Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Effect of disorder on the transverse magnetoresistance of Weyl semimetals

Effect of disorder on the transverse magnetoresistance of Weyl semimetals E ect of disorder on the transverse magnetoresistance of Weyl semimetals 1, 2, 3 1, 3 4 5, 6 Ya. I. Rodionov, K. I. Kugel, B. A. Aronzon, and Franco Nori Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Izhorskaya str. 13, Moscow, 125412 Russia National University of Science and Technology MISIS, Moscow, 119049 Russia National Research University Higher School of Economics, Moscow, 101000 Russia P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Physics Department, The University of Michigan, Ann Arbor, MI 48109-1040, USA (Dated: November 18, 2020) We study the e ect of random potentials created by di erent types of impurities on the trans- verse magnetoresistance of Weyl semimetals. We show that the magnetic eld and temperature dependence of the magnetoresistance is strongly a ected by the type of impurity potential. We analyze in detail two limiting cases: (i) the ultra-quantum limit, when the applied magnetic eld is so high that only the zeroth and rst Landau levels contribute to the magnetotransport, and (ii) the semiclassical situation, for which a large number of Landau levels comes into play. A formal diagrammatic approach allowed us to obtain expressions for the components of the electrical con- ductivity tensor in both limits. In contrast to the oversimpli ed case of the -correlated disorder, the long-range impurity potential (including that of Coulomb impurities) introduces an additional length scale, which changes the geometry and physics of the problem. We show that the magne- toresistance can deviate from the linear behavior as a function of magnetic eld for a certain class of impurity potentials. PACS numbers: 72.10.-d 72.80.-r 75.47.-m 5.60.Gg I. INTRODUCTION The results for an arbitrary position of the chemical po- tential were obtained in Ref. 19 within the framework of a similar approximation; and also in Ref. 20, where the In recent years, problems related to quantum transport approach based on the classical motion of the so-called in materials with the Dirac spectrum of charge carries, in guiding center was used. For the screened Coulomb po- particular in Weyl semimetals, have attracted consider- tential of impurities, the electron transport was also an- 1,2 able interest . Much e ort was focused on the longitu- 21,22 alyzed in the case of a gapped Dirac spectrum . dinal magnetoresistance, where the negative contribution Thus, we see that in spite of these serious e orts, some associated with the so-called chiral anomaly arising due important aspects of the analysis of the magnetoresis- the transfer of charge carries between Weyl points with tance of Weyl semimetals remain untouched. First of all, 3{8 opposite chiralities plays a dominant role . its behavior at di erent characteristic ranges of the im- At low magnetic elds, rather nontrivial manifesta- purity potential, and accurate calculations in the limit of tions of the weak localization and antilocalization e ects long-range potentials. These issues still remain unsolved 9,10 have also been addressed . No less interesting is the 15{17 even in the most well-studied ultra-quantum limit, behavior of the transverse magnetoresistance, where a when only one Landau level contributes to the magne- nonsaturating linear magnetic eld dependence is ob- toresistance. 11{14 served at high elds . The nature of such unusual In this paper, we rst focus on the calculations in behavior has been widely discussed. the ultra-quantum limit and calculate components of the The main physical mechanisms in the ultra-quantum magnetoconductivity tensor based on the accurate dia- regime were revealed in the seminal work of Abrikosov . 23 grammatic approach which we have formulated earlier . He considered a gapless semiconductor with a linear dis- Then, we present a detailed analysis of the semiclassical persion law near the chemical potential. The chemical limit, when a large number of Landau levels contributes potential itself coincides with the zeroth Landau level to the transport characteristics. and the charge carriers are scattered by impurities char- In Section II, we formulate the model and introduce all acterized by a screened Coulomb potential. the necessary parameters. In Section III, we analyze the This problem was generalized in the detailed studies components of the electrical conductivity tensor and their presented in Refs. 16{18, which were stimulated by nu- magnetic eld dependence in the ultra-quantum limit, for merous experimental observations of the linear magne- which the dominant contribution comes from the zeroth toresistance. In these papers, the main emphasis was on Landau level. In Section IV, we consider the magneto- the case of point-like impurities. transport at the semiclassical limit, for which the tem- In the opposite limit of long-range impurity potentials, perature is high enough and a large number of Landau treated by the Born and self-consistent Born approxima- levels comes into play. Both in Sections III and IV, we tions, it was possible to obtain only qualitative results. put the main emphasis on the magnetotrasport for rather arXiv:2005.08893v2 [cond-mat.mes-hall] 17 Nov 2020 2 long-range impurity potentials. In Section V, we discuss the dimensionless function g, which is introduced in mo- the obtained results. The details of our calculations are mentum space from the very beginning. Also, u is the presented in the Appendix. Fourier transform of the impurity potential, while the 1 6 disorder correlation length is r = p . The p factor is 0 0 introduced from dimensional considerations. II. MODEL AND CHARACTERISTIC PARAMETERS Our study is aimed at the analysis of the transport characteristics of the Weyl semimetal (WSM) with im- purities under the e ect of an applied transverse mag- netic eld (i.e., the magnetic eld direction is perpendic- FIG. 1: Disorder vertex for the perturbation theory. ular to that of the electric current). We start from the Here, the dashed line represents the disorder correlation low-energy Hamiltonian for the WSM in its conventional function, while solid tails are fermion lines. form H = H + H ; In this work, we are focusing on the long-range correla- 0 imp tion between disordered impurities (long-range disorder). H = v (r) p A (r)dr; 0 In the ultra-quantum case (1) maxfT; g  v=l ; (4) H = (r)u(r) (r)dr; imp where where H is the Hamiltonian of non-interacting Weyl 0 l = c=(eH ) (5) fermions and H describes the interactions with the imp is the magnetic length and  is the doping level of the impurity potential;  = ( ;  ;  ) are the Pauli ma- x y z WSM sample, and this limit corresponds to the condi- trices acting in the pseudospin space of Weyl fermions, tion: p = ir is the momentum operator, v is the Fermi velocity, and u(r) is the impurity potential. l  r  ; (6) H 0 The impurity potential is understood to be of a general where form. It is of electrostatic origin, but its speci c form as well as the form of its correlation function can be arbi- = v=fmax T; g (7) trary. Of particular importance to the experiment is the is the characteristic particle wavelength. Let us also in- screened Coulomb impurity potential. As was argued in troduce the energy scale associated with the magnetic Ref. 24, there exists a regime, in which the Coulomb im- eld strength, purity scattering dominates over the electron{electron in- teraction (see the corresponding discussion in Section V). = v 2eH=c; (8) Throughout the paper, we set ~ = k = 1. We also ne- glect the in uence of di erent Weyl cones on each other, characterizing the distance between the zeroth and rst concentrating on the low-energy physics. The vector po- Landau levels (LLs). In the opposite semiclassical limit tential of the magnetic eld H is chosen in the asymmet- T , the corresponding condition for the disorder cor- ric gauge relation length reads: r  l : (9) 0 H A = (0; Hx; 0): (2) Limit (6) intuitively appeals to the physical picture where In this paper, we will use the Kubo-type diagrammatic the center of the magnetic orbit moves along the impurity approach. The impurity potential thus enters the for- potential line, while limit (9) corresponds to the proper malism in terms of its correlation function. The relevant particle motion along the impurity potential line. diagram is shown in Fig. 1. We write the corresponding correlation function in the following form: III. MAGNETOTRANSPORT AT fT; g 2 2 n u p imp ipr 0 (ULTRA-QUANTUM LIMIT) dre hu(r)u(0)i  g(p) = g ; 6 2 p p 0 0 (3) 2 6 A. Computation of p p xx 2 0 g = ju j : 2 2 p u 0 0 As was mentioned above, the ultra-quantum limit cor- Here, n is the concentration of impurities and u is imp 0 responds to the condition the characteristic amplitude of the impurity potential. maxfT; g: (10) The disorder correlation function is written in terms of 3 This means that the rst Landau level is high enough and only the ground state (and the rst excited state) contributes to the magnetotransport. The xx component of the conductivity tensor is determined by the following analytical expression: d" dp dx df (") 2 2 = e v xx (2) d" (11) R 0 R 0 TrhImG (x; x ; "; p)ImG (x ; x; "; p)i; 11 22 where angular brackets denote the averaging over disor- der, f is the Fermi distribution function, and the retarded Green's functions are de ned as follows: FIG. 2: Three contributions to the conductivity in rst-order perturbation theory. The disorder vertices R 0 y 0 are represented by dashed lines de ned in Fig. 1. The G (x; x ; "; p) = S (x )G("; p)S (x ) n p y n p n=0 indices i; j become x or y depending on the type of conductivity which is computed. s 0 S (s) = ; (12) 0  s n1 " + v  p G("; p) = ; Ref. 15. The result is given by the following integral: 2 2 (" + i0) " 2 2 e v dq dq x = x p l : x y p y 2 2 y H = (q + q )g ; (16) xx 2;q x y xy 2 2 (2) Here,  s is the oscillator normalized wave function of where g is the e ective two-dimensional disorder cor- the nth state, 2;q xy relation function de ned as p = (0; 2n=l ; p ) (13) n H z dp g = g(p)  g(p ) : (17) 2;p xy xy 2 z=0 is the e ective momentum, p is the two-dimensional yz momentum (p ; p ). We are using perturbation theory, y z In Ref. 15, Eq. (16) was analyzed only in the case of the therefore assuming that the concentration of impurities Coulomb impurity potential. We, however, come to the is not too high. The dimensionless expansion parame- conclusion that for di erent types of disorder, the formula ter characterising the disorder strength is assumed to be abovegives a qualitatively di erent H dependence. small: Before we proceed, let us make the following observa- 1 n u tion. The integral in Eq. (16) of the 2D disorder corre- imp 1; (14) lation function determining the conductivity can become " v p divergent at high momenta (short-range case). However, where " is the characteristic energy scale for charge car- in our calculations, we used the long-range disorder ap- riers and  is its impurity scattering time (see below proximation, implying that ql  1, where q is the char- Eq. (38)). acteristic disorder momentum. Therefore, q  l is the As the analysis shows, in the ultra-quantum limit, it is natural short-range cuto scale. As we will see below, enough to keep the rst order of perturbation in the dis- the system exhibits di erent types of magnetotransport order strength. Therefore, only three possible diagrams depending on the short-range behavior of the disorder, contribute to the conductivity (see Fig. 2). Even less p  q  l . trivial is the fact that the vertex correction (diagram III in Fig. 2) is exponentially suppressed. Therefore, only the rst-order disorder corrections to the Green's func- B.  for di erent short-range behavior of the xx impurity potential tion are needed to be taken into account. This was rst stated explicitly for the case of short-range disorder in . In the ultra-quantum limit and in the long-range dis- Let us now assume that the impurity potential has the order case p l  1 (Eq. (6)), only the zeroth and rst 0 H short-range asymptotics LLs in Fig. 2 contribute to the conductivity. For the R u Green's function G in (11), it is enough to take only 11 u(r) = ; l  r  p ; 1 < < 1: (18) 1+ (p r) the contribution of the zeroth LL, 0 Here, the natural constraint < 1 means that we are not Im G ("; p) = (" p v): (15) 11 z considering pathological cases of potentials leading to the The expression for the conductivity  then goes along \falling to the center" phenomenon. On the other hand, xx the same lines for the general type of disorder as that in the < 1 constraint should exclude the unphysical case 4 of decaying at r = 0. Then, the disorder correlation We see that the H dependence of the conductivity is af- function in momentum space reads: fected by the nature of the disorder. In particular, if the correlation function has stronger than Coulomb power- 4 2 >l v ; p  p ; law growth at short distances, the corresponding expo- < 0 nent enters the conductivity. g(p) = n imp : 42 2 6 1 The parameter most relevant to many experiments is u =p (p =p) ; p  p  l : 0 0 0 0 the magnetoresistance. To calculate it, we need to know (19) the Hall conductivity  . xy The question we now address is: what is the behavior of the conductivity as a function of H for di erent values C. Hall conductivity xy of ? To answer this question, we analyze expression (16) for various cases discussed below. The Hall conductivity is given by the sum of two terms: (a) 1 < < 0: We call this the \regular disorder" case. The integral in (16) is convergent and the conver- I II =  +  : (26) xy xy xy gence region is p  p . In this case, we have The rst term in (26),  , is the so-called normal con- ec xy = n u g ; < 0; (20) xx imp 1 25 tribution, which is given by the following relation : 16Hp R 2 2 2 3 e d" df (") where g = g(x )x dx, with x = p=p , is a numerical xy 4 2 d" constant, which depends on the details of the shape of R R R R R A R A the disorder distribution function. G Im G G Im G Im G G + Im G G : 22 11 11 22 22 11 11 22 As we are going to see below, the behavior correspond- ing to Eq. (20) is identical to that characteristic of the (27) Coulomb disorder, = 0. As is seen from Eq. (27), it comes from the vicinity of (b) = 0. For the Coulomb disorder, the integral the Fermi surface, as it is proportional to df=d". In the determining the conductivity in (16) is log-divergent. In absence of disorder, it is easily veri ed that  = 0. the Coulomb case, the inverse Debye radius reads: xy Therefore, it is perturbative in the disorder strength. The p = l (21) second term in Eq. (26) is the so-called anomalous contri- bution. It is proportional to the derivative of the charge for the case fT; g , where carrier density with respect to the applied magnetic eld H and, as such, comes from the entire volume inside the = e =(~v) (22) Fermi surface. As is understood from the de nition of the anomalous part is the WSM ne structure constant and  is the permit- tivity. One recovers the result : dN (H; ; T ) II = ec ; xy dH e c 1 (28) = n ln ; = 0: (23) xx imp H N (H; ) = (")f d"; (c) 0 < < 1. We call this the \singular disorder" it is nonzero even in the absence of disorder. Here, the casedue to its short-range behavior. The integral in (16) density of states reads: is then divergent at high momenta q and an appropriate cut-o p  l should be introduced. In this case, we dp yz (") = Tr Im G("; p ; p ; x; x) have a nontrivial result for  : y z xx (2) (29) 1 dp ec eH z = ImG ("; p ): = n u ; 0 < < 1: (24) n z xx imp 0 2 2 2 2l 2 16Hp cp 0 0 The above results can be summarized by the following Thus, from perturbative arguments, we understand that II formula: =  , i.e. it is determined by the anomalous part. xy xy 8 In our case (long-range disorder), it is even possible 2 2 4 1 e v p g (c=eH )  H ; 1 < < 0; II > to compute  at all orders of perturbation theory in xy the strength of the disorder, in the limit p ! 1. The 2 2 2 1 e v ln(1= )(c=eH )  H ; = 0; result is independent of the disorder strength and is given xx by the disorder-free expression: : 2 2 2 eH 1 e v n u (c=eH )  H ; 0 < < 1: imp 2 0 e cp II 2 2 =  = = e n l ; (30) xy 0 xy H (25) 4 v 5 where n is the charge-carrier density. In experiments, n In the Appendix, we argue that in the T limit, 0 0 is usually a xed parameter stemming from the charge- the problem of disorder averaging becomes essentially a neutrality condition due to the imbalance of donor and two-dimensional one. Unlike the ultra-quantum case, the acceptor impurities in WSMs. Therefore, to compare corresponding 2D plane is y = 0 (due to asymmetry of with experiments, one needs to express the chemical po- our vector potential gauge (2)) with the e ective corre- tential in terms of n . lation potential: The formula for the resistivity is as follows: dp g = g(p)  g(p ) : (36) xx 2;p xz xz = : (31) 2p y=0 xx 0 2 2 xx xy The most crucial observation is that all the integrals Taking into account theexpressions for  (25) and xx entering the Green's functions and Dyson equations are (30), we obtain the following results for the eld xy essentially orthogonality equations sometimes spoiled by dependence of the resistivity in the ultra-quantum limit: the potential enveloping function. All the details of the perturbative analysis are summarized in the Appendix. Despite the breakdown of translational invariance (the H; 1 < < 0; Green's functions depend on x and x separately rather (32) H; = 0 Coulomb disorder; xx 0 than on x x ), it is possible to introduce an e ective 1+ H ; 0 < < 1; 2D self-energy in the momentum representation and sum up the corresponding Dyson series. The self-energy then at xed n . The expression (32) is an important result of reads: our paper. It shows that measuring  (H ) of the WSM xx in the ultra-quantum regime, one can extract information i in (p ) =  + v(p ()) ; (37) n n about disorder correlations and the form of the impurity 2 2 0 1 potential. where p is the e ective 2D momentum de ned right be- low Eq. (12). Do not confuse it with thereal p , over IV. MAGNETOTRANSPORT AT T which we integrated, when we introduced the poten- (SEMICLASSICAL LIMIT) tial (17). We also de ne the e ective two-dimensional scattering times  (l = 0, 1, 2) according to: The opposite limit, which allows for an analytical 1 " d 2 l 0 treatment, is when the temperature of the WSM is much = n u g 0 cos (  ) : (38) imp 2;p (nn ) 0 " 2 5 2 2v p 2 larger than . Here, we focus on the most experimen- tally viable case when the magnetic length isbeing much (0) (0) (0) Here, we have p = "=v and n = (cos  ; sin  ). larger than the disorder correlation length " The very fact that the whole physics of the problem maxfT; g can be reformulated in terms of the 2D potential has a l  p  : (33) v beautiful physical interpretation. Let us recall that in the Landau gauge (2), the center of the orbit is given by p l . In the case of theCoulomb potential, p is the inverse That is, the e ective scattering rates (38) entering the Debye screening length, and the right-hand side condition perturbation theory are essentially ordinary scattering in Eq. (33) is equivalent to  1 (which is true for a rates but averaged over the positions of the center of the typical WSM, like Ca As , see Refs. 28,29, where the 2 3 23 Landau orbit. With these perturbative building blocks, value is estimated as  0:05). The left-hand side we are ready to compute the conductivity tensor. condition in (33) in this case should be substituted by maxfT; g: (34) B. General expressions for conductivities Therefore, the temperatures should not be too low. The conductivity, in the leading order of the expansion parameter (14), is given by the following simple expres- A. Computation of xx sion: 2 2 Here, we need to keep in mind that the characteristic e df R A = Re Im hG G i d"dp : x;x[y] n;11 n+1;22 number of LLs contributing is large 4 v d" (39) n  maxfT; g= 1: (35) R R A A This allows us to use the large n asymptotics for LL wave Here, we discard the G G and G G terms as sub- functions (s). Their highly oscillatory behavior allows leading in the 1=(") disorder expansion. Also, by n xy one to drastically simplify the calculations. we mean the normal part  of the Hall conductivity xy 6 (see subsection IV.C below for the full computation of the Hall conductivity). We switch from the summation over LLs to integra- tion over n (in the limit T; , all the functions are smooth functions of n), we substitute 1 = dn = 2 2 v p dp = , and turn to polar coordinates: p dp dp = y y y y z p sin ddp. Now, we need to nd the nonperturbative vertex renormalization responsible for the di erence be- R A R A tween hG G i and hG ihG i. As shown in the Appendix, we are able to perform the integration over the modulus of the momentum p in (39) and end up with only an angular integral. As a result, the conductivity tensor (39) can be rewritten in the following FIG. 3: (Color online) Conductivity  given by the xx form: exact Eq. (44) (solid line) and by the approximate Z Z Eq. (45) (dashed line) expressions as a function of d" df (") d sin RA 2 dimensionless parameter =T  H: = Trf () ~ g; (40) tr x;x[y] x[y] 2 d" 2 where () is the so-called angular vertex function for any relation between the chemical potential and tem- X perature: dp dp z y RA () = (2) 2 2 3 n u T T e T (41) 0 imp = f ; where xx R A 2 4 2 g v (p v) (T ) cos  hG ("; p)  ~ G ("; p)i dx: 1 0 tr 0 0 x;x x ;x 2 2 3 4  2a 1 a + i=T 2 0 f (a) = 4 a + Re + ; 3 T  2 2 It is essentially a vertex function, integrated over the 2 3 modulus of the momentum at xed vp =" ratio. The no- u T 0 imp a = : tation  ~ in the r.h.s. of Eq. (40) stand for Pauli  ma- x;y 2 2 g (p v) 1 0 trices. Then, we plugin the vertex expressions from (A20) (44) and take the angular integral to obtain: 0 1 0 1 Here, (x) is the Euler's digamma function. The dimen- xx 2 2 2 e d"" df "  (") sionless parameter a plays the role of the relative strength tr @ A @ A = : 2 4 2 of the disorder. 2v 2 d"  (") + " I tr 2 =" tr xy The exact formula (44) can be somewhat simpli ed by (42) the interpolation expression (which becomes exact in the limit ! 0 and ! 1) making it more useful for 1 1 1 Here,     is the transport scattering rate. tr 0 2 experimental purposes: Equation (42) is quite an important result. It shows that in the long-range disorder limit, the conductivity is e ec- 2 2 4 2 e 7 2 2 tr =  max T ;  1 + : tively recast in terms of the 2D Drude-type expression. xx tr 2 2 v 5 maxfT ;  g Similar formulae for the -correlated disorder were ob- (45) tained in Ref. 17. However, the magnetoconductance in Ref. 17 is expressed in terms of the 3D scattering rates. Here, the transport scattering time  should be taken tr This is somewhat predictable, since the -corrrelated dis- at the energy " = maxf; Tg. To give the reader an order has zero correlation length and the scattering rate idea of how well the interpolation formula represents the is not a ected by any other scale, including the magnetic exact result (44), we plot it in Fig. 3. Equations (44, length, which is responsible for the change in the geom- 45) reproduce the T dependence at ! 0, obtained in etry of the problem. Ref. 30 in the zero- eld limit. The interpolation formula For the disorder of the general type, with the correla- (45) for the conductivity  e ectively recasts it in the xx tion radius independent of the characteristic energy of the form of familiar Drude-type metallic expression host system, we have for the transport scattering time: 2 2 1 /  (1 + !  ) ; (46) tr c tr 2 3 u T 0 imp 1=3 (") = ; T = n v: (43) imp tr imp 2 2 2 g " (p v) where ! = =2" is the semiclassical cyclotron fre- 1 0 c quency at energy ". 2 2 Here, g = g(x )x dx is the numerical constant de- The same kind of Drude representation but with 3D termined by the type of disorder. This way, we obtain scattering times was obtained in Ref. 17 for -correlated the general expression for the longitudinal conductivity disorder. 7 FIG. 5: (Color online) Phase diagram for the conductivity  for the non-Coulomb disorder, see xx Eqs. (25) and (44). Here, g = 0 for  0, and g = =2 for 0 < < 1. See detailed explanations in the text. sea. dp FIG. 4: (Color online) Conductivity  given by xx n (H; ; T ) = f (" ) f (" + ) : 0 n n 2 2 4 v 2 Eq. (44) as a function of =T and the dimensionless parameter =T  H . Here we assume that   T . tr (48) In the (T; ) limit, we use the Euler{MacLaurin The conductivity  (H ) for di erent values of =T xx summation formula: are shown in Fig. 4. The behavior of the conductivity  can be conve- xx F (a) niently shown in the phase diagram (see Fig. 5). The up- F (a + n)  + F (x) dx: (49) per left red corner of this phase diagram corresponds to n=0 the ultra-quantum regime, T , where depending on Then, we obtain: the characteristic exponent of the impurity potential, we expect a -dependent scaling of  . The lower right xx 1 2 corner is divided into the regimes of weak and strong dis- 2 3 2 2 n (; T ) = +  +   T : (50) 2 3 order. The brown area corresponds to a strong disorder, 4 v 3 and is described by Eq.(45) in the Only the rst term in Eq. (50) is eld dependent. Despite 2 II maxfT; g= (47) its smallness, it is this term that contributes to  . This tr xy way, we arrive at the following expression: limit. One could also refer it to as a weak magnetic eld regime, where  exhibits predominantly the T xx II = : (51) xy dependence characteristic of a zero- eld system with a 2v correction proportional to H . The green area depicts the In experiments, the charge-carrier density is constant for opposite weak-disorder limit (or that of high magnetic each sample of WSM. Hence, the chemical potential is eld), where the transport of charge carriers is strongly almost eld independent in the high-temperature regime a ected by the magnetic eld. (see Discussion section for the relevant estimates). Next, we calculate the Hall conductivity. Next, we calculate the normal part of  given xy by (42). The conductivity  can be computed exactly xy at any value of  (see the corresponding integral derived C. Hall conductivity xy in the Appendix): As usual, the Hall conductivity is split into two parts: 2 2 e T anomalous and normal ones. Let us rst calculate the = f (a); xy 1 2 2 anomalous part. As before, we make use of Eq. (28). 2 4 a T a 1 i=T To regularize the expression for the charge-carrier den- 2 2 (1) f (a) = +  a Im + + ; sity, we subtract the respective density at zero chemical T 2 2 2 2 potential, thus eliminating the contribution of the Fermi (52) 8 where a is de ned in Eq. (44). As before, we concoct a Drude-type interpolation for- mula from ! 0 to maxfT; g using for- tr mula (52): 2 2 2 c e tr ; (53) xy 2 4 v tr 1 + 2 2 maxfT ; g 2(4) where c = 1 if   T and c = 7 =3 if   T . 1;2 1;2 Here  (") is taken at " = maxf; Tg. Depending on the tr factor ! , the normal part may be a leading or sublead- tr ing contribution to the Hall conductivity. Qualitatively, can be described by the following identity: xy d ; 1; > 1 tr maxfT;g 2 2 = d ; 1 xy 2 tr tr : 2 2 maxfT ; g maxfT;g d ; 3 2 tr (54) FIG. 7: (Color online) Conductivity  given by xy Eq. (44) as a function of =T and the dimensionless Here d are numerical factors following from rela- parameter =T  H . tr tions (51) and (53). The plot illustrating the accuracy of the interpolation formula (53) is presented in Fig. 6. V. DISCUSSION Interestingly, the case of Coulomb impurities does not lead to di erent results, despite the fact that the Debye In this paper we have performed a detailed analysis of radius depends on temperature. The relatively slow de- the e ect of the long range disorder introduced by im- cay of the Coulomb correlation function leads to a trivial purities on the magnetotransport in WSMs. Our study logarithmic enhancement of the respective 2D scattering is mainly focused on the magnetic eld and temperature rate: dependence of the transverse magnetoconductivity. Two important limiting cases are considered: (i) the ultra- 2 3 imp quantum limit, corresponding to low temperatures (or = 2 ln : (55) tr high magnetic eld), for which the main contribution comes from the zeroth Landau level, and (ii) the oppo- Otherwise, the whole temperature and eld dependence site semiclassical limit, when a large number of Landau remains the same. levels is involved in the transport phenomena. We have completely discarded the e ects of the intern- ode charge transfer. In principe, this e ect can be impor- tant at suciently high elds as argued in Ref. 31. The necessary condition for the applicability of our approach 1 1 is    , where for nding the scattering rates, inter intra we can use e.g. Eqs. (38) or (43). As derived in Ref. 31, 3=2 2 this condition is equivalent to H  eQ =v, where Q is the distance between the Weyl nodes in momentum space. 32{34 However, for a typical WSM like TaAs , we extract the separation between Weyl nodes as Q = 0:01 A , while the Fermi velocity v  3  10 m/s, which gives FIG. 6: (Color online) Hall conductivity  given by xy the respective eld estimate of H  50 T even for the Eq. (52) (solid line) and by Eq. (53) (dashed line) as a 2 ne structure constant in TaAs (unknown to us at the function of the dimensionless parameter =T  H . tr moment)  1. Therefore, we safely discard this e ect. The long-range impurity potential is chosen in a rather The plot of the Hall conductivity  is presented in general form. We show that in the ultra-quantum limit xy Fig. 7 as a function of the magnetic eld H and chemical the nonlinear magnetoresistivity dependence on the mag- potential . netic eld is the manifestation of the singular (non- 9 Coulomb) short-range behavior of the impurity potential ductivity undertaken in Refs 35{37. and its correlation function. In the semiclassical limit, we have demonstrated that unlike the short-range disorder case , the long-range disorder makes the scattering in the system essentially two-dimensional. We derived gen- eral formulae for  (H ) and  (H ) valid within a wide xx xy range of values of temperature and chemical potential. In typical experiments , the doping levels in WSMs are rather high ( 10 meV), which corresponds to 100 K. The typical magnetic elds H  1 T correspond to the gap between the zeroth and rst Landau levels FIG. 8: Electron self-energy 10 K. Reference 27, however, reports the observation of WSM in an almost undoped regime   T . Therefore, both the   T and   T regimes seem experimentally viable and the relation between ; H , and T can be quite Acknowledgments general. Hence, our results obtained in both limits can be relevant. We can also mention the numerical work in Ref. 20. In This work is partly supported by the Russian Foun- this paper, Coulomb impurities are correctly identi ed as dation for Basic Research, project Nos. 19-02-00421, the long-range disorder, and the high-temperature limit 19-02-00509, 20-02-00015, and 20-52-12013, by the joint is explored. The nontrivial result of Ref. 20 is the scaling research program of the Japan Society for the Promo- 5=3 of the magnetoconductance  / H in the low- eld tion of Science and the Russian Foundation for Basic xx regime . T . Our analytical study addresses the case Research, JSPS-RFBR Grants No. 19-52-50015 and No. , and the aforementioned regime cannot be ac- JPJSBP120194828, and by Deutsche Forschungsgemein- cessed in our semiclassical computation, where =T  1 schaft (DFG), grant No. EV 30/14-1. F.N. is sup- is the essential expansion parameter. ported in part by NTT Research, Army Research Of- In a semiclassical regime T , our results match the ce (ARO) (Grant No. W911NF-18-1-0358), Japan Sci- results reported in Ref. 16 in the low impurity concentra- ence and Technology Agency (JST) (via the CREST tion regime,   T , but di er in the opposite limit. Grant No. JPMJCR1676), Japan Society for the Pro- tr We attribute this to the e ect of the long-range disorder motion of Science (JSPS) (via the KAKENHI Grant No. correlations. JP20H00134), and Grant No. FQXi-IAF19-06 from the Note also in conclusion that the problem under study Foundational Questions Institute Fund (FQXi), a donor has a deep analogy with the detailed analysis of the ef- advised fund of the Silicon Valley Community Founda- fect of interplay of quantum interference and disorder on tion. Ya.I.R. acknowledges the support of the Russian magnetoresistance in the systems with the hopping con- Science Foundation, project No. 19-42-04137. Appendix A: Perturbation theory for T 1. Self-energy and Dyson series for the Green's function The diagram representing the rst-order correction to the Green's function is presented in Fig. 8 and is given by the following analytical expression dq dq dq 1 2 x y z 0 iq (x x ) G(x; x ) = e g(q ; q ) x yz (2) n;m;k (A1) y 1 1 y 2 2 0 S (x )G (p )S (x )S (x )G (p + q )S (x )S (x )G (p )S (x ): n p n z m m z z k k z y n p p +q m p +q p k p y y y y y y y The correction to the Green's function is a matrix, each term containing a product of  (x ) functions. l p It is essential that for large n we discard the di erence between n and n+1, when computing the correlation function. y y This allows us to treat the matrices S (x ); S (x ) as proportional to unit ones: S (x ); S (x ) =  (x ) 1. n p p n p p n p y n y y n y y The leading contribution to the conductivity comes from n  1 (in fact, we will later see that the contribution comes 2 26 from n  (T= ) ) and use the asymptotic relation for the Hermite polynomials : p p 2 2n 2 n x x =2 e H (x)  2 cos x 2n 1 : (A2) e 2 2n + 1 10 The question is how to simplify the product 1 2 1 2 iq (x x ) (x ) (x )e dq (A3) m m y p +q p +q y y y y entering (A1). We split the product of two cosine functions in each Hermite polynomial into p p 1 2 1 2 cos[ 2m(x x )=l ] + cos( 2m[(x + x )=l 2q l ]: (A4) H H y H p p p p y y y y Then, we perform integration over q . The rst term gives just the integral g(q ; q )dq =2 = g : (A5) x yz y 2;xz It is simply an e ective 2D potential (17). The second term is proportional to U ( 2ml ; q ). However, the correlation H yz p p radius of the potential obeys the inequality r  l  2ml . As a result, the term proportional to cos( 2m[(x + 0 H H x )=l 2q l ]) is suppressed. H y H Now we are able to perform the next estimate: dq 0 0 G(x; x )   (x ) (x ) G (p )G (p + q )G (p ) n p k n z m z z k z y p n;m;k p p 1=4 1 2 p cos[ 2nx =l ] cos[ 2kx =l ] 1 2 dq 1 1 4 H H x p p iq (x x ) 1 2 y y 1 2 e g (q ) p cos[ 2m(x x )=l ] q q dx dx : 2 xz H p p y y 1;2 2;2 2 l nk 4 4 l 2m x x H p p H y y 1 1 2n 2k (A6) In Eq. (A6), it is important to discern the di erence between the fast-oscillating cosine-type terms in the nominator and 1 2 1 2 1 2 2 slow algebraic factors in the denominator. To perform the integration over x ; x , we change x ; x ! r = x x ; x . We obtain many fast-oscillating terms (the relevant n; m and k are large). For example, performing integration over x , we obtain: p p p p p p 2 2 1 cos[ 2nr + ( 2n + 2k)x =l ] + cos[ 2nr + ( 2n 2k)x =l ] H H dx r r : 2;2 2 2 (A7) (r+x ) 4 p 4 p y y 1 1 2 2 2nl 2kl H H As we see from the structure of the integral of (A7), the nominator is a fast-oscillating function of x . As a result, the integral is suppressed unless n = k. Thus, the integral is /  in the main order for 1= n k. Let us compute nk it for n = k. The nominator does not oscillate any more, and we should analyze the denominator. The important 2 2 r  p , which comes from g (q ). On the other hand, the important x  nl . We see that r  x for the 2 xz H 0 p p y y denominator. Therefore, we integrate over x trivially. We are then left with the following expression: dq 0 0 G(x; x )   (x )) (x ) G (p )G (p + q )G (p ) n p n n z m z z n z y p n;m (A8) p p dq dr 1 iq r e g (q ) cos( 2mr=l ) cos( 2nr=l ): 2 xz H H l 2m While integrating over r, we obtain the combination of 4 -functions. The only relevant ones are (q + 2ml p p p 1 1 1 2nl ) and (q 2ml + 2nl ). Consequently, we have the following nal formula for the correction to the H H H Green's function: h i X p p dq 1 0 0 1 G(x; x )   (x ) (x ) G (p )G (p + q )G (p ) p g ( 2m 2n)l ; q : n p n n z m z z n z 2 z y p H (A9) 2l 2m n;m Using the fact that nl  T  p , we understand that m in the last sum is actually very close to n. Indeed, we see that: p p p p m n p v m n  p l  1; while p p   1: (A10) 0 H m + n T 11 From the last inequalities, we see that the terms of the sum over m are smooth functions of m and the sum can be 0 1 2 0 turned into an integral. Introducing the e ective momentum p = 2ml ; dm  1 = q dq l , and p = p + q , y y z z y H H z we obtain: dp 0 0 0 0 G(x; x )   (x ) G(p )G(p )G(p )g (p p )  (x ): (A11) n p n n 2 n n y p 2 y (2) Here, p is introduced in (12). The expression in the square brackets in (A11) allows us to build the ordinary 2D Dyson series for the Green's function as well as vertex functions determining the conductivity tensor. Indeed, it coincides with the standard expression of perturbation theory without magnetic eld with the e ective 2D potential g (p p ). Therefore, we can write the momentum-dependent self-energy as: 2 n dp 0 0 (p ) = G(p )g (p p ): (A12) n 2 n (2) The resummed Green's function then reads: 0 1 1 0 G(x; x )   (x )[G (p ) (p )]  (x ): (A13) n p n n n y p The resulting expression yields an irrelevant part, which can be absorbed into the renormalized chemical potential and Fermi velocity, and the dissipative part. The imaginary part reads: Z Z 0 0 0 0 dp " + p  dp 1 p R 0 0 (p ) = v:p: g (p p ) i ((" " ) + (" + " )) + g (p p ) n 2 n n n 2 n 2 2 2 2 (2) (" + i0) p (2) 2 2" i in =  + vp  ; 2 2 Z Z 1 p 1 " 0 0 0 0 0 = g (nn )dn ; = (nn )g (nn )dn : 2 2 4  4 (A14) 2. Vertex renormalization and conductivity tensor The Dyson equation for the vertex is built in a more subtle way. In this case, the built-in magnetic anisotropy of the problem takes its toll. What we are going to do now is to introduce a slightly unusual de nition of the vertex. We de ne the mass-shell vertex according to the following equation: dp dp p z y z R A () =  cos  hG 0 ("; p)  G 0 ("; p)i dx: (A15) x x x;x x ;x (2) p Then, the conductivity tensor assumes the form in Eq. (40). In the zeroth-order ladder approximation, we change R A R A hG G i = hG ihG i, and the vertex becomes: X R A 1 dp 0 G G 0 11;n 22;n+1 = : (A16) x R A G G 0 2l 2 22;n 11;n1 Changing the sum and the integral using the semiclassical approximation Z Z dp p p dp z z cos  = l ; (A17) 2 p 2 we obtain: 1 1 i 0 [ + + ] RA;0   " = 2 : (A18) 1 1 i 3 1 2v [ + ] 0 In the rst order of perturbation theory, the picture changes slightly, and we obtain the rst stair of the ladder series 12 0 1 1 1 1 2 0 h i 2 2 1 1 i B C n + + RA;1   " B C = : (A19) 1 1 3 @ + A 2v 1 2 h i 1 1 i In the higher orders of the perturbation theory the pattern repeats itself. As a result, we are able to perform the full summation: 0 1 h i 1 i 0 + y B  " C tr RA = h i : (A20) @ A x 1 3 2 2v 1 i tr Finally, we are ready to obtain the expression for the conductivity: Z Z 3 2 d" df (") d sin  1 i = Re[Im] + : (A21) x;x[y] d" 2  " tr Appendix B: Calculation of the integral for the The last integral is equal to conductivity at T The integral that enters the upper matrix element in Z 1 dx 2 1 a i the expressions for the conductivities in (42) has the = Re + : 2 x 2 2 (x + ) + a a 2 2 cosh following form: (B1) 1 x dx I (a) = 2 x 2 2 x + a cosh 1 1 Z Z 2 2 2 x +  a 1 dx As a result, we recover Eq. (44) In the same manner we = dx + a : 2 x 2 x 2 2 perform the integration for  in the lower part of (42) (x + ) + a cosh cosh xy 2 2 1 1 to obtain the exact expression (52). P. Hosur and X. Qi, Recent developments in transport phe- NbP, Nature Phys. 11, 645 (2015). nomena in Weyl semimetals, C. R. Phys. 14, 857 (2013). C.Z. Li, L.X. Wang, H.W. Liu, J. Wang, Z.M. Liao, and Hai-Zhou Lu and Shun-Qing Shen, Quantum transport in D.P. Yu, Giant negative magnetoresistance induced by the topological semimetals under magnetic elds, Front. Phys. chiral anomaly in individual Cd As nanowires, Nature 3 2 12, 127201 (2017). Commun. 6, 10137 (2015). 3 8 D.T. Son and B.Z. Spivak, Chiral anomaly and classical A.V. Andreev and B.Z. Spivak, Longitudinal negative mag- negative magnetoresistance of Weyl metals, Phys. Rev. B netoresistance and magnetotransport phenomena in con- 88, 104412 (2013). ventional and topological conductors, Phys. Rev. Lett. A.A. Burkov, Chiral anomaly and di usive magnetotrans- 120, 026601 (2018). port in Weyl metals, Phys. Rev. Lett. 113, 247203 (2014). H.Z. Lu and S.Q. Shen, Weak antilocalization and localiza- Xiaochun Huang, Lingxiao Zhao, Yujia Long, Peipei Wang, tion in disordered and interacting Weyl semimetals, Phys. Dong Chen, Zhanhai Yang, Hui Liang, Mianqi Xue, Hong- Rev. B 92, 035203 (2015). ming Weng, Zhong Fang, Xi Dai, and Genfu Chen, Ob- H.-Z. Lu and S.-Q. Shen, Weak antilocalization and servation of the chiral-anomaly-induced negative magne- interaction-induced localization of Dirac and Weyl toresistance in 3D Weyl semimetal TaAs, Phys. Rev. X 5, fermions in topological insulators and semimetals, Chin. 031023 (2015) Phys. B 25, 117202 (2016). 6 11 C. Shekhar, A.K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, Tian Liang, Quinn Gibson, Mazhar N. Ali, Minhao Liu, I. Leermakers, U. Zeitler, W. Schnelle, J. Grin, C. Felser, R.J. Cava, and N.P. Ong, Ultrahigh mobility and giant and B. Yan, Extremely large magnetoresistance and ultra- magnetoresistance in the Dirac semimetal Cd As , Nature 3 2 high mobility in the topological Weyl semimetal candidate Mater. 14, 280 (2015). 13 Junya Feng, Yuan Pang, Desheng Wu, Zhijun Wang, 1972). Hongming Weng, Jianqi Li, Xi Dai, Zhong Fang, Youguo P.G. LaBarre, L. Dong, J. Trinh, T. Siegrist, and A.P. Shi, and Li Lu, Large linear magnetoresistance in Dirac Ramirez, Evidence for undoped Weyl semimetal charge semimetal Cd As with Fermi surfaces close to the Dirac transport in Y Ir O , J. Phys.: Condens. Matter 32, 3 2 2 2 7 points, Phys. Rev. B 92, 081306(R) (2015). 02LT01 (2019). 13 28 Mario Novak, Satoshi Sasaki, Kouji Segawa, and Yoichi Z. K. Liu, J. Jiang, B. Zhou, Z.J. Wang, Y. Zhang, H.M. Ando, Large linear magnetoresistance in the Dirac Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, et semimetal TlBiSSe, Phys. Rev. B 91, 041203(R) (2015). al., A stable three-dimensional topological Dirac semimetal Y. Zhao, H. Liu, C. Zhang, H. Wang, J. Wang, Z. Lin, Y. Cd As , Nature Mater. 13, 677 (2014). 3 2 Xing, H. Lu, J. Liu, Y. Wang, S.M. Brombosz, Z. Xiao, J.-P. Jay-Gerin, M.J. Aubin, and L.G. Caron, The electron S. Jia, X.C. Xie, and J. Wang, Anisotropic Fermi sur- mobility and the static dielectric constant of Cd As at 4.2 3 2 face and quantum limit transport in high mobility three- K, Solid State Commun. 21, 771 (1977). dimensional Dirac semimetal Cd As , Phys. Rev. X 5, S.D. Sarma, E.H. Hwang, and H. Min, Carrier screening, 3 2 031037 (2015). transport, and relaxation in 3D Dirac semimetals, Phys. A.A. Abrikosov, Quantum magnetoresistance, Phys. Rev. Rev. B 91, 035201 (2015). B 58, 27881 (1998). G. Bednik, K.S. Tikhonov, S.V. Syzranov, Magnetotrans- J. Klier, I.V. Gornyi, and A.D. Mirlin, Transversal magne- port and internodal tunnelling in Weyl semimetals, Phys. toresistance in Weyl semimetals, Phys. Rev. B 92, 205113 Rev. Research 2, 023124 (2020). (2015). S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, J. Klier, I.V. Gornyi, and A.D. Mirlin, Transversal magne- C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.- toresistance and Shubnikov{de Haas oscillations in Weyl M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, semimetals, Phys. Rev. B 96, 214209 (2017). A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and D.A. Pesin, E.G. Mishchenko, and A. Levchenko, Density M. Z. Hasan, Discovery of a Weyl fermion semimetal and of states and magnetotransport in Weyl semimetals with topological Fermi arcs, Science 349, 613 (2015). long-range disorder, Phys. Rev. B 92, 174202 (2015). B.Q. Lv, N. Xu, H.M. Weng, J.Z. Ma, P. Richard, X.C. Xiao Xiao, K.T. Law, and P.A. Lee, Magnetoconductiv- Huang, L.X. Zhao, G.F. Chen, C.E. Matt, F. Bisti, V.N. ity in Weyl semimetals: E ect of chemical potential and Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and temperature, Phys. Rev. B 96, 165101 (2017). H. Ding, Observation of Weyl nodes in TaAs, Nature Phys. Justin C.W. Song, Gil Refael, and Patrick A. Lee, Linear 11, 724 (2015). magnetoresistance in metals: Guiding center di usion in B.J. Ramshaw, K.A. Modic, Arkady Shekhter, Yi Zhang, a smooth random potential, Phys. Rev. B 92, 80204(R) Eun-Ah Kim, Philip J.W. Moll, Maja D. Bachmann, M.K. (2015). Chan, J.B. Betts, F. Balakirev, A. Migliori, N.J. Ghimire, V. K onye and M. Ogata, Magnetoresistance of a three- E.D. Bauer, F. Ronning, and R.D. McDonald, Quantum dimensional Dirac gas, Phys. Rev. B 98, 195420 (2018). limit transport and destruction of the Weyl nodes in TaAs, V. K onye and M. Ogata, Thermoelectric transport coe- Nature Commmun. 9, 2217 (2018). cients of a Dirac electron gas in high magnetic elds, Phys. Y.L. Lin and F. Nori, Strongly localized electrons in a Rev. B 100, 155430 (2019). magnetic eld: Exact results on quantum interference and Ya.I. Rodionov, K.I. Kugel, and Franco Nori, E ects magnetoconductance Phys. Rev. Lett. 76, 4580 (1996). of anisotropy and disorder on the conductivity of Weyl Y.L. Lin and F. Nori, Quantum interference from sums semimetals, Phys. Rev. B 92, 195117 (2015). over closed paths for electrons on a three-dimensional lat- Ya. I. Rodionov and S. V. Syzranov, Conductivity of a tice in a magnetic eld: Total energy, magnetic moment, Weyl semimetal with donor and acceptor impurities, Phys. and orbital susceptibility, Phys. Rev. B 53, 13374 (1996). Rev. B 91, 195107 (2015). Y.L. Lin and F. Nori, Analytical results on quantum in- P. St reda, Theory of quantised Hall conductivity in two terference and magnetoconductance for strongly localized dimensions, J. Phys. C: Solid State Phys. 15, L717 (1982). electrons in a magnetic eld: Exact summation of forward- M. Abramowitz and I.A. Stegun, Handbook of Mathemat- scattering paths, Phys. Rev. B 53, 15543 (1996). ical Functions with Formulas, Graphs, and Mathematical Tables (US Government Printing Oce, Washington, D.C., http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Condensed Matter arXiv (Cornell University)

Effect of disorder on the transverse magnetoresistance of Weyl semimetals

Rodionov, Ya. I.; Kugel, K. I.; Aronzon, B. A.; Nori, Franco
Condensed Matter , Volume 2020 (2005) – May 18, 2020
Download PDF
13 pages

Loading next page...
 
/lp/arxiv-cornell-university/effect-of-disorder-on-the-transverse-magnetoresistance-of-weyl-ulkbT8M0X7

References (44)

ISSN
2469-9950
eISSN
ARCH-3331
DOI
10.1103/PhysRevB.102.205105
Publisher site
See Article on Publisher Site

Abstract

E ect of disorder on the transverse magnetoresistance of Weyl semimetals 1, 2, 3 1, 3 4 5, 6 Ya. I. Rodionov, K. I. Kugel, B. A. Aronzon, and Franco Nori Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Izhorskaya str. 13, Moscow, 125412 Russia National University of Science and Technology MISIS, Moscow, 119049 Russia National Research University Higher School of Economics, Moscow, 101000 Russia P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Physics Department, The University of Michigan, Ann Arbor, MI 48109-1040, USA (Dated: November 18, 2020) We study the e ect of random potentials created by di erent types of impurities on the trans- verse magnetoresistance of Weyl semimetals. We show that the magnetic eld and temperature dependence of the magnetoresistance is strongly a ected by the type of impurity potential. We analyze in detail two limiting cases: (i) the ultra-quantum limit, when the applied magnetic eld is so high that only the zeroth and rst Landau levels contribute to the magnetotransport, and (ii) the semiclassical situation, for which a large number of Landau levels comes into play. A formal diagrammatic approach allowed us to obtain expressions for the components of the electrical con- ductivity tensor in both limits. In contrast to the oversimpli ed case of the -correlated disorder, the long-range impurity potential (including that of Coulomb impurities) introduces an additional length scale, which changes the geometry and physics of the problem. We show that the magne- toresistance can deviate from the linear behavior as a function of magnetic eld for a certain class of impurity potentials. PACS numbers: 72.10.-d 72.80.-r 75.47.-m 5.60.Gg I. INTRODUCTION The results for an arbitrary position of the chemical po- tential were obtained in Ref. 19 within the framework of a similar approximation; and also in Ref. 20, where the In recent years, problems related to quantum transport approach based on the classical motion of the so-called in materials with the Dirac spectrum of charge carries, in guiding center was used. For the screened Coulomb po- particular in Weyl semimetals, have attracted consider- tential of impurities, the electron transport was also an- 1,2 able interest . Much e ort was focused on the longitu- 21,22 alyzed in the case of a gapped Dirac spectrum . dinal magnetoresistance, where the negative contribution Thus, we see that in spite of these serious e orts, some associated with the so-called chiral anomaly arising due important aspects of the analysis of the magnetoresis- the transfer of charge carries between Weyl points with tance of Weyl semimetals remain untouched. First of all, 3{8 opposite chiralities plays a dominant role . its behavior at di erent characteristic ranges of the im- At low magnetic elds, rather nontrivial manifesta- purity potential, and accurate calculations in the limit of tions of the weak localization and antilocalization e ects long-range potentials. These issues still remain unsolved 9,10 have also been addressed . No less interesting is the 15{17 even in the most well-studied ultra-quantum limit, behavior of the transverse magnetoresistance, where a when only one Landau level contributes to the magne- nonsaturating linear magnetic eld dependence is ob- toresistance. 11{14 served at high elds . The nature of such unusual In this paper, we rst focus on the calculations in behavior has been widely discussed. the ultra-quantum limit and calculate components of the The main physical mechanisms in the ultra-quantum magnetoconductivity tensor based on the accurate dia- regime were revealed in the seminal work of Abrikosov . 23 grammatic approach which we have formulated earlier . He considered a gapless semiconductor with a linear dis- Then, we present a detailed analysis of the semiclassical persion law near the chemical potential. The chemical limit, when a large number of Landau levels contributes potential itself coincides with the zeroth Landau level to the transport characteristics. and the charge carriers are scattered by impurities char- In Section II, we formulate the model and introduce all acterized by a screened Coulomb potential. the necessary parameters. In Section III, we analyze the This problem was generalized in the detailed studies components of the electrical conductivity tensor and their presented in Refs. 16{18, which were stimulated by nu- magnetic eld dependence in the ultra-quantum limit, for merous experimental observations of the linear magne- which the dominant contribution comes from the zeroth toresistance. In these papers, the main emphasis was on Landau level. In Section IV, we consider the magneto- the case of point-like impurities. transport at the semiclassical limit, for which the tem- In the opposite limit of long-range impurity potentials, perature is high enough and a large number of Landau treated by the Born and self-consistent Born approxima- levels comes into play. Both in Sections III and IV, we tions, it was possible to obtain only qualitative results. put the main emphasis on the magnetotrasport for rather arXiv:2005.08893v2 [cond-mat.mes-hall] 17 Nov 2020 2 long-range impurity potentials. In Section V, we discuss the dimensionless function g, which is introduced in mo- the obtained results. The details of our calculations are mentum space from the very beginning. Also, u is the presented in the Appendix. Fourier transform of the impurity potential, while the 1 6 disorder correlation length is r = p . The p factor is 0 0 introduced from dimensional considerations. II. MODEL AND CHARACTERISTIC PARAMETERS Our study is aimed at the analysis of the transport characteristics of the Weyl semimetal (WSM) with im- purities under the e ect of an applied transverse mag- netic eld (i.e., the magnetic eld direction is perpendic- FIG. 1: Disorder vertex for the perturbation theory. ular to that of the electric current). We start from the Here, the dashed line represents the disorder correlation low-energy Hamiltonian for the WSM in its conventional function, while solid tails are fermion lines. form H = H + H ; In this work, we are focusing on the long-range correla- 0 imp tion between disordered impurities (long-range disorder). H = v (r) p A (r)dr; 0 In the ultra-quantum case (1) maxfT; g  v=l ; (4) H = (r)u(r) (r)dr; imp where where H is the Hamiltonian of non-interacting Weyl 0 l = c=(eH ) (5) fermions and H describes the interactions with the imp is the magnetic length and  is the doping level of the impurity potential;  = ( ;  ;  ) are the Pauli ma- x y z WSM sample, and this limit corresponds to the condi- trices acting in the pseudospin space of Weyl fermions, tion: p = ir is the momentum operator, v is the Fermi velocity, and u(r) is the impurity potential. l  r  ; (6) H 0 The impurity potential is understood to be of a general where form. It is of electrostatic origin, but its speci c form as well as the form of its correlation function can be arbi- = v=fmax T; g (7) trary. Of particular importance to the experiment is the is the characteristic particle wavelength. Let us also in- screened Coulomb impurity potential. As was argued in troduce the energy scale associated with the magnetic Ref. 24, there exists a regime, in which the Coulomb im- eld strength, purity scattering dominates over the electron{electron in- teraction (see the corresponding discussion in Section V). = v 2eH=c; (8) Throughout the paper, we set ~ = k = 1. We also ne- glect the in uence of di erent Weyl cones on each other, characterizing the distance between the zeroth and rst concentrating on the low-energy physics. The vector po- Landau levels (LLs). In the opposite semiclassical limit tential of the magnetic eld H is chosen in the asymmet- T , the corresponding condition for the disorder cor- ric gauge relation length reads: r  l : (9) 0 H A = (0; Hx; 0): (2) Limit (6) intuitively appeals to the physical picture where In this paper, we will use the Kubo-type diagrammatic the center of the magnetic orbit moves along the impurity approach. The impurity potential thus enters the for- potential line, while limit (9) corresponds to the proper malism in terms of its correlation function. The relevant particle motion along the impurity potential line. diagram is shown in Fig. 1. We write the corresponding correlation function in the following form: III. MAGNETOTRANSPORT AT fT; g 2 2 n u p imp ipr 0 (ULTRA-QUANTUM LIMIT) dre hu(r)u(0)i  g(p) = g ; 6 2 p p 0 0 (3) 2 6 A. Computation of p p xx 2 0 g = ju j : 2 2 p u 0 0 As was mentioned above, the ultra-quantum limit cor- Here, n is the concentration of impurities and u is imp 0 responds to the condition the characteristic amplitude of the impurity potential. maxfT; g: (10) The disorder correlation function is written in terms of 3 This means that the rst Landau level is high enough and only the ground state (and the rst excited state) contributes to the magnetotransport. The xx component of the conductivity tensor is determined by the following analytical expression: d" dp dx df (") 2 2 = e v xx (2) d" (11) R 0 R 0 TrhImG (x; x ; "; p)ImG (x ; x; "; p)i; 11 22 where angular brackets denote the averaging over disor- der, f is the Fermi distribution function, and the retarded Green's functions are de ned as follows: FIG. 2: Three contributions to the conductivity in rst-order perturbation theory. The disorder vertices R 0 y 0 are represented by dashed lines de ned in Fig. 1. The G (x; x ; "; p) = S (x )G("; p)S (x ) n p y n p n=0 indices i; j become x or y depending on the type of conductivity which is computed. s 0 S (s) = ; (12) 0  s n1 " + v  p G("; p) = ; Ref. 15. The result is given by the following integral: 2 2 (" + i0) " 2 2 e v dq dq x = x p l : x y p y 2 2 y H = (q + q )g ; (16) xx 2;q x y xy 2 2 (2) Here,  s is the oscillator normalized wave function of where g is the e ective two-dimensional disorder cor- the nth state, 2;q xy relation function de ned as p = (0; 2n=l ; p ) (13) n H z dp g = g(p)  g(p ) : (17) 2;p xy xy 2 z=0 is the e ective momentum, p is the two-dimensional yz momentum (p ; p ). We are using perturbation theory, y z In Ref. 15, Eq. (16) was analyzed only in the case of the therefore assuming that the concentration of impurities Coulomb impurity potential. We, however, come to the is not too high. The dimensionless expansion parame- conclusion that for di erent types of disorder, the formula ter characterising the disorder strength is assumed to be abovegives a qualitatively di erent H dependence. small: Before we proceed, let us make the following observa- 1 n u tion. The integral in Eq. (16) of the 2D disorder corre- imp 1; (14) lation function determining the conductivity can become " v p divergent at high momenta (short-range case). However, where " is the characteristic energy scale for charge car- in our calculations, we used the long-range disorder ap- riers and  is its impurity scattering time (see below proximation, implying that ql  1, where q is the char- Eq. (38)). acteristic disorder momentum. Therefore, q  l is the As the analysis shows, in the ultra-quantum limit, it is natural short-range cuto scale. As we will see below, enough to keep the rst order of perturbation in the dis- the system exhibits di erent types of magnetotransport order strength. Therefore, only three possible diagrams depending on the short-range behavior of the disorder, contribute to the conductivity (see Fig. 2). Even less p  q  l . trivial is the fact that the vertex correction (diagram III in Fig. 2) is exponentially suppressed. Therefore, only the rst-order disorder corrections to the Green's func- B.  for di erent short-range behavior of the xx impurity potential tion are needed to be taken into account. This was rst stated explicitly for the case of short-range disorder in . In the ultra-quantum limit and in the long-range dis- Let us now assume that the impurity potential has the order case p l  1 (Eq. (6)), only the zeroth and rst 0 H short-range asymptotics LLs in Fig. 2 contribute to the conductivity. For the R u Green's function G in (11), it is enough to take only 11 u(r) = ; l  r  p ; 1 < < 1: (18) 1+ (p r) the contribution of the zeroth LL, 0 Here, the natural constraint < 1 means that we are not Im G ("; p) = (" p v): (15) 11 z considering pathological cases of potentials leading to the The expression for the conductivity  then goes along \falling to the center" phenomenon. On the other hand, xx the same lines for the general type of disorder as that in the < 1 constraint should exclude the unphysical case 4 of decaying at r = 0. Then, the disorder correlation We see that the H dependence of the conductivity is af- function in momentum space reads: fected by the nature of the disorder. In particular, if the correlation function has stronger than Coulomb power- 4 2 >l v ; p  p ; law growth at short distances, the corresponding expo- < 0 nent enters the conductivity. g(p) = n imp : 42 2 6 1 The parameter most relevant to many experiments is u =p (p =p) ; p  p  l : 0 0 0 0 the magnetoresistance. To calculate it, we need to know (19) the Hall conductivity  . xy The question we now address is: what is the behavior of the conductivity as a function of H for di erent values C. Hall conductivity xy of ? To answer this question, we analyze expression (16) for various cases discussed below. The Hall conductivity is given by the sum of two terms: (a) 1 < < 0: We call this the \regular disorder" case. The integral in (16) is convergent and the conver- I II =  +  : (26) xy xy xy gence region is p  p . In this case, we have The rst term in (26),  , is the so-called normal con- ec xy = n u g ; < 0; (20) xx imp 1 25 tribution, which is given by the following relation : 16Hp R 2 2 2 3 e d" df (") where g = g(x )x dx, with x = p=p , is a numerical xy 4 2 d" constant, which depends on the details of the shape of R R R R R A R A the disorder distribution function. G Im G G Im G Im G G + Im G G : 22 11 11 22 22 11 11 22 As we are going to see below, the behavior correspond- ing to Eq. (20) is identical to that characteristic of the (27) Coulomb disorder, = 0. As is seen from Eq. (27), it comes from the vicinity of (b) = 0. For the Coulomb disorder, the integral the Fermi surface, as it is proportional to df=d". In the determining the conductivity in (16) is log-divergent. In absence of disorder, it is easily veri ed that  = 0. the Coulomb case, the inverse Debye radius reads: xy Therefore, it is perturbative in the disorder strength. The p = l (21) second term in Eq. (26) is the so-called anomalous contri- bution. It is proportional to the derivative of the charge for the case fT; g , where carrier density with respect to the applied magnetic eld H and, as such, comes from the entire volume inside the = e =(~v) (22) Fermi surface. As is understood from the de nition of the anomalous part is the WSM ne structure constant and  is the permit- tivity. One recovers the result : dN (H; ; T ) II = ec ; xy dH e c 1 (28) = n ln ; = 0: (23) xx imp H N (H; ) = (")f d"; (c) 0 < < 1. We call this the \singular disorder" it is nonzero even in the absence of disorder. Here, the casedue to its short-range behavior. The integral in (16) density of states reads: is then divergent at high momenta q and an appropriate cut-o p  l should be introduced. In this case, we dp yz (") = Tr Im G("; p ; p ; x; x) have a nontrivial result for  : y z xx (2) (29) 1 dp ec eH z = ImG ("; p ): = n u ; 0 < < 1: (24) n z xx imp 0 2 2 2 2l 2 16Hp cp 0 0 The above results can be summarized by the following Thus, from perturbative arguments, we understand that II formula: =  , i.e. it is determined by the anomalous part. xy xy 8 In our case (long-range disorder), it is even possible 2 2 4 1 e v p g (c=eH )  H ; 1 < < 0; II > to compute  at all orders of perturbation theory in xy the strength of the disorder, in the limit p ! 1. The 2 2 2 1 e v ln(1= )(c=eH )  H ; = 0; result is independent of the disorder strength and is given xx by the disorder-free expression: : 2 2 2 eH 1 e v n u (c=eH )  H ; 0 < < 1: imp 2 0 e cp II 2 2 =  = = e n l ; (30) xy 0 xy H (25) 4 v 5 where n is the charge-carrier density. In experiments, n In the Appendix, we argue that in the T limit, 0 0 is usually a xed parameter stemming from the charge- the problem of disorder averaging becomes essentially a neutrality condition due to the imbalance of donor and two-dimensional one. Unlike the ultra-quantum case, the acceptor impurities in WSMs. Therefore, to compare corresponding 2D plane is y = 0 (due to asymmetry of with experiments, one needs to express the chemical po- our vector potential gauge (2)) with the e ective corre- tential in terms of n . lation potential: The formula for the resistivity is as follows: dp g = g(p)  g(p ) : (36) xx 2;p xz xz = : (31) 2p y=0 xx 0 2 2 xx xy The most crucial observation is that all the integrals Taking into account theexpressions for  (25) and xx entering the Green's functions and Dyson equations are (30), we obtain the following results for the eld xy essentially orthogonality equations sometimes spoiled by dependence of the resistivity in the ultra-quantum limit: the potential enveloping function. All the details of the perturbative analysis are summarized in the Appendix. Despite the breakdown of translational invariance (the H; 1 < < 0; Green's functions depend on x and x separately rather (32) H; = 0 Coulomb disorder; xx 0 than on x x ), it is possible to introduce an e ective 1+ H ; 0 < < 1; 2D self-energy in the momentum representation and sum up the corresponding Dyson series. The self-energy then at xed n . The expression (32) is an important result of reads: our paper. It shows that measuring  (H ) of the WSM xx in the ultra-quantum regime, one can extract information i in (p ) =  + v(p ()) ; (37) n n about disorder correlations and the form of the impurity 2 2 0 1 potential. where p is the e ective 2D momentum de ned right be- low Eq. (12). Do not confuse it with thereal p , over IV. MAGNETOTRANSPORT AT T which we integrated, when we introduced the poten- (SEMICLASSICAL LIMIT) tial (17). We also de ne the e ective two-dimensional scattering times  (l = 0, 1, 2) according to: The opposite limit, which allows for an analytical 1 " d 2 l 0 treatment, is when the temperature of the WSM is much = n u g 0 cos (  ) : (38) imp 2;p (nn ) 0 " 2 5 2 2v p 2 larger than . Here, we focus on the most experimen- tally viable case when the magnetic length isbeing much (0) (0) (0) Here, we have p = "=v and n = (cos  ; sin  ). larger than the disorder correlation length " The very fact that the whole physics of the problem maxfT; g can be reformulated in terms of the 2D potential has a l  p  : (33) v beautiful physical interpretation. Let us recall that in the Landau gauge (2), the center of the orbit is given by p l . In the case of theCoulomb potential, p is the inverse That is, the e ective scattering rates (38) entering the Debye screening length, and the right-hand side condition perturbation theory are essentially ordinary scattering in Eq. (33) is equivalent to  1 (which is true for a rates but averaged over the positions of the center of the typical WSM, like Ca As , see Refs. 28,29, where the 2 3 23 Landau orbit. With these perturbative building blocks, value is estimated as  0:05). The left-hand side we are ready to compute the conductivity tensor. condition in (33) in this case should be substituted by maxfT; g: (34) B. General expressions for conductivities Therefore, the temperatures should not be too low. The conductivity, in the leading order of the expansion parameter (14), is given by the following simple expres- A. Computation of xx sion: 2 2 Here, we need to keep in mind that the characteristic e df R A = Re Im hG G i d"dp : x;x[y] n;11 n+1;22 number of LLs contributing is large 4 v d" (39) n  maxfT; g= 1: (35) R R A A This allows us to use the large n asymptotics for LL wave Here, we discard the G G and G G terms as sub- functions (s). Their highly oscillatory behavior allows leading in the 1=(") disorder expansion. Also, by n xy one to drastically simplify the calculations. we mean the normal part  of the Hall conductivity xy 6 (see subsection IV.C below for the full computation of the Hall conductivity). We switch from the summation over LLs to integra- tion over n (in the limit T; , all the functions are smooth functions of n), we substitute 1 = dn = 2 2 v p dp = , and turn to polar coordinates: p dp dp = y y y y z p sin ddp. Now, we need to nd the nonperturbative vertex renormalization responsible for the di erence be- R A R A tween hG G i and hG ihG i. As shown in the Appendix, we are able to perform the integration over the modulus of the momentum p in (39) and end up with only an angular integral. As a result, the conductivity tensor (39) can be rewritten in the following FIG. 3: (Color online) Conductivity  given by the xx form: exact Eq. (44) (solid line) and by the approximate Z Z Eq. (45) (dashed line) expressions as a function of d" df (") d sin RA 2 dimensionless parameter =T  H: = Trf () ~ g; (40) tr x;x[y] x[y] 2 d" 2 where () is the so-called angular vertex function for any relation between the chemical potential and tem- X perature: dp dp z y RA () = (2) 2 2 3 n u T T e T (41) 0 imp = f ; where xx R A 2 4 2 g v (p v) (T ) cos  hG ("; p)  ~ G ("; p)i dx: 1 0 tr 0 0 x;x x ;x 2 2 3 4  2a 1 a + i=T 2 0 f (a) = 4 a + Re + ; 3 T  2 2 It is essentially a vertex function, integrated over the 2 3 modulus of the momentum at xed vp =" ratio. The no- u T 0 imp a = : tation  ~ in the r.h.s. of Eq. (40) stand for Pauli  ma- x;y 2 2 g (p v) 1 0 trices. Then, we plugin the vertex expressions from (A20) (44) and take the angular integral to obtain: 0 1 0 1 Here, (x) is the Euler's digamma function. The dimen- xx 2 2 2 e d"" df "  (") sionless parameter a plays the role of the relative strength tr @ A @ A = : 2 4 2 of the disorder. 2v 2 d"  (") + " I tr 2 =" tr xy The exact formula (44) can be somewhat simpli ed by (42) the interpolation expression (which becomes exact in the limit ! 0 and ! 1) making it more useful for 1 1 1 Here,     is the transport scattering rate. tr 0 2 experimental purposes: Equation (42) is quite an important result. It shows that in the long-range disorder limit, the conductivity is e ec- 2 2 4 2 e 7 2 2 tr =  max T ;  1 + : tively recast in terms of the 2D Drude-type expression. xx tr 2 2 v 5 maxfT ;  g Similar formulae for the -correlated disorder were ob- (45) tained in Ref. 17. However, the magnetoconductance in Ref. 17 is expressed in terms of the 3D scattering rates. Here, the transport scattering time  should be taken tr This is somewhat predictable, since the -corrrelated dis- at the energy " = maxf; Tg. To give the reader an order has zero correlation length and the scattering rate idea of how well the interpolation formula represents the is not a ected by any other scale, including the magnetic exact result (44), we plot it in Fig. 3. Equations (44, length, which is responsible for the change in the geom- 45) reproduce the T dependence at ! 0, obtained in etry of the problem. Ref. 30 in the zero- eld limit. The interpolation formula For the disorder of the general type, with the correla- (45) for the conductivity  e ectively recasts it in the xx tion radius independent of the characteristic energy of the form of familiar Drude-type metallic expression host system, we have for the transport scattering time: 2 2 1 /  (1 + !  ) ; (46) tr c tr 2 3 u T 0 imp 1=3 (") = ; T = n v: (43) imp tr imp 2 2 2 g " (p v) where ! = =2" is the semiclassical cyclotron fre- 1 0 c quency at energy ". 2 2 Here, g = g(x )x dx is the numerical constant de- The same kind of Drude representation but with 3D termined by the type of disorder. This way, we obtain scattering times was obtained in Ref. 17 for -correlated the general expression for the longitudinal conductivity disorder. 7 FIG. 5: (Color online) Phase diagram for the conductivity  for the non-Coulomb disorder, see xx Eqs. (25) and (44). Here, g = 0 for  0, and g = =2 for 0 < < 1. See detailed explanations in the text. sea. dp FIG. 4: (Color online) Conductivity  given by xx n (H; ; T ) = f (" ) f (" + ) : 0 n n 2 2 4 v 2 Eq. (44) as a function of =T and the dimensionless parameter =T  H . Here we assume that   T . tr (48) In the (T; ) limit, we use the Euler{MacLaurin The conductivity  (H ) for di erent values of =T xx summation formula: are shown in Fig. 4. The behavior of the conductivity  can be conve- xx F (a) niently shown in the phase diagram (see Fig. 5). The up- F (a + n)  + F (x) dx: (49) per left red corner of this phase diagram corresponds to n=0 the ultra-quantum regime, T , where depending on Then, we obtain: the characteristic exponent of the impurity potential, we expect a -dependent scaling of  . The lower right xx 1 2 corner is divided into the regimes of weak and strong dis- 2 3 2 2 n (; T ) = +  +   T : (50) 2 3 order. The brown area corresponds to a strong disorder, 4 v 3 and is described by Eq.(45) in the Only the rst term in Eq. (50) is eld dependent. Despite 2 II maxfT; g= (47) its smallness, it is this term that contributes to  . This tr xy way, we arrive at the following expression: limit. One could also refer it to as a weak magnetic eld regime, where  exhibits predominantly the T xx II = : (51) xy dependence characteristic of a zero- eld system with a 2v correction proportional to H . The green area depicts the In experiments, the charge-carrier density is constant for opposite weak-disorder limit (or that of high magnetic each sample of WSM. Hence, the chemical potential is eld), where the transport of charge carriers is strongly almost eld independent in the high-temperature regime a ected by the magnetic eld. (see Discussion section for the relevant estimates). Next, we calculate the Hall conductivity. Next, we calculate the normal part of  given xy by (42). The conductivity  can be computed exactly xy at any value of  (see the corresponding integral derived C. Hall conductivity xy in the Appendix): As usual, the Hall conductivity is split into two parts: 2 2 e T anomalous and normal ones. Let us rst calculate the = f (a); xy 1 2 2 anomalous part. As before, we make use of Eq. (28). 2 4 a T a 1 i=T To regularize the expression for the charge-carrier den- 2 2 (1) f (a) = +  a Im + + ; sity, we subtract the respective density at zero chemical T 2 2 2 2 potential, thus eliminating the contribution of the Fermi (52) 8 where a is de ned in Eq. (44). As before, we concoct a Drude-type interpolation for- mula from ! 0 to maxfT; g using for- tr mula (52): 2 2 2 c e tr ; (53) xy 2 4 v tr 1 + 2 2 maxfT ; g 2(4) where c = 1 if   T and c = 7 =3 if   T . 1;2 1;2 Here  (") is taken at " = maxf; Tg. Depending on the tr factor ! , the normal part may be a leading or sublead- tr ing contribution to the Hall conductivity. Qualitatively, can be described by the following identity: xy d ; 1; > 1 tr maxfT;g 2 2 = d ; 1 xy 2 tr tr : 2 2 maxfT ; g maxfT;g d ; 3 2 tr (54) FIG. 7: (Color online) Conductivity  given by xy Eq. (44) as a function of =T and the dimensionless Here d are numerical factors following from rela- parameter =T  H . tr tions (51) and (53). The plot illustrating the accuracy of the interpolation formula (53) is presented in Fig. 6. V. DISCUSSION Interestingly, the case of Coulomb impurities does not lead to di erent results, despite the fact that the Debye In this paper we have performed a detailed analysis of radius depends on temperature. The relatively slow de- the e ect of the long range disorder introduced by im- cay of the Coulomb correlation function leads to a trivial purities on the magnetotransport in WSMs. Our study logarithmic enhancement of the respective 2D scattering is mainly focused on the magnetic eld and temperature rate: dependence of the transverse magnetoconductivity. Two important limiting cases are considered: (i) the ultra- 2 3 imp quantum limit, corresponding to low temperatures (or = 2 ln : (55) tr high magnetic eld), for which the main contribution comes from the zeroth Landau level, and (ii) the oppo- Otherwise, the whole temperature and eld dependence site semiclassical limit, when a large number of Landau remains the same. levels is involved in the transport phenomena. We have completely discarded the e ects of the intern- ode charge transfer. In principe, this e ect can be impor- tant at suciently high elds as argued in Ref. 31. The necessary condition for the applicability of our approach 1 1 is    , where for nding the scattering rates, inter intra we can use e.g. Eqs. (38) or (43). As derived in Ref. 31, 3=2 2 this condition is equivalent to H  eQ =v, where Q is the distance between the Weyl nodes in momentum space. 32{34 However, for a typical WSM like TaAs , we extract the separation between Weyl nodes as Q = 0:01 A , while the Fermi velocity v  3  10 m/s, which gives FIG. 6: (Color online) Hall conductivity  given by xy the respective eld estimate of H  50 T even for the Eq. (52) (solid line) and by Eq. (53) (dashed line) as a 2 ne structure constant in TaAs (unknown to us at the function of the dimensionless parameter =T  H . tr moment)  1. Therefore, we safely discard this e ect. The long-range impurity potential is chosen in a rather The plot of the Hall conductivity  is presented in general form. We show that in the ultra-quantum limit xy Fig. 7 as a function of the magnetic eld H and chemical the nonlinear magnetoresistivity dependence on the mag- potential . netic eld is the manifestation of the singular (non- 9 Coulomb) short-range behavior of the impurity potential ductivity undertaken in Refs 35{37. and its correlation function. In the semiclassical limit, we have demonstrated that unlike the short-range disorder case , the long-range disorder makes the scattering in the system essentially two-dimensional. We derived gen- eral formulae for  (H ) and  (H ) valid within a wide xx xy range of values of temperature and chemical potential. In typical experiments , the doping levels in WSMs are rather high ( 10 meV), which corresponds to 100 K. The typical magnetic elds H  1 T correspond to the gap between the zeroth and rst Landau levels FIG. 8: Electron self-energy 10 K. Reference 27, however, reports the observation of WSM in an almost undoped regime   T . Therefore, both the   T and   T regimes seem experimentally viable and the relation between ; H , and T can be quite Acknowledgments general. Hence, our results obtained in both limits can be relevant. We can also mention the numerical work in Ref. 20. In This work is partly supported by the Russian Foun- this paper, Coulomb impurities are correctly identi ed as dation for Basic Research, project Nos. 19-02-00421, the long-range disorder, and the high-temperature limit 19-02-00509, 20-02-00015, and 20-52-12013, by the joint is explored. The nontrivial result of Ref. 20 is the scaling research program of the Japan Society for the Promo- 5=3 of the magnetoconductance  / H in the low- eld tion of Science and the Russian Foundation for Basic xx regime . T . Our analytical study addresses the case Research, JSPS-RFBR Grants No. 19-52-50015 and No. , and the aforementioned regime cannot be ac- JPJSBP120194828, and by Deutsche Forschungsgemein- cessed in our semiclassical computation, where =T  1 schaft (DFG), grant No. EV 30/14-1. F.N. is sup- is the essential expansion parameter. ported in part by NTT Research, Army Research Of- In a semiclassical regime T , our results match the ce (ARO) (Grant No. W911NF-18-1-0358), Japan Sci- results reported in Ref. 16 in the low impurity concentra- ence and Technology Agency (JST) (via the CREST tion regime,   T , but di er in the opposite limit. Grant No. JPMJCR1676), Japan Society for the Pro- tr We attribute this to the e ect of the long-range disorder motion of Science (JSPS) (via the KAKENHI Grant No. correlations. JP20H00134), and Grant No. FQXi-IAF19-06 from the Note also in conclusion that the problem under study Foundational Questions Institute Fund (FQXi), a donor has a deep analogy with the detailed analysis of the ef- advised fund of the Silicon Valley Community Founda- fect of interplay of quantum interference and disorder on tion. Ya.I.R. acknowledges the support of the Russian magnetoresistance in the systems with the hopping con- Science Foundation, project No. 19-42-04137. Appendix A: Perturbation theory for T 1. Self-energy and Dyson series for the Green's function The diagram representing the rst-order correction to the Green's function is presented in Fig. 8 and is given by the following analytical expression dq dq dq 1 2 x y z 0 iq (x x ) G(x; x ) = e g(q ; q ) x yz (2) n;m;k (A1) y 1 1 y 2 2 0 S (x )G (p )S (x )S (x )G (p + q )S (x )S (x )G (p )S (x ): n p n z m m z z k k z y n p p +q m p +q p k p y y y y y y y The correction to the Green's function is a matrix, each term containing a product of  (x ) functions. l p It is essential that for large n we discard the di erence between n and n+1, when computing the correlation function. y y This allows us to treat the matrices S (x ); S (x ) as proportional to unit ones: S (x ); S (x ) =  (x ) 1. n p p n p p n p y n y y n y y The leading contribution to the conductivity comes from n  1 (in fact, we will later see that the contribution comes 2 26 from n  (T= ) ) and use the asymptotic relation for the Hermite polynomials : p p 2 2n 2 n x x =2 e H (x)  2 cos x 2n 1 : (A2) e 2 2n + 1 10 The question is how to simplify the product 1 2 1 2 iq (x x ) (x ) (x )e dq (A3) m m y p +q p +q y y y y entering (A1). We split the product of two cosine functions in each Hermite polynomial into p p 1 2 1 2 cos[ 2m(x x )=l ] + cos( 2m[(x + x )=l 2q l ]: (A4) H H y H p p p p y y y y Then, we perform integration over q . The rst term gives just the integral g(q ; q )dq =2 = g : (A5) x yz y 2;xz It is simply an e ective 2D potential (17). The second term is proportional to U ( 2ml ; q ). However, the correlation H yz p p radius of the potential obeys the inequality r  l  2ml . As a result, the term proportional to cos( 2m[(x + 0 H H x )=l 2q l ]) is suppressed. H y H Now we are able to perform the next estimate: dq 0 0 G(x; x )   (x ) (x ) G (p )G (p + q )G (p ) n p k n z m z z k z y p n;m;k p p 1=4 1 2 p cos[ 2nx =l ] cos[ 2kx =l ] 1 2 dq 1 1 4 H H x p p iq (x x ) 1 2 y y 1 2 e g (q ) p cos[ 2m(x x )=l ] q q dx dx : 2 xz H p p y y 1;2 2;2 2 l nk 4 4 l 2m x x H p p H y y 1 1 2n 2k (A6) In Eq. (A6), it is important to discern the di erence between the fast-oscillating cosine-type terms in the nominator and 1 2 1 2 1 2 2 slow algebraic factors in the denominator. To perform the integration over x ; x , we change x ; x ! r = x x ; x . We obtain many fast-oscillating terms (the relevant n; m and k are large). For example, performing integration over x , we obtain: p p p p p p 2 2 1 cos[ 2nr + ( 2n + 2k)x =l ] + cos[ 2nr + ( 2n 2k)x =l ] H H dx r r : 2;2 2 2 (A7) (r+x ) 4 p 4 p y y 1 1 2 2 2nl 2kl H H As we see from the structure of the integral of (A7), the nominator is a fast-oscillating function of x . As a result, the integral is suppressed unless n = k. Thus, the integral is /  in the main order for 1= n k. Let us compute nk it for n = k. The nominator does not oscillate any more, and we should analyze the denominator. The important 2 2 r  p , which comes from g (q ). On the other hand, the important x  nl . We see that r  x for the 2 xz H 0 p p y y denominator. Therefore, we integrate over x trivially. We are then left with the following expression: dq 0 0 G(x; x )   (x )) (x ) G (p )G (p + q )G (p ) n p n n z m z z n z y p n;m (A8) p p dq dr 1 iq r e g (q ) cos( 2mr=l ) cos( 2nr=l ): 2 xz H H l 2m While integrating over r, we obtain the combination of 4 -functions. The only relevant ones are (q + 2ml p p p 1 1 1 2nl ) and (q 2ml + 2nl ). Consequently, we have the following nal formula for the correction to the H H H Green's function: h i X p p dq 1 0 0 1 G(x; x )   (x ) (x ) G (p )G (p + q )G (p ) p g ( 2m 2n)l ; q : n p n n z m z z n z 2 z y p H (A9) 2l 2m n;m Using the fact that nl  T  p , we understand that m in the last sum is actually very close to n. Indeed, we see that: p p p p m n p v m n  p l  1; while p p   1: (A10) 0 H m + n T 11 From the last inequalities, we see that the terms of the sum over m are smooth functions of m and the sum can be 0 1 2 0 turned into an integral. Introducing the e ective momentum p = 2ml ; dm  1 = q dq l , and p = p + q , y y z z y H H z we obtain: dp 0 0 0 0 G(x; x )   (x ) G(p )G(p )G(p )g (p p )  (x ): (A11) n p n n 2 n n y p 2 y (2) Here, p is introduced in (12). The expression in the square brackets in (A11) allows us to build the ordinary 2D Dyson series for the Green's function as well as vertex functions determining the conductivity tensor. Indeed, it coincides with the standard expression of perturbation theory without magnetic eld with the e ective 2D potential g (p p ). Therefore, we can write the momentum-dependent self-energy as: 2 n dp 0 0 (p ) = G(p )g (p p ): (A12) n 2 n (2) The resummed Green's function then reads: 0 1 1 0 G(x; x )   (x )[G (p ) (p )]  (x ): (A13) n p n n n y p The resulting expression yields an irrelevant part, which can be absorbed into the renormalized chemical potential and Fermi velocity, and the dissipative part. The imaginary part reads: Z Z 0 0 0 0 dp " + p  dp 1 p R 0 0 (p ) = v:p: g (p p ) i ((" " ) + (" + " )) + g (p p ) n 2 n n n 2 n 2 2 2 2 (2) (" + i0) p (2) 2 2" i in =  + vp  ; 2 2 Z Z 1 p 1 " 0 0 0 0 0 = g (nn )dn ; = (nn )g (nn )dn : 2 2 4  4 (A14) 2. Vertex renormalization and conductivity tensor The Dyson equation for the vertex is built in a more subtle way. In this case, the built-in magnetic anisotropy of the problem takes its toll. What we are going to do now is to introduce a slightly unusual de nition of the vertex. We de ne the mass-shell vertex according to the following equation: dp dp p z y z R A () =  cos  hG 0 ("; p)  G 0 ("; p)i dx: (A15) x x x;x x ;x (2) p Then, the conductivity tensor assumes the form in Eq. (40). In the zeroth-order ladder approximation, we change R A R A hG G i = hG ihG i, and the vertex becomes: X R A 1 dp 0 G G 0 11;n 22;n+1 = : (A16) x R A G G 0 2l 2 22;n 11;n1 Changing the sum and the integral using the semiclassical approximation Z Z dp p p dp z z cos  = l ; (A17) 2 p 2 we obtain: 1 1 i 0 [ + + ] RA;0   " = 2 : (A18) 1 1 i 3 1 2v [ + ] 0 In the rst order of perturbation theory, the picture changes slightly, and we obtain the rst stair of the ladder series 12 0 1 1 1 1 2 0 h i 2 2 1 1 i B C n + + RA;1   " B C = : (A19) 1 1 3 @ + A 2v 1 2 h i 1 1 i In the higher orders of the perturbation theory the pattern repeats itself. As a result, we are able to perform the full summation: 0 1 h i 1 i 0 + y B  " C tr RA = h i : (A20) @ A x 1 3 2 2v 1 i tr Finally, we are ready to obtain the expression for the conductivity: Z Z 3 2 d" df (") d sin  1 i = Re[Im] + : (A21) x;x[y] d" 2  " tr Appendix B: Calculation of the integral for the The last integral is equal to conductivity at T The integral that enters the upper matrix element in Z 1 dx 2 1 a i the expressions for the conductivities in (42) has the = Re + : 2 x 2 2 (x + ) + a a 2 2 cosh following form: (B1) 1 x dx I (a) = 2 x 2 2 x + a cosh 1 1 Z Z 2 2 2 x +  a 1 dx As a result, we recover Eq. (44) In the same manner we = dx + a : 2 x 2 x 2 2 perform the integration for  in the lower part of (42) (x + ) + a cosh cosh xy 2 2 1 1 to obtain the exact expression (52). P. Hosur and X. Qi, Recent developments in transport phe- NbP, Nature Phys. 11, 645 (2015). nomena in Weyl semimetals, C. R. Phys. 14, 857 (2013). C.Z. Li, L.X. Wang, H.W. Liu, J. Wang, Z.M. Liao, and Hai-Zhou Lu and Shun-Qing Shen, Quantum transport in D.P. Yu, Giant negative magnetoresistance induced by the topological semimetals under magnetic elds, Front. Phys. chiral anomaly in individual Cd As nanowires, Nature 3 2 12, 127201 (2017). Commun. 6, 10137 (2015). 3 8 D.T. Son and B.Z. Spivak, Chiral anomaly and classical A.V. Andreev and B.Z. Spivak, Longitudinal negative mag- negative magnetoresistance of Weyl metals, Phys. Rev. B netoresistance and magnetotransport phenomena in con- 88, 104412 (2013). ventional and topological conductors, Phys. Rev. Lett. A.A. Burkov, Chiral anomaly and di usive magnetotrans- 120, 026601 (2018). port in Weyl metals, Phys. Rev. Lett. 113, 247203 (2014). H.Z. Lu and S.Q. Shen, Weak antilocalization and localiza- Xiaochun Huang, Lingxiao Zhao, Yujia Long, Peipei Wang, tion in disordered and interacting Weyl semimetals, Phys. Dong Chen, Zhanhai Yang, Hui Liang, Mianqi Xue, Hong- Rev. B 92, 035203 (2015). ming Weng, Zhong Fang, Xi Dai, and Genfu Chen, Ob- H.-Z. Lu and S.-Q. Shen, Weak antilocalization and servation of the chiral-anomaly-induced negative magne- interaction-induced localization of Dirac and Weyl toresistance in 3D Weyl semimetal TaAs, Phys. Rev. X 5, fermions in topological insulators and semimetals, Chin. 031023 (2015) Phys. B 25, 117202 (2016). 6 11 C. Shekhar, A.K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, Tian Liang, Quinn Gibson, Mazhar N. Ali, Minhao Liu, I. Leermakers, U. Zeitler, W. Schnelle, J. Grin, C. Felser, R.J. Cava, and N.P. Ong, Ultrahigh mobility and giant and B. Yan, Extremely large magnetoresistance and ultra- magnetoresistance in the Dirac semimetal Cd As , Nature 3 2 high mobility in the topological Weyl semimetal candidate Mater. 14, 280 (2015). 13 Junya Feng, Yuan Pang, Desheng Wu, Zhijun Wang, 1972). Hongming Weng, Jianqi Li, Xi Dai, Zhong Fang, Youguo P.G. LaBarre, L. Dong, J. Trinh, T. Siegrist, and A.P. Shi, and Li Lu, Large linear magnetoresistance in Dirac Ramirez, Evidence for undoped Weyl semimetal charge semimetal Cd As with Fermi surfaces close to the Dirac transport in Y Ir O , J. Phys.: Condens. Matter 32, 3 2 2 2 7 points, Phys. Rev. B 92, 081306(R) (2015). 02LT01 (2019). 13 28 Mario Novak, Satoshi Sasaki, Kouji Segawa, and Yoichi Z. K. Liu, J. Jiang, B. Zhou, Z.J. Wang, Y. Zhang, H.M. Ando, Large linear magnetoresistance in the Dirac Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, et semimetal TlBiSSe, Phys. Rev. B 91, 041203(R) (2015). al., A stable three-dimensional topological Dirac semimetal Y. Zhao, H. Liu, C. Zhang, H. Wang, J. Wang, Z. Lin, Y. Cd As , Nature Mater. 13, 677 (2014). 3 2 Xing, H. Lu, J. Liu, Y. Wang, S.M. Brombosz, Z. Xiao, J.-P. Jay-Gerin, M.J. Aubin, and L.G. Caron, The electron S. Jia, X.C. Xie, and J. Wang, Anisotropic Fermi sur- mobility and the static dielectric constant of Cd As at 4.2 3 2 face and quantum limit transport in high mobility three- K, Solid State Commun. 21, 771 (1977). dimensional Dirac semimetal Cd As , Phys. Rev. X 5, S.D. Sarma, E.H. Hwang, and H. Min, Carrier screening, 3 2 031037 (2015). transport, and relaxation in 3D Dirac semimetals, Phys. A.A. Abrikosov, Quantum magnetoresistance, Phys. Rev. Rev. B 91, 035201 (2015). B 58, 27881 (1998). G. Bednik, K.S. Tikhonov, S.V. Syzranov, Magnetotrans- J. Klier, I.V. Gornyi, and A.D. Mirlin, Transversal magne- port and internodal tunnelling in Weyl semimetals, Phys. toresistance in Weyl semimetals, Phys. Rev. B 92, 205113 Rev. Research 2, 023124 (2020). (2015). S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, J. Klier, I.V. Gornyi, and A.D. Mirlin, Transversal magne- C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.- toresistance and Shubnikov{de Haas oscillations in Weyl M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, semimetals, Phys. Rev. B 96, 214209 (2017). A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and D.A. Pesin, E.G. Mishchenko, and A. Levchenko, Density M. Z. Hasan, Discovery of a Weyl fermion semimetal and of states and magnetotransport in Weyl semimetals with topological Fermi arcs, Science 349, 613 (2015). long-range disorder, Phys. Rev. B 92, 174202 (2015). B.Q. Lv, N. Xu, H.M. Weng, J.Z. Ma, P. Richard, X.C. Xiao Xiao, K.T. Law, and P.A. Lee, Magnetoconductiv- Huang, L.X. Zhao, G.F. Chen, C.E. Matt, F. Bisti, V.N. ity in Weyl semimetals: E ect of chemical potential and Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and temperature, Phys. Rev. B 96, 165101 (2017). H. Ding, Observation of Weyl nodes in TaAs, Nature Phys. Justin C.W. Song, Gil Refael, and Patrick A. Lee, Linear 11, 724 (2015). magnetoresistance in metals: Guiding center di usion in B.J. Ramshaw, K.A. Modic, Arkady Shekhter, Yi Zhang, a smooth random potential, Phys. Rev. B 92, 80204(R) Eun-Ah Kim, Philip J.W. Moll, Maja D. Bachmann, M.K. (2015). Chan, J.B. Betts, F. Balakirev, A. Migliori, N.J. Ghimire, V. K onye and M. Ogata, Magnetoresistance of a three- E.D. Bauer, F. Ronning, and R.D. McDonald, Quantum dimensional Dirac gas, Phys. Rev. B 98, 195420 (2018). limit transport and destruction of the Weyl nodes in TaAs, V. K onye and M. Ogata, Thermoelectric transport coe- Nature Commmun. 9, 2217 (2018). cients of a Dirac electron gas in high magnetic elds, Phys. Y.L. Lin and F. Nori, Strongly localized electrons in a Rev. B 100, 155430 (2019). magnetic eld: Exact results on quantum interference and Ya.I. Rodionov, K.I. Kugel, and Franco Nori, E ects magnetoconductance Phys. Rev. Lett. 76, 4580 (1996). of anisotropy and disorder on the conductivity of Weyl Y.L. Lin and F. Nori, Quantum interference from sums semimetals, Phys. Rev. B 92, 195117 (2015). over closed paths for electrons on a three-dimensional lat- Ya. I. Rodionov and S. V. Syzranov, Conductivity of a tice in a magnetic eld: Total energy, magnetic moment, Weyl semimetal with donor and acceptor impurities, Phys. and orbital susceptibility, Phys. Rev. B 53, 13374 (1996). Rev. B 91, 195107 (2015). Y.L. Lin and F. Nori, Analytical results on quantum in- P. St reda, Theory of quantised Hall conductivity in two terference and magnetoconductance for strongly localized dimensions, J. Phys. C: Solid State Phys. 15, L717 (1982). electrons in a magnetic eld: Exact summation of forward- M. Abramowitz and I.A. Stegun, Handbook of Mathemat- scattering paths, Phys. Rev. B 53, 15543 (1996). ical Functions with Formulas, Graphs, and Mathematical Tables (US Government Printing Oce, Washington, D.C.,

Journal

Condensed Matter – arXiv (Cornell University)

Published: May 18, 2020

There are no references for this article.