Simulation of temperature profile for the electron- and the lattice-systems in laterally structured layered conductors
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug epl draft Simulation of temperature profile for the electron- and the lattice-systems in laterally structured layered conductors
L. Yang , , R.J. Qian , , Z.H. An , , ( a ) , S. Komiyama , , and W. Lu , Key Laboratory of Surface Physics, Institute of Nanoelectronic Devices and Quantum Computing, Key Laboratoryof Micro and Nano Photonic Structures (Ministry of Education), Department of Physics, Fudan University - Shanghai200433, China National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, The Chinese Academy of Sciences- Shanghai 200083, China Collaborative Innovation Center of Advanced Microstructures - Nanjing 210093, China Terahertz Technology Research Center, NICT, Nukui- Kitamachi 4-2-1, Koganei, Tokyo, Japan Department of Basic Science, The University of Tokyo - Komaba3-8-1, Meguro-ku, Tokyo, 153-8902, Japan School of Physical Science and Technology, ShanghaiTech University - Shanghai 201210, China
PACS – Nonequilibrium and irreversible thermodynamics
PACS – Analytical and numerical techniques
PACS – Electronic transport in nanoscale materials and structures
Abstract – Electrons in operating microelectronic semiconductor devices are accelerated by lo-cally varying strong electric field to acquire effective electron temperatures nonuniformly dis-tributing in nanoscales and largely exceeding the temperature of host crystal lattice. The thermaldynamics of electrons and the lattice are hence nontrivial and its understanding at nanoscales isdecisively important for gaining higher device performance. Here, we propose and demonstratethat in layered conductors nonequilibrium nature between the electrons and the lattice can beexplicitly pursued by simulating the conducting layer by separating it into two physical sheetsrepresenting, respectively, the electron- and the lattice-subsystems. We take, as an example ofsimulating GaAs devices, a 35nm thick 1 µm wide U-shaped conducting channel with 15nm radiusof curvature at the inner corner of the U-shaped bend, and find a remarkable hot spot to developdue to hot electron generation at the inner corner. The hot spot in terms of the electron temper-ature achieves a significantly higher temperature and is of far sharper spatial distribution whencompared to the hot spot in terms of the lattice temperature. Similar simulation calculation madeon a metal (NiCr) narrow lead of the similar geometry shows that a hot spot shows up as well atthe inner corner, but its strength and the spatial profiles are largely different from those in semi-conductor devices; viz., the amplitude and the profile of the electron system are similar to thoseof the lattice system, indicating quasi-equilibrium between the two subsystems. The remarkabledifference between the semiconductor and the metal is interpreted to be due to the large differencein the electron specific heat, rather than the difference in the electron phonon interaction. Thiswork will provide useful hints to deeper understanding of the nonequilibrium properties of elec-trical conductors, through a simple and convenient method for modeling nonequilibrium layeredconductors. Introduction. –
Electro-thermal behavior is a keyingredient for understanding charge carrier transportphenomena in semiconductor devices including two-dimensional (2D) materials, hetero Junctions, and strongcorrelated systems [1–15]. In small devices on nanoscales,hot electron generation and the resulting characteristic in- teraction with the host crystal lattice (or phonons) com-plicates the electro-thermal analysis and limits the deviceperformance [16, 17]. Whereas knowing the detailed lo-cal profile of the electron effective temperature, T e , sepa-rately from that of the lattice temperature, T L , in the pres-ence of current is prerequisite for understanding the trans-p-1. Yang et al. port characteristics on nanoscales [18–22], T e has been ex-perimentally hardly accessible [23–31] until quite recently[32, 33]. It follows that the study of electro-thermal prop-erties has so far been restricted only to the simulationmethods such as those of Monte Carlo (MC) simulationbased on the Boltzmann transport equations, hydrody-namic equations or molecular dynamics [34–38]. Unfortu-nately, however, MC simulations comprise involved calcu-lation procedures, which are not necessarily convenient togain intuitive understanding of the electro-thermal trans-port phenomena of given devices. On the other hand, thenonequilibrium condition cannot be incorporated in com-mercially available semiconductor device simulators.Here, we propose a simplified electro-thermal model forlayered conductors on the basis of the assumption thatthe electron- and the lattice-subsystems are, respectively,in quasi-equilibrium states characterized by the effectiveelectron temperature T e and the lattice temperature T L .The model is applied to a U-shaped layered conductor,where electric field is concentrated at the inner corner ofthe U-shaped bend. In a semicodncuctor device, simulat-ing GaAs, remarkable hot electron distribution ( T e ≫ T L )is found to develop at the corner, forming a sharp hot spotwith T e reaching ∼ K . Differently, in metal devices,simulating NiCr, hot electron effects are found to be ab-sent ( T e ≈ T L ), whereas a hot spot profile is visible. Thesefindings are consistent with recent experimental results re-ported on metals [39] and semiconductors [32], indicatingthe validity of the present model for simulating the electro-thermal behavior of layered conductors in nonequilibriumconditions. Simulation model. –
Figure 1 describes the modelconductor considered in this study. A layered conductorwith the electric conductivity σ e is deposited on an in-sulating substrate, which is anchored by the heat sink at300K. The lateral shape of the conductor is arbitrary, sothat the electric field E , the current density j , the elec-tron temperature T e and the lattice temperature T L inthe conductor are variables to be consistently derived asfunctions of the lateral position r for a given conductorwith a given bias voltage. In the conductor electrons gainenergy from E through P = j ( r ) · E ( r ) = σ e E and theenergy gained from the field is, in turn, released to the lat-tice via electron phonon interaction, characterized by theelectron-phonon energy relaxation time τ e − ph . The excessenergy (or heat) of electrons is transferred, as well, withinthe electron system through the electron thermal conduc-tion − κ e ▽ T e with κ e the electron thermal conductivity.The heat is transferred similarly within the lattice systemthrough lattice thermal conduction − κ L ▽ T L with κ L thelattice thermal conductivity. The heat is eventually trans-ferred to the substrate ( T L − T LS ) /h I with h I being theinterface thermal resistance and T LS ( r ) the local latticetemperature of the substrate on its top surface, and fi-nally absorbed by the heat sink. Heat is transferred aswell through electrical leads connected to the conductor, Fig. 1: Model for electro-thermal analysis. (a) Physical struc-ture of the sample. (b) Simulated structure, in which theelectron- and the lattice-systems of the conductor are separatedto thermally contacted two layers. (c) Diagram of energy/heatflow. as represented by the arrows marked with κ e and κ L infig.1 (c), which is taken into account in the model throughan appropriate boundary condition as mentioned belowfor fig.2 (a).As schematically shown in figs.1 (b) and (c), our modelrepresents the energy transfer from the electron system tothe lattice system in the conductor by the interface heattransfer between the electron sublayer at T e to the latticesublayer at T L . The energy flux released from the electronsystem to the lattice system through the electron phononinteraction is given by P e − ph = ( T e − T L ) C e /τ e − ph (1)with C e the electron specific heat per unit area, so thatthe effective interface thermal resistance h e − ph is h e − ph = τ e − ph /C e . (2)The specific heat is approximated by C e = C ec = (3 / N D k B , (3)for a classical electron system ( k B T e ≫ ε F ) and by C e = C eF = { (3 / k B T e /ε F } C ec (4)for an electron system with the Fermi energy ε F muchhigher than the thermal energy ( k B T e ≪ ε F ). Here, k B is the Boltzmann constant, and N D is the 2D electrondensity. Simulated structure. –
As schematically illustratedin fig.2 (a), we consider a 35nm thick conductor layershaped into a 1 µm -wide U-shaped channel with the radiusof inner curvature of the U-shape is 15nm and the gap be-tween the channels is S = 30 nm . For the simulation, theelectron- and the lattice-systems of the conducting chan-nel are separately represented by Layers A and B, whereLayer B is placed on the 10 µm -thick substrate (Layer C).The boundary condition of temperature is given by assum-ing T = 300 K on the bottom face Layer C and on the endp-2imulation of temperature profile for the electron- and the lattice-systems in laterally structured layered conductors Fig. 2: (a) Simulated sample structure, where the electron-and the lattice-systems of a U-shaped conducting channel ofa thickness t = 35 nm are separately represented, respectively,by Layers A and B. The conducting channels with a width of w = 1 µm ( r = 1 µm ) and a length of l = 4 µm extend from thetwo 5 × µm contact pads. The radius of curvature of the innercorner of the U-shape is 15nm, making a gap of S = 30 nm between the two channels. Bias voltage V b is defined as thevoltage difference between the end faces of Layer A (markedby yellow). (b) and (c) Simulated electric field distribution andtemperature distribution in the electron system (Layer A) ofn-GaAs. faces of Layers A and B as marked by the orange lines infig.2 (a). As to the bias condition, a constant voltage is as-sumed on each end face of the conducting channel (LayerA), and a bias voltage V b is assumed to give the voltagedifference between the two end faces.Two different conductors are considered. One is a dopedn-GaAs channel and the other is a NiCr channel, similar tothose studied, respectively, in Refs.32 and 39. Substratesare assumed to be lattice-matched GaAs/AlGaAs for n-GaAs sample [32] and single crystal Si covered with a thin SiO layer for NiCr sample [39]. The electron density inn-GaAs and NiCr samples are, respectively, N D = 3 . × /m and 1 . × /m ; in terms of the sheet electrondensity, N D = 1 . × /m and 3 . × /m . Thespecific heat is taken to be C e = 2 . × − W s/ ( Km )and 1 . × − W s/ ( Km ), respectively assuming Eqs. (3)and (4) for n-GaAs and NiCr samples. In n-GaAs sample,the interface thermal resistance ( h I ) is negligibly smallbecause the n-GaAs conducting layer is epitaxially grownon the lattice matched substrate. The electron-phononenergy relaxation time is assumed to be τ e − ph = 1 ps and3ps, respectively for n-GaAs [32] and NiCr [39]. Parametervalues used are summarized in Table 1. Fig. 3: (a) and (b) Temperature distributions of electrons andthe lattice in the NiCr sample. (c) and (d) Temperature dis-tributions of electrons and the lattice in the n-GaAs sample.White lines indicate the borders of conducting layers.
Results and discussions. –
Joule heating caused byelectric field and thermal conduction generated by tem-perature gradient or difference are consistently treatedby using a commercial multiphysics software (COMSOL),where the bias voltage is taken to be V b = 4 . V in all thecalculations described below. It is a common feature ofboth n-GaAs and NiCr samples that the electric field isconcentrated around the U-shaped inner corner as exem-plified by the result for n-GaAs sample: Electric field isnearly uniform and E = 2 ∼ kV /cm in a region awayfrom the U-shaped corner, but rapidly increases to reachabout E = 10 kV /cm in the vicinity of the U-shaped in-ner corner. As a consequence of this E -field enhancement,remarkable nonuniform hot-electron distribution is foundto be generated at the inner corner of the n-GaAs sampleas shown in fig.2 (c). While the trend of the E -field en-hancement is substantially the same in the NiCr sample,resulting temperature distribution in the electron- and thelattice-systems is largely different as described in detailbelow.Figures 3 (a), (b) and figures 4 (a)-(d) display the dis-tributions of T e (Layer A) and T L (Layers B) for the NiCrsample. The profile of T e is similar to that of T L , andboth exhibit spatially varying heating in accord with the E -field enhancement peaked at the U-shaped inner cor-ner. The highest temperature at the hot spot is about150 ◦ C above the heat sink (300K). The amplitude of thetemperature rise at the hot spot ( △ T e ≈ ◦ C) as wellas the quasi-equilibrium feature between the electron- andthe lattice-systems ( T e ≈ T L ) substantially reproduce theexperimental findings reported in Ref.39.The feature of the hot spot formation is largely differ-ent in the n-GaAs sample as shown in figs.3 (c)(d) andfigs.4 (e)-(h). The electron temperature T e is much higherthan the lattice temperature T L , indicating nonequilib-rium hot electron generation, and it assumes a very sharpprominent peak reaching as high a value as T e ∼ K .p-3. Yang et al. Table 1: Parameters used in the simulation.
Quantity σ e κ e κ L h e − ph h I κ S Unit
S/m W/ ( m · K ) W/ ( m · K ) Km /W Km /W W/ ( m · K )n-GaAs 8 . × . × − . ×
15 1 2 . × − × − Fig. 4: One-dimensional plots of T e and T L , along the blackdashed arrows shown in the inset, for the NiCr sample (a)-(d)and for the n-GaAs sample (e)-(h). On the other hand, the highest value of T L ( < K ) isat most only ∼ ◦ C above the temperature of the heatsink (300K). In addition, the hot-spot feature is practicallymissing as evident in figs.4 (f) and (h). The generation ofremarkable hot electron distribution is consistent with theexperimental finding reported on n-GaAs constriction de-vices [32].The large difference in the electro-thermal propertiesnoted between the n-GaAs sample and the NiCr samplein this study is suggested to be generally inherent to semi-conductors and metals. When energy flux P is fed to theelectron system in a steady state, the electrons are heatedabove the lattice temperature by T e − T L = ( τ e − ph /C e ) P = h e − ph P (5)if temperature gradient is ignored. For a given P , therise of T e is proportional to τ e − ph , and 1 /C e . In general, τ e − ph is not largely different between semiconductors andmetals, but the heat capacity C e is by orders of magnitudesmaller in semiconductors because the electron density isfar lower. It follows that the electron system is readilydriven away from the equilibrium with lattice in semicon-ductors. In terms of our model ( h e − ph ), thermal contactbetween the electron- and the lattice-systems are weak insemiconductors so that they are readily driven out of equi-librium. We mention that a high mobility of electrons isoften ascribed to be the cause of hot electron generationin semiconductors. The present study makes this assump-tion questionable; namely, a high mobility implies a highelectrical conductivity (and a large P ), but the electricalconductivity is usually higher in metals and does not ex-plain why semiconductor is more feasible for hot electrongeneration.In this study simulation calculation assumed lineartransport. Namely the electrical conductivity and thethermal conductivities are assumed to be constants. Inmetals, nonlinear effects may not be significant since the T e rise is not too large. In the doped n-GaAs sample atroom temperature (as in this work), nonlinear effects maynot be serious up to E ≈ kV [32], so that the findings inthe present study are supposed to be valid. In the higher E region above 10 kV /cm , however, the electron mobilitywill be reduced due to the transfer of electrons to upper(X and/or L) valleys. Even in such a higher- E region, ourmodel will provide a useful guideline at the starting point. Summary. –
We demonstrate that in layered con-ductors nonequilibrium nature between the electrons andthe lattice can be explicitly pursued by separating theelectron- and the lattice-subsystems into two physical lay-ers that exchange heat at the interface. Highly nonequi-librium distribution of electrons from that of the latticeis found in a doped n-GaAs sample. In a NiCr samplewith a similar configuration, the electron- and the lattice-systems are in quasi-equilibrium. Remarkable difference ofthe electro-thermal properties of semiconductors and met-als is suggested to arise from the difference in the electronspecific heat. This work provides a simple and convenientmethod for modeling layered conductors in a nonequilib-rium condition, and will give useful hints for deeper un-derstanding of the nonequilibrium properties of electricalconductorsp-4imulation of temperature profile for the electron- and the lattice-systems in laterally structured layered conductors ∗ ∗ ∗
We acknowledge funding support from NationalKey Research Program of China under grant No.2016YFA0302000, National Natural Science Foundationof China under grant Nos. 11674070/11427807/11634012,and Shanghai Science and Technology Committee un-der grant Nos.18JC1420402, 18JC1410300, 16JC1400400.S.K. acknowledges support by the Chinese Academy ofSciences Visiting Professorships for Senior InternationalScientists.
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