Ultrathin GaN Nanowires: Electronic, Thermal, and Thermoelectric Properties
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Ultrathin GaN Nanowires: Electronic, Thermal, and Thermoelectric Properties
A. H. Davoody, ∗ E. B. Ramayya, † L. N. Maurer, and I. Knezevic ‡ Department of Electrical and Computer Engineering, University of Wisconsin–Madison, Madison, WI 53706-1691, USA (Dated: October 10, 2018)We present a comprehensive computational study of the electronic, thermal, and thermoelectric (TE) proper-ties of gallium nitride nanowires (NWs) over a wide range of thicknesses (3–9 nm), doping densities ( – cm − ), and temperatures (300–1000 K). We calculate the low-field electron mobility based on ensemble MonteCarlo transport simulation coupled with a self-consistent solution of the Poisson and Schr¨odinger equations.We use the relaxation-time approximation and a Poisson-Schro¨dinger solver to calculate the electron Seebeckcoefficient and thermal conductivity. Lattice thermal conductivity is calculated using a phonon ensemble MonteCarlo simulation, with a real-space rough surface described by a Gaussian autocorrelation function. Through-out the temperature range, the Seebeck coefficient increases while the lattice thermal conductivity decreaseswith decreasing wire cross section, both boding well for TE applications of thin GaN NWs. However, at roomtemperature these benefits are eventually overcome by the detrimental effect of surface roughness scattering onthe electron mobility in very thin NWs. The highest room-temperature ZT of 0.2 is achieved for 4-nm-thickNWs, while further downscaling degrades it. In contrast, at 1000 K, the electron mobility varies weakly with theNW thickness owing to the dominance of polar optical phonon scattering and multiple subbands contributingto transport, so ZT increases with increasing confinement, reaching 0.8 for optimally doped 3-nm-thick NWs.The ZT of GaN NWs increases with increasing temperature beyond 1000 K, which further emphasizes theirsuitability for high-temperature TE applications. PACS numbers: 72.20.Pa, 84.60.Rb.,73.63.Nm, 65.80-g
I. INTRODUCTION
Thermoelectric (TE) devices for clean, environmentally-friendly cooling and power generation are a topic of con-siderable research activity.
The TE figure of merit, deter-mining the efficiency of a TE device, is defined as ZT = S σT /κ , where S is the Seebeck coefficient (also known asthermopower), σ is electrical conductivity, κ is thermal con-ductivity, and T is the operating temperature. Highly dopedsemiconductors are the materials with the highest ZT , be-cause heat in semiconductors is carried mostly by the lattice( κ ≈ κ l ), so electronic and thermal transport are largely de-coupled; therefore, the power factor, S σ , and thermal con-ductivity can, in principle, be separately optimized. In or-der to improve ZT , we need to increase the power factor andreduce thermal conductivity. ZT > is needed to replaceconventional chlorofluorocarbon coolers by TE coolers, butincreasing it beyond has been a challenge. Nanostructuring has the potential to both enhance thepower factor and reduce the thermal conductivity of TEdevices.
The Seebeck coefficient and the power factor couldbe higher in nanostructured TE devices than in bulk ow-ing to the density-of-states (DOS) modification, as first sug-gested by Hicks and Dresselhaus.
While this effect is ex-pected to be quite pronounced in thin nanowires (NWs),where the DOS is highly peaked around one-dimensional (1D)subband energies, surface roughness scattering (SRS) ofcharge carriers counters the beneficial DOS enhancement. The field effect has also been shown to enhance the powerfactor in nanostructures, as it provides carrier confinementand charge density control without the detrimental effects ofcarrier-dopant scattering. Moreover, nanostructured obstaclesefficiently quench heat conduction, as demonstrated on ma-terials with nanoscale inclusions of various sizes, which scatter phonons of different wavelengths, and on rough semi-conductor nanowires, in which boundary roughness scatteringof phonons reduces lattice thermal conductivity by nearly twoorders of magnitude.
Power generation based on TE energy harvesting re-quires materials that have high thermoelectric efficiency andthermal stability at high temperatures, as well as chemi-cal stability in oxide environments. Bulk III-nitrides ful-fill these criteria and have been receiving attention as poten-tial high-temperature TE materials.
Bulk GaN, in par-ticular, has excellent electron mobility, but, like other tetra-hedrally bonded semiconductors, it also has high thermalconductivity, so its overall TE performance is very modest( ZT = 0 . at 300 K and . at 1000 K, as reported byLiu and Balandin ). Recently, Sztein et al. have shownthat alloying with small amounts of In can considerably en-hance the TE performance of bulk GaN. Here, we explore adifferent scenario: considering that nanostructuring, in par-ticular fabrication of quasi-1D systems such as NWs, hasbeen shown to raise the ZT of other semiconductors, itis worth asking how well GaN NWs could perform in high-temperature TE applications. There have been a number ofadvances in the GaN NW growth and fabrication, as wellas their electronic characterization, but very few studiesof GaN NWs for TE applications. In this paper, we theoretically investigate the suitability ofrough n -type GaN NWs for high-temperature TE applica-tions. To that end, we simulate their electronic, thermal, andTE properties over a wide range of thicknesses (3–9 nm), dop-ing densities ( – cm − ), and temperatures (300–1000K). Electronic transport is simulated using ensemble MonteCarlo (EMC) coupled with a self-consistent Schr¨odinger –Poisson solver. The electronic Seebeck coefficient and ther-mal conductivity are calculated by solving the Boltzmanntransport equation (BTE) under the relaxation-time approxi-mation (RTA). Lattice thermal conductivity is calculated us-ing a phonon ensemble Monte Carlo simulation, with a real-space rough surface described by a Gaussian autocorrelationfunction. Throughout the temperature range, the Seebeck co-efficient increases while the lattice thermal conductivity de-creases with decreasing wire cross section. At room tempera-ture these benefits are eventually overcome by the detrimentaleffect of SRS on the electron mobility, so the peak ZT = 0 . is achieved at 4 nm, with further downscaling lowering the ZT . At 1000 K, however, the electron mobility varies veryweakly with the NW thickness owing to the dominance ofpolar optical phonon scattering and multiple subbands con-tributing to transport, so ZT keeps increasing with increas-ing confinement, reaching 0.8 for 3-nm-thick NWs at 1000 Kand for optimal doing. The ZT of GaN NWs increases withtemperature past 1000 K, which highlights their suitability forhigh-temperature TE applications.This paper is organized as follows: the model used to cal-culate the electron scattering rates is explained in Sec. II, fol-lowed by a discussion of the simulation results for the electronmobility (Sec. II A) and the Seebeck coefficient (Sec. II B) asa function of wire thickness, doping density, and temperature.In Section III, we discuss the phonon scattering models usedin this paper and then show the calculated values of phononicand electronic thermal conductivity. In Sec. IV, we show thecalculation of the thermoelectric figure of merit and discussits behavior. We conclude with a summary and final remarksin Sec. V. II. ELECTRONIC TRANSPORT
Bulk GaN can crystalize in zincblende or wurtzite struc-tures, the latter being more abundant. In bulk wurtziteGaN, the bottom of the conduction band is located at the Γ point. The next lowest valley, located at M , is about 1.2 eVhigher than Γ , so it does not contribute to low-field elec-tron transport. The electron band structure in the Γ -valleycan be approximated as non-parabolic, E ( k ) (1 + α E ( k )) = ~ | k | / m ∗ , where α = 0 . eV − is the non-parabolicityfactor and m ∗ = 0 . m is the isotropic electron effectivemass, given in the units of m , the free-electron rest mass.Wurtzite GaN NWs are usually grown or etchedvertically, along the bulk crystalline c -axis, and can have tri-angular, hexagonal, or quasi-circular cross-sections, depend-ing on the details of processing. In silicon, simulation ofelectronic transport in rough cylindrical, square, and atom-istically realistic nanowires yields results that are remark-ably close to one another, both qualitatively and quantita-tively, when these differently shaped wires have similar cross-sectional feature size and similar edge roughness features; forinstance, the electron mobility in a rough cylindrical wire ofdiameter equal to 8 nm is very close to the mobility in asquare NW with an 8-nm side. Therefore, in order to sim-plify the numerical simulation of electron and phonon trans-port in GaN NWs, we consider a square cross section, withthe understanding that the wire thickness or width stands infor a generic characteristic cross-sectional feature size.
Parameter Value Units Ref.Deformation potential Ξ ac ρ g / cm [51]Longitudinal sound velocity υ s m / s [51]Lattice constant a A [56]Lattice constant c A [56]Optical phonon energy E ǫ ǫ ∞ e -0.3 - [55] e -0.33 - [55] e c L . × Pa [55] c T . × Pa [55]TABLE I. GaN material parameters
In the n -type doped square GaN NWs considered here, elec-trons are confined to a square quantum well in the cross-sectional plane and are free to move along the wire axis. Inthe envelope function approximation, the three-dimensional(3D) electron wave functions have the form ψ n,k x ( x, y, z ) = φ n ( y, z ) × exp( ± ik x x ) , where φ n ( y, z ) is a two-dimensional(2D) wavefunction of the n -th 2D subband calculated fromthe Schr¨odinger-Poisson solver. The corresponding elec-tron energy is E n,k x = E n + √ αk x / m ∗ − α , where E n is the bottom-of-subband energy. The 2D Poisson’s andSchr¨odinger equations are solved in a self-consistent loop:Poisson’s equation gives the Hartree approximation for theelectrostatic potential, which is used in the Schr¨odinger equa-tion to calculate electronic wave functions and energies in thecross-sectional plane; electronic subbands are then populatedto calculate the carrier density and fed back into the Poissonsolver. More details regarding the numerical procedure can befound in Refs. 49 and 50.Electrons in GaN NWs scatter from acoustic phonons, im-purities, surface roughness, polar optical phonons (POP), andthe piezoelectric (PZ) field. GaN NWs have only a fewmonolayers of native oxide around them; therefore, for thepurpose of SRS, we will treat them as bare (i.e. surrounded byair). The scattering rates are calculated using Fermi’s goldenrule. The constants used in the scattering rate calculationsare taken from Refs. 51, 55, and 56, and shown in Table I.The acoustic phonon scattering rate from subband n to sub-band m is given by Γ acnm ( k x ) = Ξ ac D nm k B T √ m ∗ ~ ρυ s (1 + 2 α E f ) p E f (1 + α E f ) Θ( E f ) , (1a)where Ξ ac and υ s are the deformation potential and the soundvelocity, respectively. Θ( E f ) is the Heaviside step functionand the electron kinetic energy in the final state, E f , is givenby E f = E n − E m + p αk x / m ∗ − α . (1b) D nm is an overlap integral of the form D nm = Z | φ n ( y, z ) | | φ m ( y, z ) | dy dz. (1c)We used a degenerate Thomas-Fermi screening model to cal-culate the impurity scattering rates. The impurity scatteringrate from subband n to subband m is given by Γ impnm ( k x ) = Z e N d √ m ∗ √ π ~ ǫ (1 + 2 α E f ) p E f (1 + α E f ) × Z Z d R I nm ( q ± x , R ) , (2) I nm ( q x , R ) = Z φ n ( y, z )K ( q x , R ) φ m ( y, z ) dy dz, K ( q x , R ) = Z + ∞−∞ e iq x .x e − √ ( r − R )2+ x Ld p ( r − R ) + x dx, where ǫ is the static dielectric permittivity of GaN, Z = 1 isthe number of free electrons contributed by each dopant atom, N d is the doping density, and R is the position of the impurityatom in the wire. q x = | k x − k ′ x | is the magnitude of thedifference between the initial ( k x ) and final ( k ′ x ) wave vectorof the electron along the wire.The SRS rate is calculated based on enhanced Ando’smodel Γ srnm ( k x , ± ) = 2 √ m ∗ e ~ ∆ Λ2 + ( q ± x ) Λ | F nm | × (1 + 2 α E f ) p E f (1 + α E f ) Θ( E ) , (3a)where ∆ and Λ are the rms height and correlation length of thesurface roughness. q ± x = k x ± k ′ x is the difference betweenthe initial ( k x ) and final ( k ′ x ) wavevector, while the plus andminus signs correspond to forward and backward scattering,respectively. F nm is the SRS overlap integral, defined as F nm = Z Z dy dz h − ~ e W m y φ m ( y, z ) ∂ φ n ( y, z ) ∂y + φ n ( y, z ) E y ( y, z ) (cid:16) − yW (cid:17) φ m ( y, z ) (3b) + φ n ( y, z ) (cid:18) E m − E n e (cid:19) (cid:16) − yW (cid:17) ∂φ m ( y, z ) ∂y i . The overlap integral in Eq. (3b) corresponds to scatteringfrom the top surface of the wire ( y = W , W being the wirethickness and width). The SRS rate from the bottom surfacecan be calculated by shifting the origin along the y -axis. TheSRS rate from the side walls can be calculated by exchanging y and z parameters in Eq. (3b). E y ( y, z ) is the y -componentof the electric field at ( y, z ) .A detailed derivation of the POP scattering rates is shownin Appendix A. The electron scattering rate by POPs fromsubband n to subband m is given by Γ P OPnm ( k x ) = e ω (cid:0) ǫ − ∞ − ǫ − (cid:1) π N r m ∗ ~ I D ( q ± x , L y , L z ) × α E f p E f (1 + α E f ) Θ( E f ) . (4a) As before, q ± x = k x ± k ′ x is the difference between the initialand final electron wavevectors, with plus (minus) correspond-ing to forward (backward) scattering. Energy conservation de-termines the final kinetic energy as E f = E n − E m + E i ± ~ ω ,where the plus and minus signs correspond to phonon absorp-tion and emission, respectively. E i is the initial electron ki-netic energy calculated using the non-parabolic band struc-ture. ǫ ∞ and ǫ are the high-frequency and low-frequency(static) dielectric permittivities of GaN, respectively (Table I). ω is the bulk longitudinal optical phonon frequency. N isthe number of optical phonons with frequency ω , given bythe Bose-Einestein distribution N = 1 e ~ ω kBT − .I D in Eq. (4a) is the electron-phonon overlap integral definedas I D ( q x , L y , L z ) = Z Z q | I nm ( q y , q z ) | dq y dq z , (4b)where I nm ( q y , q z ) is defined in Eq. (A6). The integral istaken over the first Brillouin zone.Scattering rate due to the piezoelectric effect is derived inAppendix B as Γ P Znm ( k x ) = K av π ~ e k B Tǫ ∞ r m ∗ ~ I D ( q x , L y , L z ) × α E f p E f (1 + α E f ) Θ( E f ) , (5a)where I D is the electron-phonon overlap integral in Eq. (4b),and the final kinetic energy is E f = E n − E m + E i . ǫ ∞ isthe high-frequency effective dielectric constant, and K av isthe electromechanical coupling coefficient. For the wurtzitelattice, K av is shown to be K av = h e l i ǫ ∞ c L + h e t i ǫ ∞ c T , (5b)where h e l i = 17 e + 435 e ( e + 2 e ) + 8105 ( e + 2 e ) , h e t i = 235 ( e − e − e ) + 1635 e (5c) + 16105 e ( e − e − e ) . A. Electron Mobility
In GaN nanowires, measured electrical conductivity showsconsiderable sensitivity to variations in the wire thickness,doping density, and temperature. (For reference, thelow-field electron mobility in bulk GaN doped to cm − is of order 200–300 cm /Vs at room temperature, anddrops to 100 cm /Vs at 1000 K. ) Here, we perform a com-prehensive set of electronic Monte Carlo simulations in orderto analyze the dependence of the electron mobility in GaNNWs on the wire thickness, doping density, and tempera-ture. The calculated electron scattering rates are used in aMonte Carlo kernel to simulate electron transport and com-pute the electron mobility. In these highly doped NWs, therejection technique is used to account for the Pauli exclusionprinciple. Electronic Monte Carlo simulations are typically done with80,000-100,000 particles over timescales longer than severalpicoseconds, which is enough time to reliably achieve a steadystate. Typical ensemble time step is of order 1 fs (much shorterthan the typical relaxation times). To insure transport is dif-fusive, the wire is considered to be very long, so the elec-tronic simulation is actually not done in real space along thewire. Instead, a constant field and an effectively infinite wireare assumed, and the simulation is done in k -space. Sur-face roughness scattering of electrons from the surface witha given rms roughness and correlation length is accountedfor through the appropriate SRS matrix element. Across thewire, the Schr¨odinger and Poisson equations are solved self-consistently. A typical mesh across the wire is 67 ×
67 meshpoints for 5–9 nm wires, 57 ×
57 for the 4 nm ones. The meshis nonuniform and is denser near the wire boundary. Moredetails can be found in Ref. [15].First, we discuss the effect of the wire thickness variationon the electron mobility. Figure 1a shows the electron mobil-ity as a function of the NW thickness for a wire doped to cm − . (Doping densities of order cm − are optimal forTE applications in many semiconductors. ) The rms heightof the surface roughness is taken to be ∆ = 0 . , as one ofthe smoothest surfaces reported for GaN crystals. The cor-relation length is assumed to be 2.5 nm, a common value inSi CMOS; we have not been able to find a measured value onGaN systems. The red (black) dashed curve shows the elec-tron mobility when only intrasubband (intersubband) scatter-ing is allowed. The intersubband electron scattering processesare dominant in thicker wires. The intersubband scatteringrate decreases with decreasing thickness, as the subband spac-ing increases. Electrons have higher intrasubband scatteringrates in thin wires (red dashed curve), in which the SRS over-lap integrals are greater.Figure 1b shows the electron mobility as a function of thewire thickness for various wire doping densities. Electron mo-bility has a peak, followed by a dip, around the wire thicknessin which the transition from mostly intrasubband to mostly in-tersubband scattering happens. The dip in the mobility curvecorresponds to the onset of significant intersubband scatteringbetween the lowest two subbands, i.e. the energy differencebetween the first and second subband bottoms exceeds the po-lar optical phonon energy. As we can see in Fig. 1b, varyingthe doping density moves this transition point between mostlyinersubband and mostly intrasubband scattering regimes. Inthick NWs, similar to bulk, increasing the doping densitycauses more electron scattering with ionized dopants and theelectron mobility decreases. However, for thinner wires thebehavior is more complicated and we discuss it in more detailin the next few paragraphs.Figure 2 shows the electron mobility dependence on the
Wire thickness [nm] E l e c t r on m ob ili t y [ c m / V . s ] Intersubband scatteringIntrasubband scatteringAll scattering processes10 cm −3 doping density (a) Wire thickness [nm] E l e c t r on m ob ili t y [ c m / V . s ] × cm −3 × cm −3 × cm −3 (b) FIG. 1. Electron mobility in GaN NWs as a function of thickness.The rms roughness and correlation length of the wire interface are ∆ = 0 . and Λ = 2 . nm, respectively. Temperature is T =300 K . (a) The black dashed curve shows the electron mobility withonly intersubband scattering, while the red dashed curve correspondsto intrasubband scattering alone. The blue solid curve shows the netelectron mobility, including both inter- and intrasubband scatteringprocesses. (b) Electron mobility in GaN NWs with various dopingdensities as a function of the NW thickness. doping density for various wire thicknesses. These resultsare in good agreement with the experimental measurementsof Huang et al. for a GaN nanowire FET device of 10 nmthickness. In relatively thick NWs, we observe the expecteddecrease of the electron mobility with increasing doping den-sity. However, for a NW with a relatively small diameter, theelectron mobility shows a more complicated non-monotonicbehavior with doping density. Similar behavior has been ob-served by others in the mobility versus effective field depen-dence of gated silicon nanostructures, where its origincomes from the interplay of surface roughness and nonpolarintervalley phonon scattering in these confined systems. InGaN NWs, strong electron confinement is also key, but POPscattering plays the dominant role instead. The origin of thepeak can be readily grasped by relying on the relaxation-timeapproximation (RTA) expression for the mobility (here n, k, s Doping density [cm −3 ] E l e c t r on m ob ili t y [ c m / V . s ] FIG. 2. Electron mobility in GaN NWs as a function of doping den-sity. The rms roughness and correlation length of the wire interfaceare ∆ = 0 . and Λ = 2 . nm, respectively. Temperature is T = 300 K . are the electron quantum numbers – the subband index, mo-mentum along the NW, and the spin orientation, respectively): µ RTA = e P n,k,s υ n ( k ) τ n ( k ) [ − ∂f /∂ ( ~ k )] P n,k,s f ( k )= eN D X n,k,s υ n ( k ) τ n ( k ) ( − df /d E ) (6) = eN D Z d E X n g n ( E ) υ n ( E ) τ n ( E ) ( − df /d E ) Here, f ( E ) is the Fermi-Dirac distribution function. g n ( E ) is the density of states, τ n ( E ) the lifetime, and υ n ( E ) =( d E /d ( ~ k )) E−E n is the group velocity along the wire foran electron in the n -th subband and with the kinetic energy E − E n . We have used the fact that the denominator from thefirst line of Eq. (6), P n,k,s f ( k ) , equals the electron den-sity, which in turn equals the doping density N D . With in-creasing doping density, the Fermi level moves up in energy,and the transport window (the energy range where | df /d E| is appreciable) follows. As g n ∼ υ − n in NWs, from the in-tegrand in the numerator of Eq. (6) we see that the mobil-ity is determined by the product of τ n ( E ) and υ n ( E ) withinthe transport window. Figure 3 shows the integrand from thenumerator of Eq. (6) divided by the doping density versusenergy, for the 4-nm wire and several doping densities rang-ing cm − to × cm − ; the area under each curveis therefore proportional to the mobility for that doping den-sity. (The cumulative density of states, g ( E ) = P n g n ( E ) ,is also presented as a lightly shaded area.) Each integrandcurve has a steep drop at roughly 91 meV, corresponding tothe relaxation time drop due to the onset of intrasubband POPemission for the first subband, and a small dip at about 145meV, ∼
91 meV below the second subband bottom, which cor-responds to the onset of first-to-second subband intersubbandscattering due to POP absorption. For N D ranging from cm − to cm − , the integrand curves overlap, so the ar-eas under them are nearly the same and the mobility is nearly −3 Energy [eV] A r b i t r a r y un i t s cm −3 cm −3 cm −3 cm −3 cm −3 cm −3 cm −3 cm −3 FIG. 3. The integrand from the numerator in Eq. (6) divided by thedoping density N D for a 4-nm wire and N D ranging from to × cm − . The temperature is 300 K. The area under each curveis proportional to the electron mobility. The shaded area correspondsto the density of states, g ( E ) . E = 0 is the bottom of the lowestsubband. constant. With a further doping density increase, the transportwindow moves towards the POP emission threshold; mobil-ity reaches its maximal value when the electronic states withhigh velocities but still below the POP emission threshold arearound the middle of the transport window (doping densityabout × cm − ). As the density increases futher, thetransport window moves into the range of energies with strongintrasubband POP emission and the mobility drops.Next, we discuss the effect of a temperature increase on theelectron mobility. Figure 4a shows the electron mobility of 4-nm and 9-nm-thick NWs doped to cm − with only POPscattering and with all scattering mechanisms included. Withincreasing temperature, the POP scattering rate increase, fol-lowing the increasing number of polar optical phonons. Fortemperatures above 600 K, POP scattering becomes the dom-inant scattering process and governs the rapid decrease of theelectron mobility for the thicker, 9-nm NW. In the thinner 4-nm NWs, the mobility is less sensitive to temperature becausethe greater strength of SRS with respect to POP scattering inthin wires. Figure 4b shows the electron mobility as a functionof the wire thickness at 300 and 1000 K for two doping den-sities, while Fig. 4c presents mobility versus doping densityat 300 and 1000 K for 4-nm and 9-nm NWs. At 1000 K, POPscattering dominates over other mechanisms and the transportwindow, roughly k B T wide, contains a number of subbands;together, these two effects result in flattening of both the mo-bility vs. wire thickness (Fig. 4b) and the mobility vs. dopingdensity(Fig. 4c) dependencies. B. The Seebeck Coefficient
The Seebeck coefficient (also known as the thermopower)for bulk GaN has a value of 300 – 400 µ V/K, depending on thesample and the temperature.
The Seebeck coefficientis a sum of the electronic and the phonon-drag (also knownas phononic) contributions. For GaN NWs, our calculationshows that the phonon-drag Seebeck coefficient is about two
200 400 600 800 100010 Temperature [K] E l e c t r on m ob ili t y [ c m / V . s ] cm −3 doping density (a) Wire thickness [nm] E l e c t r on m ob ili t y [ c m / V . s ] T=300 K, 10 cm −3 T=300 K, 2 × cm −3 T=1000 K, 10 cm −3 T=1000 K, 2 × cm −3 (b) Doping density [cm −3 ] E l e c t r on m ob ili t y [ c m / V . s ] W=9 nm ,T=300 KW=9 nm ,T=1000 KW=4 nm ,T=300 KW=4 nm ,T=1000 K (c)
FIG. 4. (a) Electron mobility of 4-nm and 9-nm-thick GaN NWsdoped to cm − as a function of temperature. (b) Electron mo-bility as a function of wire thickness at temperatures of K and
K and doping densities of and × cm − . (c) Elec-tron mobility of 4-nm and 9-nm-thick NWs as a function of dopingdensity at K and
K. In all three panels, the rms roughnessand correlation length of the wire interface are ∆ = 0 . and Λ = 2 . nm, respectively. orders of magnitude smaller than the electronic one at temper-atures of interest, so we henceforth neglect the phonon-dragcontribution and equate the total and the electronic Seebeckcoefficients. In this section, we discuss the effect of the NWthickness, doping density, and temperature on the Seebeck co-efficient.Based on the 1D BTE using the relaxation-time approxima-tion (RTA), we find the Seebeck coefficient ( S e ) to be S e = − eT P n R √E ∂f ( E ) ∂ E ( E + E n − E F ) τ n ( E ) d E P n R √E ∂f ( E ) ∂ E τ n ( E ) d E , (7)where E F is the Fermi energy, f ( E ) is the equilibrium Fermi-Dirac distribution, τ n ( E ) is the relaxation time of electron insubband n , and E n is the energy of the bottom of that subband.Integration over energy is performed from zero to infinity.Note that the Seebeck coefficient is determined by the averageexcess energy with respect to Fermi energy, η F = E + E n −E F ,carried by electrons in the vicinity of the Fermi level.Figure 5a shows S e as a function of the wire thicknessfor various doping densities, which are compatible with theexperimental results by Sztein et al. . Decreasing the wirethickness increases the spacing between the subband bottomenergies and the Fermi level, which, consequently, increases η F in an average sense (Fig. 5b). The result is a rise in theSeebeck coefficient. For thicker wires, the Fermi level liesbetween subband bottoms and the interplay between the con-tributions from different subbands determines the variation of S e . As an example of this interplay, we observe a slight in-crease in the Seebeck coefficient between the 7-nm and 9-nm-thick NWs at the doping density of × cm − .Figure 6a shows the variation of the Seebeck coefficientwith doping density for GaN NWs of different thicknesses.Increasing the doping density means more subband bottomsbelow the Fermi level (Fig. 6b). This effect results in a highSeebeck coefficient for wires with lower doping densities, forwhich all subbands are above the Fermi level. In contrast,for degenerately doped wires, the Fermi level typically liesbetween subbands; S e is determined by an interplay betweenthe position of the different subbands with respect to the Fermilevel (Fig. 6b) and the S e versus doping density curve is al-most flat (Fig. 6a).Figure 7a presents the dependence of the Seebeck coeffi-cient on temperature in 4-nm and 9-nm-thick GaN NWs. Amajor effect of increasing the temperature is broadening ofthe Fermi-Dirac distribution function. With increasing tem-perature, but at a fixed doping density and wire thickness, agiven subband will be higher in energy with respect to theFermi level (thereby contributing more favorably to the See-beck coefficient) and the energy range for electrons active inelectrical conduction will widen (Fig. 7a). As seen in Fig. 7b,when the temperature is increased from 200 to 1000 K in a9 nm thick NW doped to cm − , the Seebeck coefficientincreases by a factor of 3.5.Fig. 8 shows the Seebeck coefficient as a function of dopingdensity (Fig. 8a) and wire thickness (Fig. 8b) at temperatures300 K and 1000 K. The Seebeck coefficient increases withincreasing temperature, as observed in experiment. Wire thickness [nm] S e m agn i t ude [ µ V / K ] × cm −3 × cm −3 × cm −3 (a) −0.2−0.100.10.20.30.40.50.6 E ne r g y [ e V ] E E E E F (b) FIG. 5. (a) The Seebeck coefficient as a function of NW thicknessat T = 300 K and for different doping densities. (b) Positions of thefirst and second subband bottoms with respect to the Fermi energyfor NWs of thickness 3 nm and 9 nm.
III. THERMAL TRANSPORT
In bulk GaN, most experimental measurements of the ther-mal conductivity have been done at temperature below 400K.
Typical experimental values at room temperature arein the range of 130 – 200 W/m · K; a good survey of the re-sults prior to 2010 was done by AlShaikhi et al . Recentfirst-principles theoretical calculations by Lindsey et al . givethermal conductivity values for temperatures up to 500 K,with the room-temperature value of about 200 W/m · K, inagreement with experiment.
Based on a theoretical studyby Liu and Balandin, the thermal conductivity of bulk GaNat 1000 K is expected to be about 40 W/m · K.Thermal conductivity in n -type NWs comprises two com-ponents: phonon (lattice) and electron thermal conductivities.The lattice thermal conductivity of semiconductor nanowiresis expected to be very low, based on theoretical work usingmolecular dynamics, nonequilibrium Green’s functionsin the harmonic approximation, and the Boltzmann trans-port equation addressing phonon transport. Here, we cal- Doping density [cm −3 ] S e m agn i t ude [ µ V / K ] (a) −0.500.51 E ne r g y [ e V ] × cm −3 × cm −3 × cm −3 E F ∂ f/ ∂ E (b) FIG. 6. (a) The Seebeck coefficient as a function of doping densityfor GaN NWs of thickness 4 nm, 9 nm, and 15 nm, at 300 K. (b)Subband energy bottoms with respect to the Fermi energy for a 9-nm-thick wire at three different doping densities. The green dashed lineis the negative derivative of the Fermi-Dirac distribution function, − df ( E ) /d E . culate the lattice thermal conductivity κ l using the phononensemble Monte Carlo (EMC) technique, and electronic ther-mal conductivity κ e using the RTA. While κ e is much lowerthan κ l in bulk semiconductors, the two can become compa-rable in highly doped ultrathin NWs, owing to the reductionof κ l that comes from phonon scattering from rough bound-aries and the increase in κ e with increasing doping density.The phonon Monte Carlo method used in this work is ex-plained in detail in Lacroix et al. and Ramayya et al. TheMonte Carlo kernel simulates transport of thermal energy car-ried by acoustic phonons; optical phonons are neglected due totheir short lifetime and low group velocity.
The importantacoustic phonon scattering mechanisms are phonon-phonon(normal and Umklapp), mass difference, and surface rough-ness (boundary) scattering.
We simulate wires of length greater than the typical mean-free path for bulk, in order to properly describe the diffusivetransport regime. If the wires are long enough, a linear tem-perature profile will be obtained along the wire (in contrastwith the steplike ballistic transport signature ). Typical wiresin our simulations are 200 nm long. There is no volume mesh,only a surface mesh with typically 1 angstrom mesh cell size,which captures roughness scattering. Along the wire axis, thewire is divided into cubic segments of the same length as the
400 600 800 100050100150200250300
Temperature [K] S e m agn i t ude [ µ V / K ] cm −3 × cm −3 cm −3 × cm −3 (a) −0.200.20.40.60.811.2 E ne r g y [ e V ]
200 K 1000 K (b)
FIG. 7. (a) The Seebeck coefficient as a function of temperature for a9-nm-thick NW with cm − doping density. (b) Subband energybottoms with respect to the Fermi energy for a 9-nm-thick wire at 200K and 1000 K. The blue and red dashed lines show the negative ofthe derivative of the Fermi-Dirac distribution function, − df ( E ) /d E .The black dashed line is the Fermi energy. wire thickness and width. Each segment is assumed to havea well-defined temperature, which is updated during the sim-ulation, as the phonons enter and leave. Energy that is trans-ferred through each boundary between adjacent segments perunit time is recorded and its value averaged along the wire isused to compute the thermal conductivity based on Fourier’slaw. The normal and Umklapp phonon-phonon scattering ratesare calculated using the Holland model. In contrast to thesimpler Klemens-Callaway rates, which assume a single-mode linear-dispersion (Debye) approximation, the Hollandrates are more complex as they are specifically constructedto capture the flattening of the dispersive transverse acoustic(TA) modes. For the scattering rate calculation of the TAmodes, the zone is split into two regions, such that there isonly normal scattering for small wave vectors (roughly up tohalfway towards the Brillouin zone edge), while both ump-klapp and normal scattering occur for larger wave vectors. Doping density [cm −3 ] S e m agn i t ude [ µ V / K ] W=9 nm ,T=300 KW=9 nm ,T=1000 KW=4 nm ,T=300 KW=4 nm ,T=1000 K (a)
Wire thickness [nm] S e m agn i t ude [ µ V / K ] T=300 K, 10 cm −3 T=300 K, 2 × cm −3 T=1000 K, 10 cm −3 T=1000 K, 2 × cm −3 (b) FIG. 8. (a) The Seebeck coefficient as a function of doping densityfor 4-nm and 9-nm-thick NWs at 300 K and 1000 K. (b) The Seebeckcoefficient as a function of wire thickness for NWs doped to and × cm − at 300 K and 1000 K. Therefore, (cid:0) τ NT (cid:1) − = B T N ωT , < ω < ω , (8a) (cid:0) τ UT (cid:1) − = ( , < ω < ω B TU ω sinh( ~ ω/k B T ) , ω < ω < ( ω T ) max (8b)Relaxation rates for longitudinal acoustic phonons are givenby (cid:0) τ NL (cid:1) − = B L N ω T , < ω < ( ω L ) max , (9a) (cid:0) τ UL (cid:1) − = B L U ω T , < ω < ( ω L ) max . (9b)Here, B T N , B T U , B LN , and B LU are the constants shown inTable II, which are calculated by fitting our simulation resultsfor bulk GaN to experimental results of Sichel et al. ω cor-responds to the frequency of the transverse branch at q max / point, where q max is the Brillouin zone boundary. The relaxation rate for mass difference scattering is givenby the following expression ( τ I ) − = A i ω , (10a) Parameter Value Units B LN . × − s . K − B LU . × − s . K − B TN . × − K − B TU . × − sTABLE II. Phonon-phonon scattering fitting parameters FIG. 9. A phonon hitting a rough surface spends some time bouncingaround, effectively appearing to be localized at the surface. where A i is a sample-dependent constant, given by A i = V Γ4 πυ s . (10b)Here, V is the volume per atom, equal to V = √ a c forthe wurtzite crystal. υ s is the average phase velocity givenby υ − s = [2 υ − T + υ − L ] under the isotropic phonon dis-persion approximation. Γ is the constant which indicatesthe strength of mass difference scattering. It is defined as Γ = P i f i [1 − ( M i /M )] , where f i is the fractional con-centration of the atoms type i with different mass, M i , in thelattice. M is the average atomic mass, M = P i f i M i . The Γ parameter due to isotopes for a typical sample is given in Ta-ble III. The Γ parameter due to doping with Si to cm − is about . × − , an order of magnitude smaller than forisotopes. Mass difference scattering due to dopants becomescomparable to isotope scattering at about cm − dopingdensity; therefore, thermal conductivity has a weak depen-dence on the doping density. The total relaxation rate is givenby τ − = τ − N + τ − U + τ − I .Surface roughness scattering is often modeled using theRTA and a specularity parameter that accounts for diffusescattering at the surface. Here, we have accounted for SRSmore realistically by generating a rough surface with specificrms roughness and correlation length. When a phonon hitsthe rough surface, it will reflect specularly at the point of im-pact; this approach is reminiscent of ray-tracing (see, for in-stance, Refs. 96 and 97). The phonon can undergo multiplereflections before it returns inside the wire (Fig. 9).We used a quadratic dispersion relationship, ω = υ s q + c s q , fitted to the experimental data of Ref. 92, for transverse
300 400 500 600 700 800 900 100024681012
Temperature [K] T he r m a l C ondu c t i v i t y [ W / m . K ] (a) Wire thickness [nm] T he r m a l c ondu c t i v i t y [ W / m . K ] T=300 KT=1000 K (b)
FIG. 10. (a) Thermal conductivity of 4-nm and 9-nm-thick GaNNWs as a function of temperature. (b) Thermal conductivity as afunction of NW thickness at 300 K and 1000 K. Roughness rmsheight is 0.3 nm and the correlation length is 2.5 nm. and longitudinal bulk phonons. The quadratic dispersion isquite accurate in wurtzite GaN, as shown, for example, by Ma et al . υ s is the the sound velocity (i.e. the phonon groupvelocity at the Γ point). The material parameters are listed inTable III.Figure 10 shows thermal conductivity as a function of tem-perature and wire thickness, for rms roughness of 0.3 nm anda correlation length of 2.5 nm. For these roughness parame-ters, thermal conductivity in GaN NWs shows a reduction bya factor of 20 with respect to bulk at 300 K, which empha-sizes the dominance of SRS in phonon transport over otherprocesses.The slight waviness in the 1000 K in Fig. 10b is of nu-merical origin; with increasing temperature the number ofreal phonons represented by one numerical phonon increasesrapidly, which affects accuracy. While the error bars on the300 K data are too small to be visible, the 1000 K values areof order a few percent (Figure 10b).0 Parameter ValueLattice constant a . ˚ALattice constant c . ˚AL branch phonon group velocity at point Γ υ L . × cm / s T branch phonon group velocity at point Γ υ T . × cm / s Longitudinal phonon frequency at point
M f L THzTransverse phonon frequency at point
M f T . THzTransverse phonon dispersion curve fitting parameter c T − . × − m / s Longitudinal phonon dispersion curve fitting parameter c L − . × − m / s Isotope scattering parameter
Γ 2 × − TABLE III. GaN material parameters, from Ref. 92. Γ is assumed to be for sample 2 in Ref. 93. IV. FIGURE-OF-MERIT CALCULATION
Using the calculated electron mobility, Seebeck coefficient,and lattice thermal conductivity, we compute the TE figure ofmerit. Figure 11 shows the variation of room-temperature ZT as a function of wire thickness (Fig. 11a), doping density (Fig.11b), and temperature (Fig. 11c).The highest room-temperature ZT values in GaN NWsof approximately 0.2 are two-orders-of-magnitude greaterthan the bulk ZT value of 0.0017 reported by Liu andBalandin, an increase that stems both from the thermalconductivity reduction (Fig. 10b) and the Seebeck coeffi-cient increase (Fig. 8b) with decreasing wire thickness. Wireswith characteristic cross-sectional features of about 4 nm havethe highest ZT values at room temperature; the decrease in ZT with further reduction in thickness comes from the over-all detrimental effect of SRS on the electron mobility, whichovershadows the beneficial effects of thermal conductivity re-duction and S e increase.In contrast, at 1000 K, the transport window contains mul-tiple subbands and POP scattering is the dominant scatteringmechanism, so the electron mobility is nearly independent ofboth thickness and doping density. As a result, the ZT ofGaN NWs continues to increase with decreasing thickness,and reaches 0.8 in 3-nm-thick GaN NWs for the × cm − doping density. (For wires thinner than 3 nm, changes in thephonon dispersion and electronic band structure become con-siderable and atomistic approaches ought to be employed, which may quantitatively change the TE figure of merit.) The ZT of GaN NWs continues to rise with increasing temper-ature beyond 1000 K, which should ensure efficient energyharvesting with these devices up to high temperatures. V. CONCLUSION
We presented a comprehensive computational study of theelectronic, thermal, and thermoelectric properties of GaNNWs over a broad range of thicknesses, doping densities, andtemperatures.At room temperature, SRS of electrons in thin GaN NWscompetes with polar optical phonon scattering, and resultsin a decrease of the electron mobility with decreasing thick- ness. Roughness also decreases thermal conductivity in thinwires, which is beneficial in thermoelectric applications. Re-duced wire thickness improves the Seebeck coefficient, whichis considerably higher in thin wires over in bulk, owing tothe combined effects of the 1D subband density-of-states andthe increasing subband separation that follows a reductionin the wire cross section. Cumulatively, reducing the wirecross-sectional features down to 4 nm results in the room-temperature ZT increasing, with a maximum of 0.2 obtainedfor wires of 4-nm thickness doped to × cm − , a two-orders-of-magnitude increase over bulk. Below 4 nm, theroom-temperature ZT does not improve with further confine-ment, as the detrimental surface-roughness-scattering of elec-trons and the drop in mobility win over the beneficial effectsthat confinement has on the Seebeck coefficient and thermalconductivity.At high temperatures, the highest in this study being 1000K, the electron mobility flattens as a function of thickness, asmany subbands start to contribute to transport and POP scat-tering wins over the temperature-insensitive SRS. The See-beck coefficient is higher at 1000 K than at 300 K and in-creases with decreasing wire thickness, although less dramat-ically than at lower temperatures, while thermal conductivitybeneficially decreases with increased confinement. Overall,at 1000 K the thermoelectric figure of merit increases with in-creasing confinement (i.e. decreasing NW thickness), reach-ing a value of 0.8 for 3 nm wires.The ZT of GaN NWs continues to increase with the tem-perature increasing beyond 1000 K, owing to the negligibleminority carrier generation across the large gap, which under-scores the suitability of these structures for high-temperatureenergy-harvesting applications. Extrapolation of the trendwould yield ZT = 1 at 2000 K for 4-nm-thick NWs. Fur-ther improvements in ZT might be achieved by additional al-loy scattering of phonons by introducing In, as demonstratedin Ref. 30. Combined with nanostructuring, InGaN NWsmight prove to be a particularly interesting choice for high-temperature power generation. VI. ACKNOWLEDGEMENT
The authors thank Z. Aksamija and T. Kuech for helpfuldiscussions. This work was primarily supported by the NSF1
Wire thickness [nm] F i gu r e o f m e r i t T=300 K, 10 cm −3 T=300 K, 2 × cm −3 T=1000 K, 10 cm −3 T=1000 K, 2 × cm −3 (a) Doping density [cm −3 ] F i gu r e o f m e r i t W=9 nm ,T=300 KW=9 nm ,T=1000 KW=4 nm ,T=300 KW=4 nm ,T=1000 K (b)
300 400 500 600 700 800 900 100000.10.20.30.4
Temperature [K] F i gu r e o f m e r i t cm −3 × cm −3 cm −3 × cm −3 (c) FIG. 11. The TE figure of merit of GaN NWs as function of (a) wirethickness (doping density and × cm − , temperature 300and 1000 K), (b) doping density (4 and 9 nm thickness, temperatures300 and 1000 K), and (c) temperature (4 and 9 nm thickness, dopingdensity and × cm − ). grant No. 1121288 (the University of Wisconsin MRSEC onStructured Interfaces, IRG2, funded A.H.D.), with partial sup-port by the AFOSR grant No. FA9550-09-1-0230 (fundedE.B.R. and I.K.) and by the NSF grant No. 1201311 (fundedL.N.M.). Appendix A: Polar Optical Phonon Scattering
In wurtzite crystals, there is no clear distinction betweenthe longitudinal and transverse optical phonon modes. Basedon careful calculations, Yamakawa et al. have shown thatelectrons have two orders of magnitude higher scattering rateswith the LO-like modes than the TO-like ones, so it is suffi-cient to consider the LO-like modes alone in electronic trans-port calculations. Furthermore, there is a profound anisotropyin the bulk electron-phonon scattering rate with respect to theelectron momentum (see Yamakawa et al. , Ref. 55). As ourwires are assumed to be along the wurtzite c -axis, we consideronly LO-like phonons interacting with electrons whose initialand final momenta are along the c -axis. In this case, there isa single relevant phonon energy, whose value of 91.2 meV istaken after Ref. 51 and is also given in Table I.Here, we show the detailed calculation of the electron-longitudinal polar optical phonon scattering rate. The electricfield due to the propagation of a longitudinal optical phononis given by ~E ( q ) = s ~ γV ω (cid:0) a q e i~q · ~r + a † q e − i~q · ~r (cid:1) · ~e q , (A1)where ~e q is the polarization vector, a q ( a † q ) is the phonon cre-ation (annihilation) operator, and ω is the optical phonon fre-quency. γ is the effective interaction parameter given by γ = ω (cid:18) ǫ ∞ − ǫ (cid:19) . (A2)Here, ǫ ∞ and ǫ are the high-frequency and low-frequencydielectric permittivities, respectively. From Eq. (A1), the per-turbing Hamiltonian is equal to H pop = X ~q Cq (cid:0) a q e i~q · ~r − a † q e − i~q · ~r (cid:1) , (A3)where C = i r ~ e ω V (cid:16) ǫ ∞ − ǫ (cid:17) .The matrix element for scattering from the initial electronicstate | k x , n i to the final state | k ′ x , m i is given by M nm ( k x , k ′ x , q ) = h k ′ x , m | H pop ( q ) | k x , n i = Cq r N + 12 ± (A4) × Z ψ n ( y, z ) e i ( q y y + q z z ) ψ m ( y, z ) dy dz × L x Z e i ( k x − k ′ x ∓ q x ) x dx, N is the number of optical phononsgiven by the Bose-Einstein distribution function N = 1 e ~ ω kBT − . (A5)We define the function I nm ( q y , q z ) as I nm ( q y , q z ) = Z h ψ n ( y, z ) e i ( q y y + q z z ) ψ m ( y, z ) i dy dz, (A6)and the Eq. (A4) yields | M nm ( k x , k ′ x , q ) | = | C | q (cid:18) N + 12 ± (cid:19) (A7) × | I nm ( q y , q z ) | δ ( k x − k ′ x ∓ q x ) . According to Fermi’s golden rule, the polar optical phononscattering rate is given by Γ popnm = 2 π ~ X q k ,k ′ x | M nm ( k x , k ′ x ) | δ ( E ′ − E ± ~ ω ) . (A8)By changing the sum to integral we get Γ popnm ( k x ) = | C | V π ~ (cid:18) N + 12 ± (cid:19) (A9) × Z dk ′ x Z | I nm ( q y , q z ) | dq y dq z × δ ( k x − k ′ x ∓ q x ) δ ( E ′ − E ± ~ ω ) . Next, we define the overlap integral I D ( q x , L y , L z ) as I D ( q x , L y , L z ) = Z q | I nm ( q y , q z ) | dq y dq z . (A10)After substituting Eq. (A10) into Eq. (A8), and convertingthe integration over wave vector ( k ′ x ) to the integration overenergy ( E ′ ), the final POP scattering rate is written as Γ popnm ( k x ) = | C | V π ~ N r m ∗ ~ I D ( q x , L y , L z ) × α E f p E f (1 + α E f ) Θ( E f ) , (A11)where q x = k x ± k ′ x is the optical phonon wave vector alongthe NW axis. E f is the final electron kinetic energy, which isgiven by E f = E n − E m + E i ± ~ ω . (A12) Appendix B: Piezoelectric Scattering
The creation of a built-in electric field by strain is called thepiezoelectric effect, and this field causes piezoelectric scatter-ing of charge carriers. Here, we show a detailed derivation ofthe piezoelectric scattering rate in GaN NWs. The purturbingHamiltonian due to the piezoelectric effect is given by H pz = X q ee ∗ pz ǫ ∞ s ~ ρV ω q (cid:0) a q e i~q · ~r − a † q e − i~q · ~r (cid:1) , (B1)where e ∗ pz and ǫ ∞ are the effective piezoelectric constant andthe high-frequency effective dielectric constant, respectively.The matrix element for scattering from the initial electronicstate | k x , n i to the final state | k ′ x , m i is given by M nm ( k x , k ′ x ) = ee ∗ pz ǫ ∞ √ ~ ρV ω q p N q × Z h ψ n ( y, z ) e i ( q y y + q z z ) ψ m ( y, z ) i dy dz × L x Z e i ( k x − k ′ x ∓ q x ) x dx, (B2)where we used the equipartition approximation for the acous-tic phonon population, N q ≃ N q + 1 ≃ k B T ~ ω q .By assuming the linear dispersion relation for acousticphonons, i.e. ω q = υ s q , Eq. (B2) yields | M nm ( k x , k ′ x ) | = K av e k B T V ǫ ∞ q (B3) × | I nm ( q y , q z ) | δ ( k x − k ′ x ∓ q x ) , where I nm ( q y , q z ) is the overlap integral defined in Eq.(A6). 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