Linear magnetoresistance due to multipleelectron scattering by low-mobility islands in an inhomogeneous conductor

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1 Reeived 2 Jun 212 Aepted 31 Aug 212 Pulished 2 Ot 212 DOI: 1.138/nomms216 Liner mgnetoresistne due to multipleeletron sttering y low-moility islnds in n inhomogeneous ondutor N.V. Kolov 1,2, N. Mori 3, O. Mkrovsky 1, L. Eves 1, Q.D. Zhung 4, A. Krier 4 & A. Ptnè 1 Liner trnsverse mgnetoresistne is ommonly oserved in mny mteril systems inluding semimetls, nrrow nd-gp semiondutors, multi-lyer grphene nd topologil insultors. It n originte in n inhomogeneous ondutor from distortions in the urrent pths indued y mrosopi sptil flututions in the rrier moility nd it hs een explined using phenomenologil semilssil rndom resistor network model. However, the link etween the liner mgnetoresistne nd the mirosopi nture of the eletron dynmis remins unknown. Here we demonstrte how the liner mgnetoresistne rises from the stohsti ehviour of the eletroni yloidl trjetories round low-moility islnds in high-moility inhomogeneous ondutors nd tht this proess is only wekly ffeted y the pplied eletri field strength. Also, we estlish quntittive link etween the islnd morphology nd the strength of liner mgnetoresistne of relevne for future pplitions. 1 Shool of Physis nd Astronomy, The University of Nottinghm, Nottinghm NG7 2RD, UK. 2 IFW Dresden, Institute for Metlli Mterils, PO ox 27116, D-1171 Dresden, Germny. 3 Grdute Shool of Engineering, Osk University, 2-1 Ymd-Ok, Suit City, Osk , Jpn. 4 Deprtment of Physis, Lnster University, Lnster LA1 4Y, UK. Correspondene nd requests for mterils should e ddressed to A.P. (emil: mli.ptne@nottinghm..uk). nture ommunitions 3:197 DOI: 1.138/nomms Mmilln Pulishers Limited. All rights reserved.

2 nture ommunitions DOI: 1.138/nomms216 Liner trnsverse mgnetoresistne (MR) is phenomenon ommonly oserved in mny mteril systems, inluding semimetls 1, nrrow nd-gp semiondutors 2,3, multi-lyer grphene 4 nd topologil insultors 5. Liner MR n rise from either quntum 6 or lssil effets 7,8. In prtiulr, it n originte in n inhomogeneous ondutor from distortions in the urrent pths indued y mrosopi sptil flututions in the rrier moility; this type of non-sturting liner MR hs een oserved in severl mterils nd explined in terms of phenomenologil semilssil rndom resistor network model 7,8. However, the link etween the liner MR nd the mirosopi nture of the eletron dynmis remins still unexplored. Existing theories of lssil liner MR pply to systems with ohmi ondutivity, wheres the effets of hot eletron dynmis on the liner MR re lrgely unknown. On the other hnd, signifint enhnement of MR hs een oserved in the non-omhi spe-hrge limited trnsport regime 9 nd under vlnhe rekdown 1, thus rising prolems of fundmentl interest nd tehnologil relevne, tht is, wht is the speifi rrier dynmis giving rise to the liner MR, is it preserved in the presene of lrge pplied eletri field nd n it e ontrolled nd exploited in sensitive Hll sensors or mgnetoresistors 11? Here, we exmine the physil origin of the lssil liner MR in high-moility inhomogeneous ondutor. Our mesurements nd Monte Crlo simultions fous on the n-type ompound semiondutor InAs nd onsider n extended rnge of pplied eletri (E) nd mgneti () fields, nd different smple geometries. y modelling our dt with Monte Crlo simultions of the eletron dynmis, we demonstrte tht the liner MR rises from multiple sttering of the urrent-rrying eletrons y low-moility islnds within the onduting lyer. This proess is only wekly ffeted y the detils of the eletroni nd struture nd eletri field strength. Also, it persists t mgneti fields s high s ~5 T orresponding to mgneti length s smll s 4 nm, signifintly smller thn the eletron men free pth (~1 nm) nd islnd sie. The link etween the liner MR nd the stohsti ehviour of the eletroni yloidl trjetories round low-moility islnds is finding of generl interest s it is relevnt to ll high-moility inhomogeneous ondutors. The identifition of quntittive link etween the islnd morphology nd the strength of liner MR is lso potentilly useful for tiloring the liner MR in future pplitions. Results Mteril systems nd devies. Our InAs nd In(AsN) epilyers were grown y moleulr em epitxy (ME) on semi-insulting (1)- oriented GAs sustrte. Owing to the mismth etween the sustrte nd epilyer ltties, inhomogeneities n rise due to the presene of dislotions nd defets; lso, the N-toms t s sttering entres for the ondution eletrons 12,13. Thus with inresing N-ontent from to 1%, the eletron Hll moility dereses from µ H = 3.3 m 2 V 1 s 1 (3.1 m 2 V 1 s 1 ) to.6 m 2 V 1 s 1 (.5 m 2 V 1 s 1 ) t T = 3 K (4.2 K). Correspondingly, the eletron density inreses from n e = m 3 in InAs to n e = m 3 in In(AsN), with oth vlues eing very wekly ffeted y temperture. The presene of inhomogeneous disorder omined with the high eletron moility mkes these InAs nd In(AsN) lyers prtiulrly well suited to the study of lssil liner MR: even in the low-moility lyers the high vlues of µ 1, whih re neessry for oserving lrge MR, n e hieved with ville onventionl (superonduting) solenoids. To investigte the influene of smple geometry nd eletri field on the MR, two-terminl plnr devies with InAs hnnels of length L = 2, 5 nd 1 µm nd of width W = 1, 2 nd 5 µm were lso studied. High mgneti field studies. In our experiments, the mgneti field ws pplied prllel to the growth xis, tht is, = [,, ]. 1,2 6 I T =3 K T =1 K T = 4.2 K (T) 6 3 V xx L w (T) The trnsverse MR rtio defined s ρ xx /ρ xx = [ρ xx ( ) ρ xx ()]/ρ xx () ws studied with mgneti fields,, up to 14 T, generted y superonduting mgnet, or in pulsed solenoid t up to 47 T with totl pulse length of 1 ms. Figure 1 shows the -dependene of ρ xx /ρ xx for n Hll-r sed on n-type InAs. A positive liner MR is oserved t > 1 T nd is wekly ffeted y temperture in the rnge T = K. A wek osilltory omponent of ρ xx is oserved t low temperture (T < 2 K), whih we ttriute to the Shunikov de Hs effet in the two-dimensionl eletron gs tht umultes on the surfe of InAs due to the ending of the ondution nd y hrged ntive surfe defets 12. The resistne of the surfe two-dimensionl eletron gs t = T is t lest one order of mgnitude higher thn ρ xx nd, t ll vlues of nd T, the mplitude of the mgneto-osilltions is muh smller thn the mesured monotonilly inresing MR. Liner MR n our when the eletron distriution enters the so-lled quntum limit, in whih only the lowest Lndu level is oupied, tht is, when the ylotron energy, w, exeeds the Fermi energy, ε F, nd when w, ε F k T (ref. 6). As shown in Fig. 1, (T) t L = 5 µm W = 5 µm L =1 µm W = 2 µm Figure 1 Liner trnsverse MR in Hll rs nd short-hnnel devies. () Trnsverse MR, ρ xx /ρ xx, s funtion of mgneti field,, for n InAs Hll-r t T = 4.2, 1 nd 3 K. () ρ xx /ρ xx s funtion of for InAs (lk) nd In(AsN) (green) Hll-rs (T = 3 K). Dshed lines nd rrows illustrte the trnsition of the dependene of ρ xx /ρ xx on from qudrti to liner t the hrteristi mgneti field t. A liner fit to the MR dt y the reltion ρ xx /ρ xx = µ (s/2 l) gives n verge islnd overge ftor f~.17 nd.32 in InAs nd In(AsN), respetively. Here, s is the verge sie of the low-moility regions, l is the inter-islnd seprtion nd µ is the ero field moility. () ρ xx /ρ xx versus for short InAs hnnels of length, L, nd width, W, t T = 1 K (line) nd 3 K (symols). The insets in pnels () nd () sketh Hll-r nd twoterminl plnr devie, respetively. nture ommunitions 3:197 DOI: 1.138/nomms Mmilln Pulishers Limited. All rights reserved.

3 nture ommunitions DOI: 1.138/nomms216 ARTICLE d Figure 2 Monte Crlo simultions of eletron trjetories in mgneti field. Clulted eletron trjetories (red lines) in InAs for n pplied eletri field E x =.5 kv m 1 nd under vrious pplied mgneti fields : () =, () 1, () 1 nd (d) 2 T. The lk res orrespond to regions of low moility. As shown in the mgnified trjetories in the insets of pnels () nd (d), in the viinity of low-moility islnd, the eletron undergoes multiple sttering events, whih t to deflet the yloidl motion. The r in pnel () orresponds to 1 µm. the trnsition of the MR to liner -dependene ours t ~1 T elow the vlue stisfying the quntum limit ondition w = ef, 2 2/ 3 tht is, = ( 3p n e ) / 2e. This limit orresponds to fields >3 T for n eletron density n e = m 3, whih we derive from the grdient of the Hll resistivity, ρ xy, in the limit. Also, we note tht the ondition for single oupny of the lowest Lndu level requires tht k T w. This ondition is stisfied for tempertures < 5 K t = 1 T nd for the eletron ylotron mss of InAs (m e = (.25±.1) m, where m is the eletron mss in vuum) 14. However, the liner MR is oserved well ove T = 5 K (see Fig. 1), inditing tht it does not hve quntum origin, ut is lssil effet. Clssil liner MR hs een explined using phenomenologil model in whih the distorted urrent pths rising from disorder-indued ihnomogeneities nd mrosopi vritions, µ, in the eletron moility re simulted y network of rndom resistors. Eh resistor element hs four terminls to tke into ount not only the lol resistive voltge drop ut lso the Hll voltge ssoited with the lol moility, µ, of prtiulr region of the smple 7,8. The rossover of the MR from qudrti to liner -dependene is predited to our t hrteristi mgneti t field = m 1 for m / m < 1, or t t = µ 1 for m / m > 1, where m orresponds to n verge of the sptilly vrying moility. A omprison of the -dependene of ρ xx /ρ xx in our InAs nd In(AsN) Hll-rs t T = 3 K (see Fig. 1) shows tht oth the vlue of 1/ t nd the slope of the MR follow the systemti derese of µ from 3.3 m 2 V 1 s 1 in InAs to.6 m 2 V 1 s 1 in In(AsN). The hrteristi mgneti field t provides n estimte of the verge moilities m = 1/ 1 m 2 V 1 s 1 nd.5 m 2 V 1 s 1 in the t InAs nd In(AsN) smples, respetively, whih re omprle with those mesured y the Hll effet. Also, the MR tends to follow the temperture dependene of µ: in the InAs smple, n inrese of T > 4.2 K nd up to T ~1 K leds to n inrese of µ H from 3.1 to 3.8 m 2 V 1 s 1 followed y derese t T > 15 K. This vrition is used y the interply etween impurity nd phonon sttering nd is ompnied y similr T-dependene of the MR (see Fig. 1). Liner MR is lso oserved in the two-terminl smll hnnel devies, see Fig. 1. Monte Crlo simultions. As our InAs nd In(AsN) epilyers re grown on highly lttie-mismthed GAs, threding dislotions tend to form t the epilyer/sustrte interfe. The mrosopi ( >.1 µm) inhomogeneities oserved y trnsmission eletron mirosopy of our smples 15 re likely to indue vritions in the eletroni properties. We model the effet of suh inhomogenities on the eletron dynmis y Monte Crlo simultions 16. As the mgneti field is pplied long the -diretion, the mgneti field ffets the omponents of the eletron motion in the x y plne. In our simultion of the two-dimensionl eletron trjetories, inhomogeneity is introdued y pling low-moility islnds rndomly within the x y plne (Fig. 2). The rndom sptil profile of the islnds, (x,y), is generted from the power spetrum of the utoorreltion funtion (r) (r ) = 2 exp[ r r 2 / 2 ], where is the orreltion length, r = (x, y) nd r = (x, y ). The low-moility regions R low { } re then defined ording to the reltion R low = r ( r) < th. This ondition determines the frtionl re, f, of the smple with low moility in whih eletrons undergo sttering t rte higher thn in the remining high-moility prt of the smple. In threedimensionl model, the vlue of f would result from n verge of the f-vlues from vrious x y plnes of the three-dimensionl onduting hnnel. In the low-moility regions, we use sttering rte r islnd = r i + r e, where r i = s 1 nd r e = s 1 re the elsti nd inelsti phonon sttering rtes, respetively; elsewhere, the eletron drift is minly limited y inelsti sttering with r~r i. The non-proliity of the eletron energy dispersion of InAs [ ] = k m e, where m e =.26m nd ( ) = is modelled s e( k) 1+ e( k) α = 2.2 ev 1. This form of the energy dispersion ours in numerous semiondutors, inluding the nrrow gp InS for whih oservtions of liner MR hve een lso reported 2,3. The eletron velo- ity, given y v k 1 ke, nd the wve vetor, k, hnge with time under the tion of the Lorent fore, F = e[ E + v ], where E = [E x, E y, ] nd = [,, ] 17. The Hll field omponent, E y, is set so tht the y-omponent of the verge eletron veloity is ero, tht is, v = [ v x,, ], onsistent with the urrent flow for our Hll rs. The min findings of our simultions re not hnged if we onsider different geometry in whih the Hll field is effetively shorted out, tht is, E y =. Thus, our nlysis pplies not only to Hll-rs ut lso to short hnnels nd Corino geometries, in whih the Hll eletri field is fully short-iruited y the ontt eletrodes. Figure 2 d illustrtes the lulted eletron trjetories t E x =.5 kv m -1 nd =, 1, 1 nd 2 T. Here, we hve ssumed orreltion length =.5 µm, f = 5% nd low-moility regions of verge sie s =.4 µm nd inter-islnd seprtion l = 8 µm. It n e seen tht in the viinity of the low-moility islnds, the eletron undergoes multiple sttering events, whih t to deflet the yloidl motion (see inset of Fig. 2). This islnd-indued sttering is oserved t low (Fig. 2) nd high (Fig. 2d) pplied mgneti fields. It is sent in homogenous system nd is responsile for the liner mgneti field dependene of the inverse moility omponent, µ 1 xx, whose strength inreses with f, s shown in Fig. 3. nture ommunitions 3:197 DOI: 1.138/nomms Mmilln Pulishers Limited. All rights reserved.

4 nture ommunitions DOI: 1.138/nomms216 xx 1 (Vs m 2 ) y x (T) A Disussion We interpret the results of the Monte Crlo simultions y onsidering simplified Drude model of the verge eletron veloity v x nd verge hopping distne trvelled y the ondution eletrons long the x-diretion. In uniform system, tht is, with no islnds, nd for E y = nd v y, the verge veloity long x is v x = X/τ, where X m e E x /(e 2 ) is the typil hopping distne nd τ is the sttering time. In the presene of inhomogeneity, n eletron trvels n dditionl distne X ~s/2 round low-moility islnd. As the eletron hits n islnd every time intervl τ islnd ~l/v H, where v H = E x / is the eletron Hll veloity long y nd l is the distne etween the islnds, the verge eletron veloity v x nd moility omponent µ xx inrese s v x = X/τ + X /τ islnd nd µ xx = m e /(τe 2 2 ) + s/(2l ), respetively. Thus, if we ompre this result with the se of n homogenous system for whih µ xx = m e /(τe 2 2 ) nd v x = m e E x /(τe 2 ), we onlude tht t high islnd sttering tends to filitte eletron drift long the x-diretion, perpendiulr to the yloidl motion long y (see eletron sttering round islnd A in Fig. 3). For the geometry in whih E y = nd v y, the resistivity omponent ρ xx is given y rxx = Ex Jx = 1/ nemxx. Using the dependene of µ xx on t lrge fields, tht is, µ xx ~ X /τ islnd = s/(2l ), we derive liner MR, tht is, ρ xx /ρ xx ~ µ*, where µ* = µ (s/2 l) is nd effetive moility nd µ is the ero field moility. A similr onlusion n e drwn for the Hll geometry, tht is, for E y nd v y =. In this se the resistivity omponent ρ xx is given y 2 2 rxx = mxx /[ ne( mxx + mxy)], where µ xy = 1. Using this expression in the limits of high, we otin lso in this se tht ρ xx /ρ xx ~ µ*. Note, however, tht in this geometry, the islnd sttering ts to derese the verge eletron veloity, v x, y defleting the y 1 x E x = 3. kv m 1 E x = 2. kv m 1 E x = 1. kv m 1 E x =.5 kv m 1 (T) Figure 3 Monte Crlo simultions of the eletron moility nd its mgneti field dependene. () Dependene of the inverse of the moility, µ xx 1 on mgneti field,, in InAs t vrious vlues of the islnd overge ftor f =.1 (lk), f =.5 (red) nd f =.2 (lue). () Clulted eletron trjetories (red lines) in InAs t = 1 T, E x =.5 kv m 1 nd different eletri field omponents E y (left: E y = nd v y = m s 1 ; right: E y = 1.5 kv m 1 nd v y = ). The lk res show regions of low moility. A nd indite two typil islnds. () Dependene of µ xx 1 on E x for f =.5. eletron motion long y (see eletron sttering round islnd- in Fig. 3). In summry, the mgneti field dependene of the moility omponent µ xx due to islnd sttering genertes hrteristi liner dependene of the MR on mgneti field, whih is the sme in oth geometries, tht is, E y nd E y =. This nlysis identifies the physil origin of the rndom vrition of the resistive nd Hll effet omponents of eh four-terminl resistive element of the network model 7,8 : the rndomness rises from the stohsti ehviour of the eletroni yloidl trjetories round the low-moility islnds. From liner fit to the mesured MR dt for the lrge InAs Hll-r of Fig. 1 y the reltion ρ xx /ρ xx = µ*, we otin µ* =.33 m 2 V 1 s 1. For µ = µ H = 3.3 m 2 V 1 s 1, this orresponds to s/(2 l) = µ*/µ =.1 nd n verge islnd overge ftor f = s/(s + l)~.17. Similrly, for the In(AsN) Hll-r of Fig. 1 we find tht s/(2 l) = µ*/µ =.23 nd f = s/(s + l)~.32. Thus, in oth ses the verge islnd seprtion, l, is signifintly lrger thn the verge islnd sie, s. Although we nnot dedue the ext vlue of l or s, we n infer resonle estimte from our MR experiments in smll hnnel devies, see Fig. 1. Here, it n e seen tht devies with smll vlues of the hnnel length L (2 1 µm) nd/or width W (1 5 µm) exhiit muh smller vlues of ρ xx /ρ xx nd in the smllest devies the MR tends to sturte t high. The liner MR dereses systemtilly s the hnnel width W nd length L derese to vlues of the order of 1 µm. Thus, we infer tht the dimensions of these smll devies re omprle to the verge islnd seprtion. The Monte Crlo simultions indite tht different vlues of the nd non-proliity prmeter α produe similr liner MR ehviour. Thus, the speifi detils of the eletroni nd struture do not hve ritil role in the oservtion of liner MR. Perhps surprisingly, lthough µ xx 1 tends to inrese with inresing E x, its liner dependene on is found to e wekly ffeted y eletri field (Fig. 3) nd only t low-mgneti fields ( < 1 T) nd high E x does the vlue of µ xx 1 tend to derese with deresing, thus leding to wek negtive MR. This is used y the interply etween eletron ylotron motion nd islnd-indued sttering, whih tends to e inresed y E x, tht is, eletrons hit low-moility islnds more frequently t higher speeds. However, overll this effet tends to e wek nd, t lrge, the defletion of the yloidl motion y the low-moility islnds nd orresponding liner MR re preserved. The wek dependene of the liner MR on pplied eletri field is onfirmed y our MR experiments in the short hnnel InAs smples. Figure 4 shows the dependene of the urrent density J on E x t low temperture (T = 4.2 K) nd t vrious (up to 14 T) for n InAs hnnel of length L = 1 µm nd width W = 2 µm. Similr dependenes of J on E x nd were found in devies with different vlues of L nd W. At = T, the J(E x ) urve is pproximtely liner t low is. The ohmi ehviour is then followed y slight suliner is dependene nd, t lrger eletri fields, y shrp rise of the urrent tht is hrteristi of systems swithing from stte of low to high ondutivity due to impt ionition nd vlnhe rekdown 15,18. At ll vlues of E x, the mgneti field ts to suppress the urrent nd to shift the threshold eletri field for impt ionition to signifintly higher vlues. From Fig. 4, it n e seen tht the MR remins liner over n extended rnge of E x up to ~3 kv m 1, just elow the threshold eletri field for impt ionition. In ontrst, in the regime of vlnhe rekdown the MR is strongly enhned nd tends to deprt from simple liner dependene due to the effet of the mgneti field in suppressing the vlnhe multiplition of rriers 1. To onlude, our dt nd nlysis indite tht the presene of low-moility islnds in high-moility ondutor nd their effet on the eletron trjetories underpin the oservtion of lssil nture ommunitions 3:197 DOI: 1.138/nomms Mmilln Pulishers Limited. All rights reserved.

5 nture ommunitions DOI: 1.138/nomms216 ARTICLE J (A m 2 ) T = 4.2 K E x (kv m 1 ) 3 L w = T E x = 4. kv m 1 E x = 3. kv m 1 E x = 2. kv m 1 E x =.5 kv m 1 liner MR. We hve estlished ler link etween the stohsti dynmis of the eletron trjetory round low-moility islnds used y multiple sttering t the islnds nd the lssil liner MR in devies with different geometry nd sie. Although our oservtions hve foused on the high-moility inhomogeneous n-type InAs nd In(AsN) epilyers grown on GAs, our min findings nd onlusions re relevnt to wide vriety of strutures, devie geometries nd mteril systems. Our results re therefore of generl interest nd will help guiding future reserh into the mnipultion of the liner MR y tuning of the eletron trjetories in tilor-mde inhomogeneous mterils nd devie strutures. Methods Growth nd frition. Our InAs epilyer (thikness t = 1. 5 µm) ws grown y ME on semi-insulting (1)-oriented GAs sustrte. An In(AsN) epilyer (t = 1.3 µm) ontining smll onentrtion (N = 1%) of sustitutionl N-toms ws lso grown y ME using rdio frequeny (RF) plsm N-soure. The surfe reonstrution ws monitored y in situ refletion high-energy eletron diffrtion, while the sustrte temperture ws mesured using n infrred pyrometer lirted with the surfe reonstrution trnsitions t fixed As-flux. A Veeo UNI-ul RF plsm soure ws used s nitrogen soure. It ws set t RF power of 16 W with nitrogen flux of mr for ll epilyers. The N-ontent in the In(AsN) epilyers nd the rystl qulity of the lyers were optimied y setting the growth temperture, T G, nd the growth rte, r G, to given vlues in the rnges T G = 4 46 C nd r G =.3 1 µm h 1, nd y keeping minimum As-flux. The InAs epilyer ws grown t T G = 46 C. High-resolution X-ry diffrtion mesurements using ede QC2 diffrtometer onfirmed relxtion of the rystl lttie y 98% in the InAs nd In(AsN) epilyers. The wfers were spun with photoresist nd ptterned y photolithogrphy into Hll-rs of length L = 1,25 µm nd width W = 25 µm. Two-terminl plnr devies with InAs hnnels of length L = 2, 5 nd 1 µm nd of width W = 1, 2 nd 5 µm were lso frited y eletron em lithogrphy. Eh hnnel ws formed y dry-ething the epilyer down to the GAs sustrte. Metl ontts onsisting of 1 nm of Ti followed y 2 nm of Au were deposited onto the smples to provide Ti Au ohmi ontts. 1 T 2 T 3 T 5 T 1 T 14 T (T) Figure 4 Liner trnsverse MR t high eletri fields. () Dependene of the urrent density J on eletri field E x t vrious mgneti fields for n InAs hnnel of length L = 1 µm nd width W = 2 µm (T = 4.2 K). Mesurements of the J(E x ) hrteristis were performed in oth the diret urrent (lines) nd pulsed mode (symols). The inset skethes two-terminl plnr devie. () Trnsverse MR, ρ xx /ρ xx, s funtion of t vrious vlues of E x. The resistivity ρ xx is lulted s ρ xx = E x /J. Trnsport studies. Mesurements of the urrent voltge hrteristis were performed in oth diret urrent nd pulsed mode. For the pulsed mode of quisition of the urrent, voltge pulse genertor ws used to generte swtooth single voltge pulses of durtion of 1 µs, nd digitl osillosope ws used for the synhronied nd fst temporl reording of the pplied is nd urrent, thus reduing exessive smple heting t high-operting urrents. The urrent voltge urves in pulsed mgneti field were tken within 1 ms t the top of the mgneti field pulse, during whih the mgneti field remins onstnt to within ±.5%. Referenes 1. Xu, R. et l. Lrge mgnetoresistne in non-mgneti silver hlogenides. Nture 39, 57 6 (1997). 2. Hu, J. & Rosenum, T. F. Clssil nd quntum routes to liner mgnetoresistne. Nt. Mter. 7, (28). 3. rnford, W. R. et l. Geometri mnipultion of the high-field liner mgnetoresistne in InS epilyers on GAs (1). Appl. Phys. Lett. 86, (25). 4. Friedmn, A. L. et l. Quntum liner mgnetoresistne in multilyer epitxil grphene. Nno Lett. 1, (21). 5. Zhng, W. et l. Topologil spet nd quntum mgnetoresistne of β-ag 2 Te. Phys. Rev. Lett. 16, (211). 6. Arikosov, A. A. Quntum liner mgnetoresistne; solution of n old mystery. J. Phys. A Mth. Gen. 36, (23). 7. Prish, M. M. & Littlewood, P.. Non-sturting mgnetoresistne in hevily disordered semiondutors. Nture 426, (23). 8. Prish, M. M. & Littlewood, P.. Clssil mgnetotrnsport of inhomogeneous ondutors. Phys. Rev. 72, (25). 9. Demo, M. P., Ymmoto, S., Ksi, S., Ono, T. & Koyshi, K. Lrge positive mgnetoresistive effet in silion indued y the spe-hrge effet. Nture 457, (29). 1. Sun, Z. G., Miuguhi, M., Mngo, T. & Aking, H. Mgneti-fieldontrollle vlnhe rekdown nd gint mgnetoresistive effets in gold/ semi-insulting-gas Shottky diode. Appl. Phys. Lett. 85, (24). 11. Heremns, J. Solid stte mgneti field sensors nd pplitions. J. Appl. Phys. 26, (1993). 12. Ptnè, A. et l. Effet of low nitrogen onentrtions on the eletroni properties of InAs 1 x N x. Phys. Rev. 8, (29). 13. Fhy, S., Lindsy, A., Ouerdne, H. & O Reilly, E. P. Alloy sttering of n-type rriers in GN x As 1 x. Phys. Rev. 74, 3523 (26). 14. Drhenko, O. et l. Cylotron resonne mss nd Fermi energy pinning in the In(AsN) lloy. Appl. Phys. Lett. 98, (211). 15. Mkrovsky, O. et l. Hot eletron trnsport nd impt ionition in the nrrow energy gp InAs 1 x N x lloy. Appl. Phys. Lett. 96, (21). 16. Jooni, C. & Reggini, L. The Monte Crlo method for the solution of hrge trnsport in semiondutors with pplitions to ovlent mterils. Rev. Mod. Phys. 55, (1983). 17. Hmguhi, C. si Semiondutor Physis 46 (Springer-Verlg, London, 26). 18. Fishetti, M. V. Monte Crlo simultion of trnsport in tehnologilly signifint semiondutors of the dimond nd in-lende strutures. I. Homogeneous trnsport. IEEE Trns. Eletron Dev. 38, (1991). Aknowledgements This work ws supported y the Royl Soiety (UK), DFG KO 3743/1-1, AOJ: We knowledge Dr R. Airey (EPSRC, III V Ntionl Fility, Sheffield) for proessing our smples. Author ontriutions A.P., N.V.K. nd O.M. designed nd performed the experiments; N.M. performed the Monte Crlo simultions; L.E. took prt in the disussions of the Monte Crlo studies; A.K. nd Q.D.Z. grew the smples; A.P. wrote the pper; nd ll uthors took prt in the disussion/nlysis of the dt. Additionl informtion Competing finnil interests: The uthors delre no ompeting finnil interests. Reprints nd permission informtion is ville online t reprintsndpermissions/ How to ite this rtile: Kolov, N.V. et l. Liner mgnetoresistne due to multipleeletron sttering y low-moility islnds in n inhomogeneous ondutor. Nt. Commun. 3:197 doi: 1.138/nomms216 (212). Liense: This work is liensed under Cretive Commons Attriution-NonCommeril- Shre Alike 3. Unported Liense. To view opy of this liense, visit nture ommunitions 3:197 DOI: 1.138/nomms Mmilln Pulishers Limited. All rights reserved.