Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 ACOUSTIC PHONON-LIMITED MOBILITY IN GAN QUANTUM WIRES: EFFECT OF INELASTICITY N. S. SANKESHWAR* SAMEER. M. GALAGALI** *Profeor, Dept. of Phyic, Karnata Univerity, Dharwad, Karnataa, India **At. Profeor, Dept. of Phyic, K.L.S V.D.R. Intitute of Technology, Haliyal, Karnataa, India ABSTRACT A tudy of the acoutic phonon-limited electron mobility in rectangular quantum wire (QWR) for T< 30K i reported. Taing into account the inelaticity of the cattering via piezoelectric () coupling, an epreion for the phonon cattering rate i given. Numerical calculation of the cattering rate, conidering both and coupling, and preented for GaN QWR auming electron to occupy the lowet ubband, bring out their relative importance. The dependence of mobility on temperature, electron concentration and thicne of the QWR i tudied. For temperature greater (le) than ~5 K, the mobility in GaN i dominated by () cattering. The mobility i found to increae with increae in QWR thicne and how a monotonic dependence on electron concentration. Within the elatic cattering approimation, uually ued in literature, the mobility i found to be underetimated. A detailed invetigation of low-temperature mobility in ultrapure ample will provide a better undertanding of electron-phonon interaction in QWR. KEYWORDS: Acoutic Phonon, Inelatic Scattering, Mobility, Quantum Wire, GaN I. INTRODUCTION Semiconductor quantum wire (QWR) owing to their enhanced functionalitie have, in recent year, attracted great attention in variou area uch a nanoelectronic, optoelectronic, bioening and energy harveting []. Of particular ignificance, from both theoretical and application point of view, i the low-temperature tranport propertie of a quai-one-dimenional (QD) electron ga realied in emiconducting QWR. Fabrication technique uch a MBE and MOCVD, which have made poible the growth of high-purity QWR ytem, in which the cattering by unwanted defect i reduced, have augmented the interet in low-temperature tranport in QWR. The low-field mobility of electron in a typical emiconducting QWR i determined, at any finite temperature by the variou cattering mechanim operative, namely electronimpurity, electron-phonon, and electron-urface roughne. Among the intrinic cattering 40
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 mechanim, the acoutic phonon are nown to be important in limiting the low-temperature (T< ~50 K) mobility while the optical phonon are ecited at higher temperature (T > ~00 K) []. In polar emiconductor, uch a GaN, the electron-acoutic phonon interaction tae place via the deformation potential () and piezoelectric () coupling. In literature, invetigation of the electron-acoutic phonon cattering in QD electron ga are often made conidering the electron-acoutic phonon interaction to be elatic [3-5], where acoutic phonon energy i neglected. Thi approimation fail in QWR ytem. Ridley [6], to include the inelaticity of the electron-phonon interaction, propoed the momentum conervation approimation (MCA), in which the form factor decribing confinement i ubtituted by a um of Kronecer delta. Conidering electron-acoutic phonon cattering in rectangular quantum well wire to be elatic, Arora [3] ha preented a theory for conductivity in the ize quantum limit, where electron are retricted to the lowet ubband. Employing MCA, Lee and Vael [4] obtained approimate analytical epreion for the momentum relaation time for electron-impurity and electron-phonon interaction, and tudied low-field drift mobility in rectangular quantum wire. The mobility i found to decreae with temperature and increae with tranvere dimenion of the QWR, and at higher temperature, the phonon cattering i the dominant mechanim. Kundu and Sarar [5], auming the electron to be confined by a nearly triangular potential well in one direction and a quare quantum well in the other tranvere direction, tudied the effect of creening on the low-field mobility in QWR. Fihman [], auming envelope wave function to be contant inide the cylindrical wire, calculated low- temperature mobility. However, the electron-acoutic phonon interaction in QWR i eentially inelatic [7]. The inelaticity caued by momentum conervation uncertainty i due to lac of tranlation ymmetry in direction perpendicular to a QD tructure. Miceviciu and Mitin [7] have tudied electron mobility in GaA rectangular QWR conidering the inelaticity of the cattering by acoutic phonon via only coupling. Karpu and Lehman [8] tudied acoutic-phonon limited cattering via and coupling in the Bloch-Gruneien regime (T< ~0 K) and They aume a parabolic-fang-howard confinement along the tranvere direction and etimate the low-temperature (~ K) mobility in GaA QWR to be ~0 8 cm V - -, at low electron concentration (~ 0 6 m - ). Recently, employing Ridley [6] momentum conervation approimation, Bhattacharya et al [9] developed analytic epreion for normalized acoutic phonon limited cattering rate and mobility via only coupling in GaA/GaAlA QWR. They have compared their reult with calculation of [7]. 40
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 In the preent wor, following the formalim of [7], an epreion for the cattering rate of the electron-acoutic-phonon interaction via phonon coupling i given taing into account the inelaticity of the interaction. The role of inelaticity on the dependence of acoutic phonon limited mobility on temperature, electron concentration and effective thicne of the wire, for GaN QWR ytem, i dicued in Sec.III. II. Theory We conider a QD electron ga confined in a rectangular quantum wire, and aume for implicity the confining potential along the tranvere (y and z) direction to be of infinite height. The electron wave function and energy eigenvalue are given by mn ( r) i my nz e in in V L y L z () and E mn m n m L y L * z () E m E n E where, m, n =,,3,... are ubband indice in the y and z quantied direction, repectively, V= L Ly Lz i the volume of the quantum wire and i the electron wave vector. Auming the electron, in defect-free, ultrapure QWR ytem at low temperature to be cattered by only the intrinic acoutic phonon, the tranition probability of electron cattering from the tate (m, n, ) to the tate (m,n, ) by the acoutic phonon can be epreed a W C j, q q,, ', q N G q, V v mm ' q q y G nn ' q z E E ' E q, ', q Here, j (,), ω q, i the energy of an acoutic phonon of wave vector q q and mode ( l,t) with q / q y q z (3), q, N q, i the phonon ditribution, with the upper (lower) ign correponding to emiion (aborption) of acoutic phonon, E m* m m' Ly n n' Lz i the energy eparation between ubband,, and G mm (q y ) and G nn (q z ) are the form factor decribing the confinement [7], and q i the creening factor. 403
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 In equation (3), the matri element j q C,, for electron- phonon interaction may be epreed a,, 4 q eh A q/ q C (4) where, h 4 i the piezoelectric tenor component and A (q) i the dimenionle aniotropy 4 6 factor[0] with A 9 q q q l 4 6 6 q A q 8 q q q 4 q of phonon of mode. The epreion for the phonon cattering rate become, E dq C q N G q G q E E E v q,, q, v t mm' y nn' z q and v i the velocity To evaluate Eq. (5), we carry out the integration over one of the tranvere direction, ay q z. / Uing delta function and ubtituting q q q z y (5) where E E E v q q q ma eh ma y 4, E, dq dqy Nq, B v q min q y min Eq.(5) can then be epreed a,, 6 q q q / y G q (6) where, 9 l q q, 4 Bt (, q) ( t q ) 8 q ( t q ), and G G mm' q y Gnn' qz Bl (, q) 4., It may be noted that, the epreion for the electron cattering rate E acoutic phonon via coupling, i given by Eq.(5) with the factor C, l a q E, q, due to replaced by, C, where E a i the contant. It may be mentioned that only the longitudinal component, A l (q), the longitudinal component of the aniotropic factor, i aociated with coupling. The limited cattering rate i given by [7] / q q q ma q ma E y y, a G l E dq dq y N q y vl q min q y min q. (7) In the cae of QD ytem, one ditinguihe between bacward cattering event, in which electron wave vector change it ign during the cattering, and forward cattering event, in which electron wave vector doe not change it ign. Accordingly, in equation (6) and (7), the requirement that the epreion under the quare root i greater than zero, impoe the limit of integration for forward and bacward emiion and aborption procee [7]. Simplified epreion for the cattering rate (6) and (7) can be obtained if one aume the electron-acoutic-phonon cattering to be elatic, a at high temperature, 404
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 wherein vq BT and Nq Nq BT vq, o that the phonon energie can be neglected. In the momentum conervation approimation of [6], for the form factor G i ued in the form of um of the weighted ronecer delta. Further, one may aume that q <<q, where q q y q z, i the tranvere component of q [7]. Both thee aumption require that energie, vqma bt, where / q m m' Ly n n' Lz. For large electron ma E E vqma, one may neglect the phonon energy. For intraubband electron cattering (m=m and n=n ), E=0, and, E Eq v quantum limit (m=m =n=n =), the form factor G i given by [0]. 8 in G Qy Qz y Q in Q y Q Q z z. Further, in ize where Q y =q y L y / and Q z = q z L z /. Integrating over q and neglecting phonon energy, Equation (5), for cattering retricted to lowet ubband may then be written a, (8) / eh b T m 4, E dy dz v L y L * z 0 0 where, E (9) 8 m* 6 4 q E 4 E,, E. l t / 9 3 q E Similarly Eq.(7), within elatic approimation, become / * 9 E a b T m E. (0), l v l L y L z E The total cattering rate for and phonon of mode can be calculated uing Mattheien rule, E E, () j, i j,, i where the ummation i over all the relevant cattering event i f a, f e,b a,b e due to forward aborption( f a ), forward emiion (f e ), bacward aborption (b a ) and bacward emiion (b e ) of acoutic phonon. The overall cattering rate due to and phonon i written a E E E () j, j, 405
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 Since the forward cattering event do not contribute to the momentum relaation, the bacward cattering play an important role in determining the momentum relaation time, and the tranport parameter. m E The epreion for mobility i given by * E / m e, (3) ac m where m E i the momentum relaation time due to acoutic phonon cattering averaged over equilibrium electron ditribution function: / o E. me.( df / de ) de ( E) (4) m / o E.( df / de ) de III. Reult and Dicuion: We have performed numerical calculation of acoutic phonon-limited mobility uing Eq.(3) for a QD electron ga confined in a GaN QWR, conidering contribution from both and coupling. With a view to tudy the relative importance of the variou contribution and tudy the influence of the inelaticity of the electron- phonon cattering in QWR, we firt preent calculation of the cattering rate for coupling for parameter characteritic of GaN [,]: m*=0. m 0, E a =8.3eV, v l =6.50 3 m -, v t =3.040 3 m -, h 4 =6.660 9 Vm -. For implicity, we conider electron concentration in the range N D =0 7-0 8 m -, enuring carrier confinement to the lowet ubband. Figure how the energy dependence of the variou contribution to the electronphonon cattering rate auming Ԑ(q )= for a QWR of cro ection 4040Å for T=30 K. - (E ) ( - ) 0 5 0 4 0 3 0 0 0 0 (a) GaN 7 6 5 4 3 ac - (E ) ( - ) 0 4 0 3 0 0 (b) GaN 0 9 0. 0 00 E (mev) 0 0 0. 0 00 E (mev) Figure.: Variation of acoutic phonon ( and ) cattering rate a a function of electron energy in a GaN QWR of cro ection 4040Å at 30 K. (a) Curve and, repectively, denote the cattering rate due to emiion and aborption of longitudinal phonon. Curve 3 denote the total contribution. Curve 4 and 5, repectively, denote the cattering rate due to aborption and emiion of tranvere phonon. Curve 6 denote the total contribution. 406
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 Curve 7 depict the overall cattering rate. (b) Curve denote the overall contribution from (dahed) and (dah-dotted) phonon. Curve repreent overall rate in elatic approimation. In figure (a), curve 3 denote the total contribution to the cattering rate due to longitudinal phonon by aborption (curve ) and emiion (curve ) procee. Curve 6 denote the total contribution from tranvere phonon due to emiion (curve 5) and aborption (curve 4) procee. Curve 7 depict the overall contribution from longitudinal and tranvere phonon. For the range of energie conidered, the rate are found to decreae with increae in energy, the dominant contribution to overall cattering rate (curve 7) i from tranvere mode. In cae of longitudinal phonon, the dominant contribution i from aborption procee and for tranvere phonon, emiion procee are important. Further, in cae of longitudinal phonon, bacward emiion dominate the emiion procee where a forward aborption dominate for aborption procee. In cae of tranvere phonon, forward emiion dominate the emiion procee where a forward aborption dominate for aborption procee at all energie. To bring out the relative importance of phonon coupling interaction, with repect to coupling interaction, we now preent our calculation of the cattering rate due to both phonon (uing Eq. 6) and phonon (uing Eq. 7). In Figure (b), the dahed and dah- 0 4 (a) GaN 0 (b) GaN ac (m V - - ) 0 3 0 0 0 0 ac (m V - - ) 0 0 0-5 0 5 0 5 30 T(K) 0-5 0 5 0 5 30 T(K) Figure. : Temperature dependence of acoutic phonon limited electron mobility in GaN QWR of cro ection 4040 Å with N D =0 7 m -. The dahed and dah-dotted curve repreent repectively, and phonon-limited mobilitie. Curve how the total (+) mobility. Correponding curve in Figure (b) refer to the elatic cattering. dotted curve repreent repectively, the energy dependence of the individual contribution from and phonon to the overall electron- acoutic phonon cattering rate (curve ) calculated uing Eq.(): E E E E. It i een that the, l cattering rate for phonon decreae with increae in energy, where a for coupling, the energy dependence ehibit a maimum at E=E c =v l q ma [7], which for cattering, l, t 407
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 retricted to the lowet ubband depend on the phonon velociiey v, and on the effective thicne L=(L - y +L - z ) -/ of the wire; E c 9.6meV for GaN. We find that the overall rate i dominated by the contribution, although it i influenced by cattering for E> 3meV. Curve repreent total (+) electron-phonon cattering rate calculated uing Eq. (9) and Eq (0). We find that in thi approimation the coupling till play a major role in carrier cattering. Figure. (a) how, the temperature dependence of electron mobility calculated uing Eq. (3) for a GaN QWR. In figure (a), the dahed and dah-dotted curve, repectively, repreent contribution from and phonon to the total acoutic-phonon (+) limited mobility (curve ). In thi cae, it i een that cattering i dominant for temperature up to about 0 K, after which cattering become effective. Figure. (b) how, temperature dependence of contribution from (dahed) and (dah-dotted) phonon to the total acoutic-phonon (+) limited mobility (curve ) calculated within (a) GaN ac (m V - - ) 0 3 0 0 0 0 ac (m V - - ) 0 0 0 (b) GaN 0-4 6 8 0 4 L (nm) 0-4 6 8 0 4 L(nm) Figure. 3: Variation of acoutic phonon limited electron mobility in GaN QWR at 30K with N D =0 7 m - a a function of effective thicne of the wire. The dahed and dah-dotted curve, how, contribution from and phonon, repectively. Curve denote overall (+) mobility. Curve in Figure 3(b) refer to the elatic cattering. elatic approimation. A comparion of Figure (a) and (b) how that thi approimation not only underetimate mobility but alo alter it temperature dependence, and that the contribution i dominant for the whole temperature range conidered. A imilar underetimation ha been oberved by Miceviciu and Mitin [7] for cattering in GaA QWR. In order to tudy the dependence of mobility on the tranvere dimenion of the wire, we have performed calculation a a function of the effective thicne, L, of the wire. Figure 3(a), depict the variation of the electron mobility in GaN QWR with N D =0 7 m - at 30K. The dahed and dah-dotted curve repreent contribution from and cattering to the total acouticphonon (+) mobility (Curve ). For the value of the parameter conidered, it i een 408
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 that, the total mobility increae with increae in QWR thicne a reported in literature [4]. It i alo oberved that the contribution due to cattering i ignificant for QWR of low thicne (L< nm). Figure 3(b) repreent the variation of mobility calculated uing Eq.(3), within elatic approimation. It i een that the overall mobility at firt (L < 4 nm) increae rapidly with L and then tend to aturate for L >0 nm. A comparion of figure 3(a) and 3(b) how that the magnitude of mobility i underetimated in the elatic approimation. Further, the cattering i the dominant mechanim for L > 3nm. It may be noted that Bhattacharya et.al [9] calculated mobility of a GaA QWR conidering the contribution from only phonon. They find the mobility calculated within 0 (a) GaN T=4. K 0 (b) GaN 4.K ac (m V - - ) 0 0 0 ac (m V - - ) 0 0 4 6 8 0 N D ( 0 7 m - ) 4 6 8 0 N D ( 0 7 m - ) Figure. 4: Acoutic phonon limited mobility variation a a function of electron concentration in GaN QWR 4. K with L y =L z =40Å. The dahed and dah-dotted curve repreent repectively, the contribution from and phonon to the overall (curve ) mobility. Curve in Figure 4(b) refer to the elatic cattering. Elatic approimation reult in overetimated magnitude of mobility for higher temperature. To bring out the relative importance of and coupling, at very low temperature, we have performed calculation of mobility in GaN QWR at 4. K. Figure 4(a), depict the concentration dependence of mobility. The dahed and dah-dotted curve repreent, repectively, the contribution from and limited cattering to the overall (curve ) mobility. It i found that, with increaing concentration, the total mobility (curve ) follow a dependence due to phonon contribution, decreaing with increae in N D. However, for N D >80 7 m -. The contribution due to phonon dominate. In the cae of elatic approimation a depicted in figure 4(b), it i een that, the total mobility calculated uing Eq.(3) increae with increae in N D and phonon play a dominant role in limiting mobility over the whole range of concentration conidered. It may be noted that for GaA QWR Fihman et. al. find that mobility increae with increae in N D with major contribution from phonon and that for higher value of radiu of quantum wire cattering i ignificant. However room 409
Impact Factor.393, ISSN: 30-5083, Volume, Iue 6, July 04 temperature data [3] for 0 nm GaN QWR how a decreaing trend, mainly attributed to impurity and urface roughne cattering. IV. Concluion In concluion, we have eamined the role of the inelaticity of the electron-acoutic phonon interaction via both and coupling on low temperature mobility of a GaN quantum wire. We have given an epreion for -phonon limited cattering rate and tudied the variation of electron mobility a a function of temperature, electron concentration and thicne of the QWR. Our calculation how that, at low temperature (T < 4 K), acoutic phonon cattering via coupling i dominant in GaN QWR, and that the acouticphonon limited mobility i underetimated if the elatic cattering approimation i employed. The effect of creening of the electron-phonon interaction i, a our preliminary tudie how, to enhance the mobilitie by more than one order of magnitude at T=30 K Detailed invetigation of low-temperature mobility in ultrapure, thin ample are required to better undertand electron-phonon interaction in QWR. Acnowledgement Thi wor i upported by UGC, India. Reference:. C. M. Lieber, Z.L.Wang, MRS Bull. 3,(007) 99.. G.Fihman, Phy. Rev.B, 36, (987)7448. 3. V.K. Arora, Phy.Rev.B 3, (98) 56. 4. J.Lee, M.O.Vael, J.Phy. C, 7,(984) 55. 5. S. Kundu, C.K. Sarar and P.K.Bau, J.Appl. Phy. 68 (990) 070. 6. B.K. Ridley, Journal of Phyic C: Solid State Phyic 5 (98) 5899. 7. R.Miceviciu and V.Mitin, Phy.Rev. B 48, (993) 794. 8. V Karpu and D. Lehman, Semicond. Sci.Technol., (997) 78. 9. D.P.Bhattacharya, S.Midday, S.Nag, A.Biwa, Phyic E, 47 (03) 64. 0. P.J. Price, Ann. Phy. (N.Y) 33, (98) 7.. S.Shrehta, C.K.Sarar and A. Charaborty, J.Appl.Phy, 00, (006), 03705.. M.Catti, Y.Noel, R.Dovei, J.Phy. and Chem.of olid, 64, (003) 83. 3. Yu Huang, Xiangfeng Duan, Yi Cui, and Charle M. Lieber, Nano Lett., Vol., No., 00. 40