, comparable to the relaxation time of ome of the phonon mode. Therefore, it i important to conider whether there i any accumulation of phonon from one witch to another. To keep the problem tractable, we ignore the device geometry completely and conider only a tep-like phonon ource in one dimenion, a hown chematically in Fig.. The ource i ymmetric about x=0 where x i the coordinate direction. We place phonon ink at x=±l. The time dependent BTE to be olved i n k, t + v x n k, x where f t i the witching function = n k, k, + ṅ k, f t 29 Journal of Heat Tranfer JULY 2006, Vol. 28 / 645

9 Fig. 2 Contour of the normalized phonon number denity, paced by 0.0, are hown a a function of poition and time for the longitudinal optical phonon at 4 THz. No wave retardation i evident ince the emitted phonon have large enough group velocitie. The accumulation of LO phonon near the ource during the time period of power diipation i clearly viible. f t = t mod t o =0 t mod t o 30 with t o being the witching period and being the duty cycle or the fraction of the time period that the device i on. Further, the boundary condition are n + k, x =0, v x,t = n k, x =0, v x,t n k, x = l, v x,t =0 T x = l,t = T o 3 + where n k, i the departure function for phonon traveling to the right in Fig. and n k, i the function for phonon traveling to the left. Equation 29 i linearized by computing the relaxation time at the temperature field obtained from the heat diffuion equation. The olution to the tranient problem i n + k, = e x x ṅ v x v 0 x f t + x x e x v v x x dx l + 0 ṅ v x f t x + x e v x x v x dx Fig. 3 Contour of the normalized phonon number denity, paced by 0.0, are hown a a function of frequency and time at x=0. There i no phonon accumulation for a witching period of 00 p with a duty cycle of 30%. n k, = e x l ṅ v x v x x f t x x e x v v x x dx 32 We now decribe the numerical reult for a hotpot with a power denity of 5 W/ m 3. The numerical value for l, a, and the boundary temperature, T o, ued in thee calculation are given in Fig.. A phonon emitted during device witching may how a wave retardation in time if the time cale for witching i comparable to a characteritic time, obtained by dividing the length cale for emiion by the group velocity of the emitted phonon. Retardation would caue the phonon ditribution to be hitory dependent. In our computation, however, we do not find any evidence of retarded phonon. The patial and temporal ditribution for the 4 THz longitudinal optical phonon i hown in Fig. 2. We attribute the abence of retardation effect to the fact that phonon dominating the emiion pectrum have non-zero group velocitie on the order of 000 m/. Another important iue i whether the witching i fat enough to caue phonon to remain in perpetually trong nonequilibrium or to even caue accumulation over time. We do not find thi to be the cae for typical clock cycle. Thi i evident from Fig. 3 which how the number denity contour at x=0 a a function of phonon frequency and witching time. The emitted phonon thermalize within the off tate of the device and there i no accumulation. In the abence of phonon retardation, the accumulation of phonon from one logic tate to another will only occur when the time between ucceive tate approache the relaxation time of the dominant LO phonon. Thi number i about 0 p to the bet of our knowledge. Thu, unle the witch period approache uch hort time, we do not anticipate any phonon accumulation. However, we note that thi aertion depend on the accuracy of the relaxation time. A detailed invetigation into the accuracy of cattering rate i thu neceary before making a definitive concluion about thi apect. 6 Concluding Remark In thi paper, we have preented a new model for determining the non-equilibrium phonon ditribution function in emiconductor device, tarting from the phonon BTE. We olved a twodimenional form of the BTE to compare our model with previou thermal reitance data on hotpot in ilicon. The thermal reitance i een to cale a the ratio of the peak power denity to the lattice heat capacity. We conidered a teady-tate hotpot in a 90 nm gate-length bulk ilicon tranitor. The ource ditribution i taken from our prior work on Monte Carlo imulation of electron-phonon cattering. The peak equivalent temperature rie i nearly 3% more than the temperature rie from heat diffuion, with mot of the exce energy reident in longitudinal optical phonon. The olution to the tranient BTE how that the emitted phonon are able to relax completely at current witching peed but may accumulate if the witching period i reduced by half. We find that though ballitic tranport near the hotpot introduce an additional thermal reitance to that predicted by diffuion theory, the dominating contribution to the total reitance i till from the package. The ub-continuum contribution can be ignificant at higher peak electric field and current denitie, both being poible in future nanotranitor. Longitudinal optical phonon dominate the emiion pectrum from electron-phonon 646 / Vol. 28, JULY 2006 Tranaction of the ASME

10 cattering and, conequently, the energy denity and heat flux at the hotpot. In a clear departure from the commonly ued aumption, the dominating LO mode do not have a near-zero group velocity. Alo, the mean free path of the emitted phonon i ignificantly horter than that for thermal phonon at 300 K. Thi erve to decreae the ub-continuum ize effect. Additionally, the thickne of the ilicon film in a ilicon-on-inulator device i important in determining the onet of ub-continuum ource-ize effect. For a temperature lip to occur at the heat ource, the mean free path of the emitted phonon hould be comparable to the film thickne. The dimenion of the ource play a econdary role in determining the ize effect. We are unable to gauge the impact of ub-continuum phonon conduction on leakage current ince current method for evaluating leakage do not account for evere nonequilibrium. Thi i an important area for future work. Finally, we remark that, although there ha been coniderable progre in technique to olve the phonon BTE, our knowledge of phonon relaxation time at large energy denitie remain poor. Thi, however, i a prerequiite for accurately predicting non-local effect in future tranitor. The relaxation rate ued in thi work were all derived for nearequilibrium tranport and, hence, their validity at uch large excitation a near a hotpot remain unknown. The author have invetigated thi apect through molecular dynamic imulation in a eparate work 22. Acknowledgment The author acknowledge upport from the Semiconductor Reearch Corporation through tak 043. S.S. wa upported through graduate fellowhip from the Intel Corporation and the Powell Foundation. S.S. thank Dr. Ravi Praher at Intel Corporation, Arizona, for helpful dicuion on device boundary condition. Reference Lai, J., and Majumdar, A., 996, Concurrent Thermal and Electrical Modeling of Sub-Micrometer Silicon Device, J. Appl. Phy., 79, pp Sverdrup, P. G., Ju, Y. S., and Goodon, K. E., 200, Sub-Continuum Simulation of Heat Conduction in Silicon-on-Inulator Tranitor, ASME J. Heat Tranfer, 23, pp Narumanchi, S. V. J., Murthy, J. Y., and Amon, C. H., 2004, Submicron Heat Tranport Model in Silicon Accounting for Phonon Diperion and Polarization, ASME J. Heat Tranfer, 26, pp Yang, R., Chen, G., Laroche, M., and Taur, Y., 2005, Simulation of Nanocale Multidimenional Tranient Heat Conduction Problem Uing Ballitic- Diffuive Equation and Phonon Boltzmann Equation, ASME J. Heat Tranfer, 27, pp Sinha, S., and Goodon, K. E., 2002, Phonon Heat Conduction From Nanocale Hot Spot in Semiconductor, 2th International Heat Tranfer Conference, Grenoble, France. 6 Mahan, G. D., and Claro, F., 988, Nonlocal Theory of Thermal Conductivity, Phy. Rev. B, 38, pp Chen, G., 996, Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticle, ASME J. Heat Tranfer, 8, pp Yeo, Y. C., Subramanian, V., Kedzierki, J., Xuan, P., King, T.-J., Bokor, J., and Hu, C., 2000, Nanocale Ultra-Thin-Body Silicon-on-Inulator P-MOSFET With a SiGe/Si Heterotructure Channel, IEEE Electron Device Lett., 2, pp Klemen, P. G., 95, The Thermal Conductivity of Dielectric Solid at Low Temperature Theoretical, Proc. R. Soc. London, Ser. A, 208, pp Claro, F., and Mahan, G. D., 989, Tranient Heat Tranport in Solid, J. Appl. Phy., 66, pp Ferry, D. K., 2000, Semiconductor Tranport, Taylor and Franci, New York. 2 Menèndez, J., and Cardona, M., 984, Temperature Dependence of the Firt- Order Raman Scattering by Phonon in Si, Ge, and alpha-sn: Anharmonic effect, Phy. Rev. B, 29, pp Holland, M. G., 963, Analyi of Lattice Thermal Conductivity, Phy. Rev., 32, pp Sverdrup, P. G., Sinha, S., Aheghi, M., Srinivaan, U., and Goodon, K. E., 200, Meaurement of Ballitic Phonon Conduction Near Hot Spot in Silicon, Appl. Phy. Lett., 78, pp Hahne, E., and Grigull, U., 975, Shape Factor and Shape Reitance for Steady Multidimenional Heat Conduction, Int. J. Heat Ma Tranfer, 8, pp Chen, G., 200, Ballitic-Diffuive Heat-Conduction Equation, Phy. Rev. Lett., 86, pp Goodon, K. E., and Flik, M. I., 992, Effect of Microcale Thermal Conduction on the Packing Limit of Silicon-on-Inulator Electronic Device, IEEE Tran. Compon., Hybrid, Manuf. Technol., 5, pp Pop, E., Dutton, R. W., and Goodon, K. E., 2004, Analytic Band Monte Carlo Model for Electron Tranport in Silicon Including Acoutic and Optical Phonon Diperion, J. Appl. Phy., 96, pp Pop, E., Dutton, R. W., and Goodon, K. E., 2005, Monte Carlo Simulation of Joule Heating in Bulk and Strained Silicon, Appl. Phy. Lett., 86, p Dingle, R. B., 950, The Electrical Conductivity of Thin Wire, Proc. R. Soc. London, Ser. A, 20, pp Groeeneken, G., Colinge, J. P., Mae, H. E., Alderman, J. C., and Holt, S., 990, Temperature Dependence of Threhold Voltage in Thin-Film SOI MOSFET, IEEE Electron Device Lett.,, pp Sinha, S., Schelling, P. K., Phillpot, S., and Goodon, K. E., 2005, Scattering of g-proce LO Phonon at Hotpot in Silicon, J. Appl. Phy., 97, pp Dolling, G., 963, Lattice Vibration in Crytal With the Diamond Structure, Sympoium on Inelatic Scattering of Neutron in Solid and Liquid, Chalk River, Canada, pp Journal of Heat Tranfer JULY 2006, Vol. 28 / 647