Electron-phonon interaction in a spin-orbit coupled quantum wire with a gap
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Electron-phonon interaction in a spin-orbit coupled quantum wire with a gap
Tutul Biswas and Tarun Kanti Ghosh
Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India (Dated: August 31, 2018)Interaction between electron and acoustic phonon in an in-plane magnetic field induced gappedquantum wire with Rashba spin-orbit interaction is studied. We calculate acoustic phonon limitedresistivity ( ρ ) and phonon-drag thermopower ( S g ) due to two well known mechanisms of electron-phonon interaction namely, deformation potential (DP) and piezoelectric (PE) scattering. In the socalled Bloch-Gruneisen temperature limit both ρ and S g depend on temperature ( T ) in a power lawfashion i.e. ρ or S g ∼ T ν T . For resistivity, ν T takes the value 5 and 3 due to DP and PE scatteringrespectively. On the other hand, ν T is 4 and 2 due to DP and PE scattering, respectively for phonon-drag thermopower. Additionally, we find numerically that ν T depends on Rashba parameter ( α )and electron density ( n ). The dependence of ν T on α becomes more prominent at lower density. Wealso study the variations of ρ and S g with carrier density in the Bloch-Gruneisen regime. Througha numerical analysis a similar power law dependence ρ or S g ∼ n − ν n is established in which theeffective exponent ν n undergoes a smooth transition from a low density behavior to a high densitybehavior. At a higher density regime, ν n matches excellently with the value obtained from theoreticalarguments. Approximate analytical expressions for both resistivity and phonon-drag thermopowerin the Bloch-Gruneisen regime are given. PACS numbers: 71.70.Ej, 73.21.Hb, 63.20.kd, 72.20.Pa.
I. INTRODUCTION
Due to the promising applications in the area of quan-tum information processing and device technology, spindependent phenomena in low dimensional structureshave been of major interest in scientific communities forseveral years. The main route of spin related phenomenais the well known spin-orbit interaction (SOI). In semi-conductor structures, SOI originates due to the inversionsymmetry breaking either in the bulk or at the hetero-interface. Band bending in heterostructure gives rise toan electric field to produce an asymmetric confining po-tential which itself is responsible for generating a SOIof Rashba type. The strength of Rashba SOI is propor-tional to the magnitude of the electric field generated andhence is tunable with the aid of an external gate volt-age. Another kind of SOI, usually termed as DresselhausSOI, originates due to the breaking of inversion symme-try in the bulk crystal. In general the strength of theDresselhaus SOI is smaller than that of Rashba SOI inheterostructure. The case of equal strength of both SOIsis of particular importance for future scope of developingnon-ballistic spin field effect transistor. In a quantum well, restriction of carrier’s motion by anadditional confinement in a particular direction essen-tially leads to the formation of a quantum wire (QW).The width of QW is of the order of the Fermi wavelength in order to allow ballistic transport. Semicon-ductor QW with SOI is considered as a building block fora better implementation of spin-field effect transistor. An in-plane magnetic field along the wire direction liftsthe k = 0 degeneracy in a spin-orbit coupled QW andas a result a gap is induced in the energy spectrum. Im-mense interests have been grown on the gapped spin-orbit coupled QW because of several proposals of asym- metric spin filtering, controlling impurities by a mag-netic field, topological superconducting phase, heli-cal states etc. Recently, magnetic field inducedspin-orbit gap in an one dimensional hole gas has beenrealized experimentally. In the present study we mainly focus on various con-sequences of electron-phonon interaction in a spin-orbitcoupled gapped QW by calculating phonon limited re-sistivity and phonon-drag thermopower in the Bloch-Gruneisen (BG) regime. In the BG regime, the resistivitydiffers abruptly from its equipartition behavior. An up-per bound of the BG regime can be defined by the char-acteristic temperature T BG = 2 ~ v s k F /k B , where v s isthe sound velocity and k F is the Fermi wave vector. Be-low T BG the acoustic phonon energy is comparable withthe thermal energy. Due to the smallness of Fermi sur-face in semiconductor structures, acoustic phonon withwave vector q ≃ k F can not be excited appreciablywhich in turn leads to a complicated temperature de-pendence of resistivity below T BG . The existence ofthe BG regime in semiconductor quantum well has beenconfirmed experimentally. Another important quantitythat can be used to probe the electron-phonon inter-action in semiconductor nanostructure is the phonon-drag contribution to the thermoelectric power. A num-ber studies have been performed to understand the be-havior of phonon limited mobility and phonon-dragthermopower in quantum well and wire for severalyears. Recently, acoustic phonon limited resistivity andphonon-drag thermopower in a Rashba spin-orbit cou-pled two dimensional electron gas have been studied. Itis revealed through a numerical analysis that the effec-tive exponents of the temperature dependence of bothresistivity and thermopower depend significantly on thestrength of the Rashba SOI.In BG regime, we find analytically that both resistiv-ity and phonon-drag thermopower in a spin-orbit coupledgapped QW maintains a power law dependence with tem-perature i.e. ρ or S g ∼ T ν T . The effective exponent ν T is5 and 3 due to deformation potential (DP) and piezoelec-tric (PE) scattering, respectively in the case of phononlimited resistivity. For phonon-drag thermopower ν T be-comes 4 and 2 due to DP and PE scattering, respectively.The exponent ν T has also been extracted numericallywhich clearly undergoes a transition from BG regime toequipartition limit. At a relatively higher density, ν T is inexcellent agreement with the analytical results. Temper-ature variation of ν T for different Rashba parameter ( α )at a fixed density has also been shown. The effect of α onthe temperature dependence of ν T becomes less promi-nent as one approaches towards higher density. Addition-ally, the dependence of both ρ and S g on carrier densityhas been shown in which both quantities undergo a tran-sition from a relatively low to a higher density behavior.We organize this paper in the following way. In sectionII we present all the theoretical details. Numerical resultsand discussions have been reported in section III. Wesummarize our work in section IV. II. THEORYA. Physical system
We consider a semiconductor QW of radius R in whichelectrons are free to move in the z - direction. RashbaSOI in a QW essentially breaks the spin-degeneracy by shifting the non-degenerate subbands laterally alongthe wave vector. But the spectrum is still degenerate at k = 0. An external magnetic field B = B ˆ z along thewire direction can be used to lift this degeneracy furtherby inducing a gap ∆ in the energy spectrum. Nowthe single particle Hamiltonian describing a gapped QWwith Rashba SOI is written as H = (cid:16) p m ∗ + V ( r ) (cid:17) σ + α ~ σ y p + ∆ σ z , (1)where p = ~ k is the electron momentum in the z - di-rection, m ∗ is effective mass of electron, σ is the unitmatrix, σ i ’s are the Pauli spin matrices, and α is thestrength of RSOI. Also ∆ = g ∗ µ B B/ g ∗ and µ B as the effective Lande g - factorand Bohr magneton, respectively. Finally, V ( r ) is theconfining potential in the transverse direction r ≡ ( x, y ).The wire is assumed to be thin enough so that onlythe lowest subband in the transverse direction is occupiedby electrons. By diagonalizing Eq. (1), the eigen ener-gies corresponding to the present physical system can befound in the following form ǫ λk = ~ k m ∗ + λ p α k + ∆ , (2)where λ = ± describes the branch index. Note that theenergy is measured from the bottom of the lowest sub- band energy ε = ~ k / (2 m ∗ ) with k as the sub-bandwave vector.The eigen functions corresponding to the + and − branches are respectively given byΨ + ( r, z ) = e ikz √ π (cid:18) cos φ k i sin φ k (cid:19) Φ ( r ) , (3)and Ψ − ( r, z ) = e ikz √ π (cid:18) sin φ k − i cos φ k (cid:19) Φ ( r ) , (4)where tan φ k = αk/ ∆. The lowest subbandwave function for r ≤ R is given by Φ ( r ) = J ( k r ) / {√ πR J ( k R ) } with J ν ( x ) as the Bessel func-tion of order ν . It is important to mention that k R isthe first zero of J ( k R ). Out side the quantum wireΦ ( r ) vanishes.At a fixed energy namely, Fermi energy ǫ F , one canhave the following expression for the Fermi wave vectors k λF = ((cid:18) − λk α + r k α + 2 m ∗ ǫ F ~ + ∆ α (cid:19) − ∆ α ) , (5)where k α = m ∗ α/ ~ . In the B → v λk = ~ km ∗ + λ α k ~ √ α k + ∆ . (6) B. Phonon limited resistivity
In this section we shall calculate resistivity due to theelectron-phonon interaction using Boltzmann transporttheory. We restrict ourselves to consider only the lon-gitudinal and transverse acoustic phonon modes. UsingDrude’s formula, the resistivity is simply written as ρ = m ∗ ne D τ E , (7)where h /τ i is the inverse relaxation time (IRT) averagedover energy and n is the density of electron.The energy averaged IRT for a specific energy branch λ is given by D τ λ E = 2 k B T Z dǫ λk f ( ǫ λk ) { − f ( ǫ λk ) } τ ( ǫ λk ) , (8)where f ( ǫ λk ) = [1 + e β ( ǫ λk − ǫ F ) ] − is the Fermi-Dirac dis-tribution function with β = ( k B T ) − . Here 2- factor ap-pears to consider the k < k = 0.According to the Boltzmann transport theory, the IRTcan be found in the following semi-classical form1 τ ( ǫ λk ) = X k ′ ,λ ′ (cid:16) − k ′ k (cid:17) W λ,λ ′ k,k ′ − f ( ǫ λ ′ k ′ )1 − f ( ǫ λk ) . (9) The transition probability from an initial state | k, λ i to a final state | k ′ , λ ′ i is given by the Fermi’s Golden ruleas W λ,λ ′ k,k ′ = 2 π ~ X Q | C Q | | F ( q ⊥ ) | | ξ λ,λ ′ k,k ′ | n N Q δ (cid:0) ǫ λ ′ k ′ − ǫ λk − ~ ω Q (cid:1) δ k ′ ,k + q + (cid:0) N Q + 1 (cid:1) δ (cid:0) ǫ λ ′ k ′ − ǫ λk + ~ ω Q (cid:1) δ k ′ ,k − q o , (10)where Q = ( q ⊥ , q ) is the phonon wave vector with q ⊥ = ( q x , q y ) and q = q z , N Q = [ e β ~ ω Q − − is thephonon distribution function, C Q is the matrix elementcorresponding to the electron-phonon interaction, F ( q ⊥ )is the form factor arising due to the transverse confine-ment. The first and second terms in the braces of Eq.(10) correspond to the absorption and emission of acous-tic phonons, respectively. Finally, the overlap integral | ξ λ,λ ′ k,k ′ | coming from the spinor part of the wave functionis given by | ξ λ,λ ′ k,k ′ | = 1 + λλ ′ cos( φ k ′ − φ k )2 . (11)The square of the electron-phonon matrix elementscorresponding to DP and PE scatterings are respectivelygiven by | C Q | = D ~ Q ρ m v sl , (12) | C Q,l ( t ) | = ( eh ) ~ ρ m v sl ( t ) A l ( t ) Q , (13)where D is deformation potential strength, h is therelevant PE tensor component, ρ m is the mass density,and v sl ( t ) is the longitudinal (transverse) component ofsound velocity. The anisotropic factors are given by A l =9 q q ⊥ / (2 Q ) and A t = (8 q q ⊥ + q ⊥ ) / (4 Q ).Finally, the square of the form factor is defined as | F ( q ⊥ ) | = (cid:12)(cid:12)(cid:12) Z Φ ∗ ( r ) e i q ⊥ · r Φ ( r ) d r (cid:12)(cid:12)(cid:12) . (14)In literature, it is assumed that the electrondensity is sufficiently high near the axis of the wireand vanishes everywhere and consequently one replaces | Φ ( r ) | ∼ / ( πR ), where R < R defines some con-finement region. So in this approximation the square ofthe form factor can be readily obtained in the followingexact form as | F ( q ⊥ ) | = 4 { J ( q ⊥ R ) / ( q ⊥ R ) } . In therest of the paper we will be using this expression of theform factor. Let us now discuss the possibility of intra and inter-branch scatterings. Generally, at low temperature intra-branch scatterings are dominant. To occur inter-branchscattering large momentum transfer is needed. Since inthe BG regime q ≪ k F so the possibility of inter-branchscatterings are ruled out. In a spin-orbit coupled QWwithout the gap, the inter-branch scattering is strictlyforbidden as readily understood from Eq. (11) sincecos( φ k − φ k ′ ) becomes unity as B →
0. But for B = 0inter-branch scattering are possible. We have checked nu-merically that the inter-branch contribution is very smallin comparison with the intra-branch one. So henceforthwe will consider only the intra-branch scattering.Now, the summation over k ′ in Eq. (9) can be per-formed with the aid of δ k ′ ,k ± q given in Eq. (10). Inthe BG regime, phonon energy is comparable with thethermal energy but much less than the Fermi energyi.e. ~ ω Q ≃ k B T ≪ ǫ F . So one can safely makethe following approximation f ( ǫ k ) { − f ( ǫ k ± ~ ω Q ) } ≃ ~ ω Q ( N Q + 1 / ± / δ ( ǫ k − ǫ F ), where + and − signscorrespond to the absorption and emission of acousticphonons, respectively. Using the above mentioned sim-plification and approximation one can find the energyaveraged IRT in the following form D τ λ E = 4 πk B T X Q | C Q | | F ( q ⊥ ) | qω Q k λF N Q ( N Q + 1) × ( | ξ λ,λk F ,k F − q | δ (cid:0) ǫ λk F − q − ǫ λk F + ~ ω Q (cid:1) − | ξ λ,λk F ,k F + q | δ (cid:0) ǫ λk F + q − ǫ λk F − ~ ω Q (cid:1)) . (15)The summation over Q in Eq. (15) can be trans-formed into an integration over q and q ⊥ as P Q → (1 / π ) R q ⊥ dq ⊥ dq . At very low temperature (BGregime) phonon states with wave vector q ≪ k F are pop-ulated. The delta functions given in Eq. (15) can beapproximated in the following form as (see Appendix Afor details) δ ( ǫ λk F ± q − ǫ λk F ∓ ~ ω Q ) ≃ m ∗ ~ ˜ k λF (cid:26) ∓ m ∗ v s ~ ˜ k λ F ˜ g λα q ⊥ (cid:27) × δ (cid:16) q − m ∗ v s ~ ˜ k λF q ⊥ (cid:17) , (16)where ˜ k λF = k λF (1 + 2 λε α /ε k λF ) with ε k λF = q α k λ F + ∆ , ε α = m ∗ α / (2 ~ ) and ˜ g λα = 1 + 2 λ ˜ ε α /ε k λF with ˜ ε α = m ∗ ˜ α / (2 ~ ). Here ˜ α is defined as ˜ α = α ∆ /ε k λF .Now inserting Eq. (16) into Eq. (15) and putting theexpressions for | ξ λ,λk F ,k F ± q | we have D τ λ E = m ∗ v s π ~ k B T k λF ˜ k λF Z dqdq ⊥ | C Q | | F ( q ⊥ ) | qq ⊥ Q × N Q ( N Q + 1) δ (cid:16) q − m ∗ v s ~ ˜ k λF q ⊥ (cid:17)( cos φ − k F − cos φ + k F m ∗ v s ~ ˜ k λ F ˜ g λα q ⊥ h φ − k F + cos φ + k F i) , (17)where φ ± k F = φ k F − φ k F ± q . C. Phonon-drag thermopower
In presence of a temperature gradient diffusion motionof electron takes place. As a result of electron-phononinteraction, phonon gains a finite heat flux which in turndrags electron along with it from the hot to the cold end.In this way a phonon-drag contribution to the thermo-electric power is generated. Phonon-drag thermopower ismore fundamental quantity in probing electron-phononstrength experimentally.To calculate phonon-drag thermopower two ap-proaches named as Q and Π-approach are mainly fol-lowed. In the rest we follow only the Q -approach. Wenow start with the following expression for the phonon-drag thermopower S λg = eτ p σLk B T X λ ′ X k , k ′ , Q ~ ω Q f ( ǫ λk ) h − f ( ǫ λ ′ k ′ ) i × W λλ ′ Q, Ab ( k , k ′ ) n τ ( ǫ λk ) v λk − τ ( ǫ λ ′ k ′ ) v λ ′ k ′ o · v p , (18)where τ p is the phonon mean free time, L is the lengthof the sample, σ is the Drude conductivity, τ ( ǫ k ) is elec-tron’s momentum relaxation time, v λk is the velocity ofan electron as given in Eq. (6), v p = v s ˆ Q is the velocityof phonon. Finally, W λλ ′ Q, Ab ( k , k ′ ) is the transition prob-ability by which an electron makes a transition from aninitial state | k , λ i to a final state | k ′ , λ ′ i with the absorp-tion of an acoustic phonon.The transition probability can be written as W λλ ′ QAb = 2 π ~ | C Q | | F ( q ⊥ ) | | ξ λ,λ ′ k,k ′ | N Q δ (cid:16) ǫ λ ′ k ′ − ǫ λk − ~ ω Q (cid:17) × δ k ′ ,k + q . (19)In Eq. (18) summation over k ′ is readily done using δ k ′ ,k + q given in Eq. (19). Further a slow variation of τ ( ǫ k ) over an energy scale ∼ ~ ω Q is assumed. So one canuse the approximation τ ( ǫ k + ~ ω Q ) ≃ τ ( ǫ k ).The summation over k in Eq. (18) can be convertedinto an integral over ǫ k by the following transformation X k → m ∗ L π ~ Z k λ − λ k α q k α + m ∗ ǫ k ~ + ∆ α ! dǫ k , (20)where k λ can be obtained from Eq. (5).Using Eq. (18-20) one can finally obtain the follow-ing expression for the phonon-drag thermopower for aspecific branch λ in the BG regime as S λg = − m ∗ l p v s π ne ~ k B T N λF k λF ˜ k λF Z dqdq ⊥ | C Q | | F ( q ⊥ ) | q ⊥ × Q N Q ( N Q + 1) (cid:16) − m ∗ v s ~ ˜ k λ F ˜ g λα q ⊥ (cid:17) δ (cid:16) q − m ∗ v s ~ ˜ k λF q ⊥ (cid:17) × | ξ λ,λk F ,k F + q | h v λk F + q − v λk F i · v p , (21)where N λF = (cid:16) − λk α / q k α + m ∗ ǫ F ~ + ∆ α (cid:17) . Note thatin deriving Eq. (21) we have used the approximation f ( ǫ k ) { − f ( ǫ k + ~ ω Q ) } ≃ ~ ω Q ( N Q + 1) δ ( ǫ k − ǫ F ) asearlier. D. Approximate analytical results in BG regime
We shall now derive some approximate analytical ex-pressions for phonon limited resistivity and phonon-dragthermopower in the BG regime.In the BG regime we have q << k F . In this limitone can use following approximation cos φ ± k F ≃
1. Againphonon energy is comparable with the thermal energyin the BG regime i.e. ~ v s q ⊥ ∼ k B T . So we have q ⊥ R = k B T R / ~ v s which is much lower than unity andconsequently one can approximate the form factor as | F ( q ⊥ ) | ≃ D τ λ E ≃ π ~ k B T (cid:16) m ∗ v s ~ ˜ k λF (cid:17) ˜ g λα k λF ˜ k λF × Z dq ⊥ q ⊥ | C q ⊥ | N q ⊥ ( N q ⊥ + 1) . (22)Now inserting | C q ⊥ | as given in Eq. (12-13) and us-ing the standard integral R ∞ x p e x / ( e x − = ζ ( p ) p ! inEq. (22) one can derive the following expressions for theenergy averaged IRT corresponding to DP, longitudinaland transverse PE scatterings, respectively as D τ λ E DP ≃ D ˜ g λα ζ (5) πρ m ~ v sl k λF ˜ k λF (cid:16) m ∗ v sl ~ ˜ k λF (cid:17) ( k B T ) , (23) D τ λ E PE , l ≃ eh ) ˜ g λα ζ (3)2 πρ m ~ v sl k λF ˜ k λF (cid:16) m ∗ v sl ~ ˜ k λF (cid:17) ( k B T ) , (24)and D τ λ E PE , t ≃ ( eh ) ˜ g λα ζ (3)4 πρ m ~ v st k λF ˜ k λF (cid:16) m ∗ v st ~ ˜ k λF (cid:17) ( k B T ) × n (cid:16) m ∗ v st ~ ˜ k λF (cid:17) o . (25)In deriving approximate analytical results for phonon-drag thermopower we expand velocity in Eq. (6) andretain terms up to q since q ≪ k F in the BG regime. Wefind the approximate expression for the following quan-tity as v λk F + q − v λk F = (cid:16) ~ m ∗ + λ ˜ α ~ α ∆ (cid:17) q. (26)Inserting Eq. (26) in Eq. (21) and doing the integra-tion over q we arrive at the following approximate resultfor the phonon-drag thermopower S λg ≃ − m ∗ l p v s N λF π ne ~ k λF k B T (cid:16) m ∗ v s ~ ˜ k λF (cid:17) (cid:16) ~ m ∗ + λ ˜ α ~ α ∆ (cid:17) × Z dq ⊥ q ⊥ | C q ⊥ | N q ⊥ ( N q ⊥ + 1) . (27)After doing the integration over q ⊥ we finally ob-tain the following expressions for the phonon-drag ther-mopower due to DP, longitudinal, and transverse PEscatterings, respectively S λg (cid:12)(cid:12)(cid:12) DP ≃ − k B e D ~ v sl (cid:16) m ∗ v sl ~ ˜ k λF (cid:17) P λl ζ (5)( k B T ) , (28) S λg (cid:12)(cid:12)(cid:12) PE , l ≃ − k B e (cid:16) m ∗ v sl ~ ˜ k λF (cid:17) ( eh ) P λl ζ (3)( k B T ) , (29)and S λg (cid:12)(cid:12)(cid:12) PE , t ≃ − k B e v st v sl (cid:16) m ∗ v st ~ ˜ k λF (cid:17) n (cid:16) m ∗ v st ~ ˜ k λF (cid:17) o × ( eh ) P λt ζ (3)( k B T ) . (30)Here P λl ( t ) is defined as P λl ( t ) = m ∗ l p N λF π nρ m ~ v sl ( t ) k λF (cid:16) ~ m ∗ + λ ˜ α ~ α ∆ (cid:17) . (31) Let us now provide here a systematic comparison be-tween the results obtained for a QW (present case) andtwo-dimensional electron system (2DES) with RashbaSOI in BG regime. In this context, we calculate the fol-lowing quantity S g ρ − for both quasi-2DES and QW. Inthe case of a quasi-2DES, using approximate analyti-cal expressions for ρ and S g in the BG regime, one can ob-tain the following result: S g ρ − (cid:12)(cid:12) d ≃ − Γ k f κ d /T for DPand longitudinal PE scattering. Here, Γ = el p v sl / (4 π ), κ d = 1 − k α /k f and finally, k f is the Fermi wavevector obtained via k f = √ πn d with n d as the car-rier concentration in 2D. The result corresponding to thetransverse PE scattering is easily obtained by multiply-ing the above result by a factor of v st /v sl . Note that ρ ( S g ) represents the total resistivity (phonon-drag ther-mopower) which is obtained by summing up the con-tributions coming from individual energy branches. Inthe present case, with B = 0, the total resistivity orphonon-drag thermopower can not be obtained becauseof the complicated structure of k λF and other quanti-ties as evident from Eqs. (23)-(25) and Eqs. (28)-(30).However, in the B → S g ρ − (cid:12)(cid:12) d ≃− Γ k F κ d / (2 T ) due to DP and longitudinal PE scatter-ing, where κ d = 1 − ( k α /k F ) and k F = nπ/ n as the electron density in 1D. Multiplying S g ρ − (cid:12)(cid:12) d by v st /v sl , one can obtain the corresponding result fortransverse PE case. However, the functional forms of S g ρ − (cid:12)(cid:12) d and S g ρ − (cid:12)(cid:12) d are different but in both cases weessentially obtain S g ρ − ∼ T − which confirms Herring’slaw. III. RESULTS AND DISCUSSIONS
From Eqs. (23-25) and Eqs. (28-30) it is revealed thatphonon limited resistivity and phonon-drag thermopowerin the BG regime depends on temperature in a power lawfashion i.e. we have ρ or S g ∼ T ν T . The effective expo-nent ν T becomes 5 and 3 due to DP and PE scattering,respectively in the case of resistivity. On the other hand,for phonon-drag thermopower ν T is 4 and 2 correspond-ing to DP and PE scattering, respectively. However, theintegrals over q ⊥ in Eqs. (17) and (21) have been eval-uated numerically for both DP and PE scattering mech-anisms to the show the explicit temperature dependenceof ρ and S g .For the numerical calculation following material pa-rameters, appropriate for an InAs quantum wire, havebeen considered: m ∗ = 0 . m e with free electron mass m e , g ∗ = − ρ m = 5 . × Kg m − , v sl = 4 . × ms − , v st = 2 . × ms − , D = − .
08 eV, h =3 . × Vm − , n = 10 m − , R = 10 nm, and α = 10 − eVm. The value of the external magneticfield is taken to be B = 0 . n = −4 −2 T (K) ρ ( Ω m − ) −2 −1 T (K) ρ ( Ω m − ) T ν T T ν T n=3n n=5n n=7n n=9n (b): PE (a): DP FIG. 1: (Color online) The dependence of phonon limitedresistivity on temperature is shown. Panel (a) and (b) repre-sent DP and PE scattering contribution, respectively. Differ-ent values of density namely n = 3 n , 5 n , 7 n , and 9 n areconsidered. The strength of RSOI is fixed to α = 3 α . Thetemperature variations of the exponent ν T are shown in theinsets of both panels. −15 −14 −13 −12 −11 −10 −9 T (K) − S g ( V / K ) −13 −12 −11 −10 −9 T (K) − S g ( V / K ) T ν T T ν T n=3n n=5n n=7n n=9n (b): PE(a): DP FIG. 2: (Color online) Temperature dependence of phonon-drag thermopower due to DP and PE scatterings are shown.Different values of density namely n = 3 n , 5 n , 7 n , and 9 n are considered. The strength of RSOI is fixed to α = 3 α .The temperature dependencies of the effective exponent ofphonon-drag thermopower are shown in the insets of bothpanels. n , n , n and 9 n . The value of Rashba parameteris considered as α = 3 α . Fig. 1 clearly demonstrates acrossover from the low temperature BG regime to hightemperature equipartition regime (in which ρ ∼ T ). For both DP and PE scattering mechanisms ρ decreases as n increases. The resistivity due to PE scattering is higherin magnitude than DP scattering. The exponent ν T of the temperature dependence of ρ can be defined as ν T = d log ρ/d log T which is extracted numerically and itsvariation with temperature has been shown in the insetsof Fig. 1. It is clear that the temperature variation of ν T depends on electron density. At lower density, namely n = 3 n the exponent ν T shows a clear deviation fromthe limiting case (i.e. ν T = 5 due to DP and ν T = 3 forPE scattering). As density increases the BG temperatureregime becomes more stable. This numerically obtainedBG regime is in excellent agreement with the approxi-mated analytical results. As temperature increases ν T approaches to its equipartition value i.e. ν T = 1. T (K) ν T T (K) ν T α =0 α = α α =3 α α =5 α n=2n n=2n n=6n (i): DP (ii): PE n=6n FIG. 3: (Color online) The temperature variation of the effec-tive exponent of resistivity i.e. ν T = d log ρ/d log T for variousvalues of α , namely, α = 0, α , 3 α and 5 α are shown. Leftpanel is considered for DP scattering in which upper and lowerpanel correspond to n = 2 n and 6 n . Similarly right paneldescribes PE scattering for n = 2 n and 6 n . In Fig. 2 we show the temperature dependence ofphonon-drag thermopower due to DP and PE scattering. S g decreases with the increase of density. In this casewe also extract the exponent ν T = d log S g /d log T of thetemperature dependence of S g . Similar to the resistivitycase the temperature dependence of ν T also depends onthe density as depicted in the insets. At higher densityBG regime is obtained in which S g ∼ T due to DP and S g ∼ T due to PE scattering. The magnitude of S g duePE scattering is higher than that of DP scattering.Let us now discuss the following important point. Theboundary of the BG regime is defined by the character-istic temperature T BG = 2 ~ v s k F /k B . For a typical valueof electron density, say n = 5 n we have T BG ∼ ν T not only dependson the density but also on the Rashba parameter α .These facts are depicted in Figs. 3 and 4 in which the ν T T (K) ν T T (K) α =0 α = α α =3 α α =5 α n=6n n=6n n=2n n=2n (ii): PE(i): DP FIG. 4: (Color online) The temperature variation of theeffective exponent of phonon-drag thermopower i.e. ν T = d log S g /d log T for various values of α , namely, α = 0, α , 3 α and 5 α are shown. Left panel is considered for DP scatteringin which upper and lower panel correspond to n = 2 n and6 n . Similarly right panel describes PE scattering for n = 2 n and 6 n . n (10 m −1 ) ρ ( Ω m − ) −2 n (10 m −1 ) ρ ( Ω m − ) m −1 ) ν n m −1 ) ν n T=0.5 KT=1 KT=2 K (c)(a): DP (b): PE(d)
FIG. 5: (Color online) The dependence of phonon limited re-sistivity on carrier density for different temperatures namely, T = 0 . α = 2 α .Panels (a) and (b) are due to DP and PE scattering. Inpanels (c) and (d) we show the variation of the quantity ν n = − d log ρ/d log n with n due to DP and PE scatteringrespectively. temperature dependencies of ν T corresponding to ρ and S g for different α are shown. When density is low thetemperature variation depends significantly on α . At arelatively higher density, the effect of α on this tempera-ture dependence is not so prominent for both ρ and S g .Similar effect of α on the temperature dependence of ρ or −14 −12 −10 −8 n (10 m −1 ) − S g ( V / K ) m −1 ) ν n m −1 ) ν n −13 −12 −11 −10 −9 n (10 m −1 ) − S g ( V / K ) T=0.5 KT=1 KT=2 K (a): DP (b): PE(c) (d)
FIG. 6: (Color online) The dependence of phonon-drag ther-mopower on carrier density for different temperatures namely, T = 0 . α = 2 α .Panels (a) and (b) are due to DP and PE scattering. Inpanels (c) and (d) we show the variation of the quantity ν n = − d log S g /d log n with n due to DP and PE scatteringrespectively. S g in a Rashba spin-orbit coupled two dimensional elec-tron gas in BG regime has been addressed recently. In Figs. (5) and (6) we have shown how ρ and S g depend on the electron density at a fixed temperaturein BG regime. In Eqs. (23-25) and (28-30) one can no-tice that both IRT (and consequently ρ ) and phonon-dragthermopower show a power law dependence with electrondensity through the Fermi wave vectors at a fixed temper-ature. So in general we can write ρ or S g ∼ n − ν n , wherethe exponent ν n corresponding to ρ and S g can be ob-tained by taking negative logarithmic differentiation of ρ or S g with respect to n i.e. ν n = − d log ρ ( S g ) /d log n . Letus now estimate ν n from Eqs. (23-25) and (28-30). It iswell known that the Fermi wave vector scales with densityas k F ∼ n in one dimension. In our case k λF depends ondensity in a complicated way as seen from Eq. (5). Never-theless, we can find k λF ∼ n since k α , ∆ /α ≪ m ∗ ǫ F / ~ .From Eqs. (23-25) one finds h τ − i ∼ n − and as a re-sult we have ρ ∼ n − . The phonon-drag thermopowerdepends on density as S g ∼ n − as seen from Eqs. (28-30). However solving Eqs. (17) and (21) numerically wefind that ν n undergoes a crossover from a relatively lowerdensity behavior to a higher density behavior for both ρ and S g . As density increases ν n approaches towards thevalues obtained from asymptotic expressions i.e. ν n = 6for ρ and ν n = 5 for S g . Note that at higher densitysame values of ν n are obtained due to DP and PE scat-tering for both case of ρ and S g . But at lower densities ν n differs significantly due to DP and PE scattering.Although a gap ∆ is considered in the energy spectrumbut its magnitude is much smaller than that correspond-ing to the Rashba spin-splitting i.e. ∆ ≪ αk F . The mainpurpose for considering ∆ is to see whether inter-branchtransitions are happening or not. But in the BG regime,the possibility of inter-branch scattering has been ruledout. So the qualitative results do not change significantlydue to the presence of ∆ in the energy spectrum. IV. SUMMARY
In summary we have studied various features of acous-tic phonon limited resistivity and phonon-drag ther-mopower in a Rashba spin-orbit coupled semiconductorQW with an in-plane magnetic field induced gap. Twomechanisms of electron-phonon interaction, namely, DPand PE scatterings are taken into consideration. In theBG regime a power law dependence of both resistivityand phonon-drag thermopower with temperature havebeen obtained analytically. We find the exponent ( ν T )of the temperature dependence which takes the value 5and 3 corresponding to the DP and PE scattering, re-spectively in the case of resistivity. ν T becomes 4 and 2in the case of phonon-drag thermopower due to DP andPE scattering, respectively. Through a numerical calcu-lation, we have shown a transition in resistivity from BGto equipartition regime. Numerically, it is also found that ν T depends on both density and Rashba parameter. Athigher density ν T matches well with that obtained fromthe analytical calculation for both ρ and S g or in otherwords a BG regime is established at higher density. Theeffect of spin-orbit interaction on ν T is found to be moreprominent in low density regime. Finally the dependenceof ρ and S g on the carrier density are also discussed. Anapproximate analytical calculation shows that ρ ∼ n − and S g ∼ n − in the BG regime. These dependence on n have been confirmed through a numerical analysis athigher densities. The results obtained in the present casehave also been compared with the corresponding resultsfor spin-orbit coupled two-dimensional electron systemand we obtain in both cases S g ρ − ∼ T − which affirmsHerring’s law. Appendix A
In this Appendix we shall perform an explicit deriva-tion of the term δ ( ǫ λk F ± q − ǫ λk F ∓ ~ ω Q ) as given in Eq.(16.) From Eq. (2) one can write ǫ λk F + q = ~ ( k λF + q ) m ∗ + λ q α ( k λF + q ) + ∆ . (A1)Now defining ε k λF = q ( αk λF ) + ∆ and assuming q ≪ k λF , the second term in Eq. (A1) can be expanded up to q as ε k λF + q = ε k λF + α k λF qε k λF + α q ε k λF (cid:16) − α k λ F ε k λF (cid:17) . (A2)We then have ǫ λk F + q − ǫ λk F = ~ m ∗ ˜ g λα (cid:16) q + 2 q ˜ k λF ˜ g λα (cid:17) , (A3)where ˜ g λα and ˜ k λF are defined earlier.Since we are dealing with the BG regime in which q ≪ k λF , the term q in Eq. (A3) can be neglected. So fromthe energy conservation ǫ λk F ± q − ǫ λk F ∓ ~ ω Q = 0, one canobtain q = (cid:0) m ∗ v s / ~ ˜ k λF (cid:1) Q with Q = p q ⊥ + q . Since thecoefficient m ∗ v s / ( ~ ˜ k λF ) ≪ q ≪ Q which in turn forces us to write the followingexpression q = m ∗ v s ~ ˜ k λF q ⊥ . 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