Energy scales and magnetoresistance at a quantum critical point
V.R. Shaginyan, M.Ya. Amusia, A.Z. Msezane, K.G. Popov, V.A. Stephanovich
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Energy scales and magnetoresistance at a quantum critical point
V.R. Shaginyan,
1, 2, 3, ∗ M.Ya. Amusia, A.Z. Msezane, K.G. Popov, and V.A. Stephanovich Petersburg Nuclear Physics Institute, RAS, Gatchina, 188300, Russia Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel CTSPS, Clark Atlanta University, Atlanta, Georgia 30314, USA Komi Science Center, Ural Division, RAS, 3a, Chernova street Syktyvkar, 167982, Russia Opole University, Institute of Mathematics and Informatics, Opole, 45-052, Poland
The magnetoresistance (MR) of CeCoIn is notably different from that in many conventional met-als. We show that a pronounced crossover from negative to positive MR at elevated temperaturesand fixed magnetic fields is determined by the scaling behavior of quasiparticle effective mass. At aquantum critical point (QCP) this dependence generates kinks (crossover points from fast to slowgrowth) in thermodynamic characteristics (like specific heat, magnetization etc) at some tempera-tures when a strongly correlated electron system transits from the magnetic field induced LandauFermi liquid (LFL) regime to the non-Fermi liquid (NFL) one taking place at rising temperatures.We show that the above kink-like peculiarity separates two distinct energy scales in QCP vicinity -low temperature LFL scale and high temperature one related to NFL regime. Our comprehensivetheoretical analysis of experimental data permits to reveal for the first time new MR and kinksscaling behavior as well as to identify the physical reasons for above energy scales. PACS numbers: 71.27.+a, 73.43.Qt, 64.70.Tg
Keywords: Quantum criticality; Heavy-fermion metals; Magnetoresistance
INTRODUCTION
An explanation of rich and striking behavior ofstrongly correlated electron system in heavy fermion(HF) metals is, as years before, among the main problemsof condensed matter physics. One of the most interestingand puzzling issues in the research of HF metals is theiranomalous normal-state transport properties. Measure-ments of magnetoresistance (MR) on CeCoIn [1, 2] haveshown that it is notably different from ordinary weak-field orbital MR described by Kohler’s rule which holdsin many conventional metals, see e.g. [3]. At fixed mag-netic fields B , MR of CeCoIn exhibits a crossover fromnegative (low temperatures) to positive (hight tempera-tures) one at temperature growth [1, 2]. This crossover ishard to explain within both conventional Fermi liquid ap-proach for metals and in terms of Kondo systems [4]. Toexplain this effect, it has been assumed that the crossovercan be attributed to some distinct energy scales revealedby kinks (crossover points from fast to slow growth) inthermodynamic characteristics (like specific heat, magne-tization etc) and leading to a change of spin fluctuationscharacter with increasing of the applied magnetic fieldstrength [1, 2, 4, 5, 6, 7, 8, 9].Here we investigate the NFL-LFL transition region(we call it below crossover region), where MR changesits sign. The modified Kohler’s rule (MR versus tan-gent of Hall angle) have been utilized to describe MRdata [8, 9]. In this region, both Kohler’s rule and itsmodified version do not work. In Landau Fermi liquid(LFL) regime, the quasiparticles were observed in mea-surements of transport properties in CeCoIn [10]. Ananalysis of above thermodynamic quantities shows thatquasiparticles exist in both LFL and crossover regimes when strongly correlated Fermi systems like HF met-als or two-dimensional (2D) He [11, 12, 13, 14, 15, 16]transit from LFL to NFL behavior. It is of crucial im-portance to verify whether quasiparticles with effectivemass M ∗ still exist and determine the transport prop-erties and energy scales in HF metals in crossover re-gion. On the other hand, even early measurements onHF metals gave evidences in favor of the quasiparticlesexistence. For example, the application of magnetic field B restores LFL behavior of HF metals which demon-strate NFL properties in the absence of the field . Inthat case the empirical Kadowaki-Woods (KW) ratio K is conserved, K = A ( B ) /γ ( B ) ∝ A ( B ) /χ ( B ) = const [17, 18, 19] where γ = C/T , C is a heat capacity, χ isa magnetic susceptibility and A ( B ) is a coefficient de-termining the temperature dependence of the resistivity ρ = ρ + A ( B ) T . Here ρ is the residual resistance.The observed conservation of K can be hardly interpretedwithin scenarios when quasiparticles are suppressed, forthere is no reason to expect that γ ( B ), χ ( T ), A ( B ) andother transport and thermodynamic quantities like ther-mal expansion coefficient α ( B ) are affected by the fluc-tuations or localization in a correlated fashion. As wewill see below, the MR measurements in the crossoverregion can present indicative data on the quasiparticlesavailability. Such MR measurements were carried outin CeCoIn when the system transits from LFL to NFLregime at elevated temperatures and fixed magnetic fields[1, 2].In this Letter, we analyze MR of CeCoIn and showthat the crossover from negative to positive MR at ele-vated temperatures and fixed magnetic fields can be wellcaptured utilizing fermion condensation quantum phasetransition (FCQPT) concept based on the quasiparti-cles paradigm [12, 20, 21, 22]. We demonstrate thatthe crossover is regulated by the universal behavior ofthe effective mass M ∗ ( B, T ) observed in many HF met-als. It is exhibited by M ∗ ( B, T ) when HF metal transitsfrom LFL regime (induced by the application of magneticfield) to NFL one taking place at rising temperatures.The above behavior of the effective mass also generateskinks (crossover points from fast to slow growth at el-evated temperatures) in thermodynamic characteristics(like specific heat, magnetization etc). We show that theabove kink-like peculiarity separates two distinct energyscales - low temperature LFL scale and high tempera-ture one related to NFL regime. Our calculations of MRare in good agreement with observations and allow us toreveal new scaling behavior of both MR and the kinks.
SCALING BEHAVIOR OF THE KINKS
To study universal low temperature features of HFmetals, we use the model of homogeneous heavy-fermionliquid with the effective mass M ∗ ( T, B, x ), where x = p F / π is a number density and p F is Fermi momentum[23]. This model permits to avoid complications associ-ated with the crystalline anisotropy of solids [13]. Wefirst outline the case when at T → x . Atelevated temperatures the system transits to the NFLstate. The dependence M ∗ ( T, x ) is governed by Landauequation [23]1 M ∗ ( T, x ) = 1 M + Z p F p p F F ( p F , p ) ∂n ( p , T, x ) ∂p d p (2 π ) , (1)where n ( p , T, x ) is the distribution function of quasipar-ticles and F ( p F , p ) is Landau interaction amplitude, M is a free electron mass. At T = 0, eq. (1) reads [23] M ∗ /M = 1 / (1 − N F ( p F , p F ) / N is the den-sity of states of a free electron gas, F ( p F , p F ) is the p -wave component of Landau interaction amplitude F .Taking into account that x = p F / π , we rewrite the am-plitude as F ( p F , p F ) = F ( x ). When at some criticalpoint x = x F C , F ( x ) achieves certain threshold value,the denominator tends to zero and the system under-goes FCQPT related to divergency of the effective mass[12, 16, 20, 22], M ∗ ( x ) M = A + Bx F C − x . (2)Equation (2) is valid in both 3D and 2D cases, while thevalues of factors A and B depend on the dimensionality.The approximate solution of Eq. (1) is of the form [14] MM ∗ ( T ) = MM ∗ ( x ) + βf (0) ln { − /β ) } + λ β + λ β + ..., (3) where λ > λ < β = T M ∗ ( T ) /p F and f (0) ∼ F ( x F C ). It follows fromEq. (3) that the effective mass M ∗ as a function of T and x reveals three different regimes at growing temper-ature. At the lowest temperatures we have LFL regimewith M ∗ ( T, x ) ≃ M ∗ ( x ) + aT with a < λ > T decays down to aminimum and after grows, reaching its maximum M ∗ M at some temperature T M ( x ) then subsequently dimin-ishing as T − / [11, 12]. Moreover, the closer is thenumber density x to its threshold value x F C , the higheris the rate of the growth. The peak value M ∗ M growsalso, but the maximum temperature T M lowers. Nearthis temperature the last ”traces” of LFL regime disap-pear, manifesting themselves in the divergence of abovelow-temperature series and substantial growth of M ∗ ( x ).Numerical calculations based on Eqs. (1) and (3) showthat at rising temperatures T > T / ( T / is a charac-teristic temperature determining the validity of regime(4), see Ref. [14] for details) the linear term ∝ β givesthe main contribution and leads to new regime when Eq.(3) reads M/M ∗ ( T ) ∝ β yielding M ∗ ( T ) ∝ T − / . (4)We remark that Eq. (4) ensures that at T ≥ T / theresistivity behaves as ρ ( T ) ∝ T [12]. Near the critical ~ T -1/2 ~ T -2/3 N o r m a li z e d m ass Normalized temperature
Transition region
NFL
LFL
Control parameter,
SC,FM,AFM T e m p e r a t u r e , a r b . un i t s NFL FC LFL
FIG. 1: Schematic phase diagram of the systems under consid-eration. Control parameter ζ represents number density (ordoping) x , magnetic field B , pressure P etc. ζ FC denotes thepoint of effective mass divergence. SC , FM , AFM denote thesuperconducting, ferromagnetic and antiferromagnetic states,respectively. The vertical arrow shows LFL-NFL transitionsat rising temperatures and fixed ζ . Inset shows a schematicplot of the normalized effective mass M ∗ N = M ∗ ( T /T M ) /M ∗ M ( M ∗ M is its maximal value at T = T M ) versus the normalizedtemperature T N = T /T M . Several regions are shown. Firstgoes the LFL regime ( M ∗ N ( T N ) ∼ const) at T N ≪
1, thenthe transition regime (the hatched area) where M ∗ N reachesits maximum. At elevated temperatures T − / regime occursfollowed by T − / behavior, see Eq. (4). point x F C ( M/M ∗ ( x → x F C ) → M ∗ in units of M (as M/M ∗ ( x → x F C ) →
0) and we have to measure M ∗ in units of M ∗ M and T in units of T M . Latter scales can be viewed asnatural ones.The schematic phase diagram of HF liquid is reportedFig. 1. The control parameter ζ can be pressure P , mag-netic field B , or doping (density) x . At ζ = ζ F C , FCQPTtakes place leading to a strongly degenerated state. Thisstate is captured by the superconducting (SC), ferromag-netic (FM), antiferromagnetic (AFM) etc. states liftingthe degeneracy [12]. The variation of ζ drives the sys-tem from NFL region to LFL one. For example, in thecase of magnetic field B , ζ F C = B c , where B c is a crit-ical magnetic field, such that at B > B c the system isdriven towards its LFL regime. Below we consider thecase with ζ > ζ F C when the system is on the LFL sideof FCQPT. The inset demonstrates the behavior of thenormalized effective mass M ∗ N = M ∗ /M ∗ M versus normal-ized temperature T N = T /T M . Both T − / and T − / regimes are marked as NFL ones since the effective massdepends strongly on temperature. The temperature re-gion T ≃ T M signifies the crossover between the LFLregime with almost constant effective mass and NFL be-havior, given by T − / dependence. Thus temperatures T ∼ T M can be regarded as the crossover region betweenLFL and NFL regimes.It turns out that M ∗ ( T, x ) in the entire T ≤ T / rangecan be well approximated by a simple universal inter-polating function [11, 12, 14]. The interpolation occursbetween the LFL ( M ∗ ∝ T ) and NFL ( M ∗ ∝ T − / ,see Eq. (4)) regimes thus describing the above crossover.Introducing the dimensionless variable y = T N = T /T M ,we obtain the desired expression M ∗ ( T /T M , x ) M ∗ M = M ∗ N ( y ) ≈ M ∗ ( x ) M ∗ M c y c y / . (5)Here M ∗ N ( y ) is the normalized effective mass, c and c are parameters, obtained from the condition of best fit toexperiment. To correct the behavior of M ∗ N ( y ) at risingtemperatures M ∗ ∼ T − / , we add a term to Eq. (5)and obtain M ∗ N ( y ) ≈ M ∗ ( x ) M ∗ M (cid:20) c y c y / + c exp( − /y ) √ y (cid:21) , (6)where c is a parameter. The last term on the right handside of Eq. (6) makes M ∗ N satisfy Eq. (4) at temperatures T /T M > B (that means that Zeemansplitting is small), the effective mass does not depend onspin variable and B enters Eq. (1) as Bµ B /T ( µ B isBohr magneton) making T M ∝ Bµ B [11, 12, 14]. The application of magnetic field restores the LFL behavior,and at T ≤ T M the effective mass depends on B as [11,12] M ∗ ( B ) ∝ ( B − B c ) − / . (7)Note that in some cases B c = 0. For example, the HFmetal CeRu Si is characterized by B c = 0 and showsneither evidence of the magnetic ordering or supercon-ductivity nor the LFL behavior down to the lowest tem-peratures [24]. In our simple model B c is taken as aparameter. We conclude that under the application ofmagnetic field the variable y = T /T M ∝ Tµ B ( B − B c ) (8)remains the same and the normalized effective mass isagain governed by Eqs. (5) and (6) which are the finalresult of our analytical calculations. We note that theobtained results are in agreement with numerical calcu-lations [11, 12].The normalized effective mass M ∗ N ( y ) can be ex-tracted from experiments on HF metals. For example, M ∗ ( T, B ) ∝ C ( T ) /T ∝ S ( T ) /T ∝ χ AC ( T ), where S ( T )is the entropy, C ( T ) is the specific heat and χ AC ( T ) isac magnetic susceptibility [11, 12]. If the correspondingmeasurements are carried out at fixed magnetic field B (or at fixed x and B ) then, as it seen from Fig. 1, theeffective mass reaches the maximum at some tempera-ture T M . Upon normalizing both the effective mass byits peak value M ∗ M at each field B and the temperatureby T M , we observe that all the curves merge into a sin-gle one, given by Eqs. (5) and (6) thus demonstrating ascaling behavior.To verify Eq. (4), we use measurements of χ AC ( T )in CeRu Si at B = 0 .
02 mT at which this HF metaldemonstrates the NFL behavior [24]. It is seen from Fig.2 that Eq. (4) gives good description of the facts in theextremely wide range of temperatures. The inset to Fig.2 exhibits a fit for M ∗ N ( y ) extracted from measurementsof χ AC ( T ) at different magnetic fields, clearly indicatingthat the function given by Eq. (5) represents a goodapproximation for M ∗ N ( y ) when the system transits fromthe LFL regime to NFL one. M ∗ N ( y ) extracted from the entropy S ( T ) /T and mag-netization M measurements on the He film [25] at dif-ferent densities x is reported in the left panel of Fig. 3.In the same panel, the data extracted from the heat ca-pacity of the ferromagnet CePd . Rh . [26] and the ACmagnetic susceptibility of the paramagnet CeRu Si [24]are plotted for different magnetic fields. It is seen thatthe universal behavior of the normalized effective massgiven by Eq. (5) and shown by the solid curve is in ac-cord with the experimental facts. All 2D He substancesare located at ζ > ζ
F C (see Fig. 1), where the systemprogressively disrupts its LFL behavior at elevated tem-peratures. In that case the control parameter, driving
Normalized temperature N o r m a li z ed da t a CeRu Si AC susceptibility
B=0.2 mT B=0.39 mT B=0.94 mT
CeRu Si AC susceptibilityCeRu Si , B=0.02 mT ( T ) [ a r b . U n i t] T[mK]
FIG. 2: Temperature dependence of the ac susceptibility χ AC for CeRu Si . The solid curve is a fit for the data shownby the triangles at B = 0 .
02 mT and represented by thefunction χ ( T ) = a/ √ T given by Eq. (4) with a being a fittingparameter. Inset shows the normalized effective mass versusnormalized temperature y extracted from χ AC measured atdifferent fields as indicated in the inset [24]. The solid curvetraces the universal behavior of M ∗ N ( y ) determined by Eq.(5). Parameters c and c are adjusted to fit the averagebehavior of the normalized effective mass M ∗ N ( y ). the system towards its quantum critical point (QCP) ismerely the number density x . It is seen that the behav-ior of M ∗ N ( y ), extracted from S ( T ) /T and magnetization M of 2D He looks very much like that of 3D HF com-pounds. In the right panel of Fig. 3, the normalized dataon C ( y ), S ( y ), yχ ( y ) and M ( y ) + yχ ( y ) extracted fromdata collected on CePd − x Rh x [26] , He [25], CeRu Si [24], CeCoIn [27] and YbRu Si [7] respectively are pre-sented. Note that in the case of YbRu Si , the variable y = ( B − B c ) µ B /T M can be viewed as effective normal-ized temperature. As seen from Eq. (5), this representa-tion of the variable y is correct when the temperature isa fixed parameter.It is seen from the right panel of Fig. 3 that all thedata exhibit the kink (shown by arrow) at y ≥ y ) is characterized by the fast growth andthe high temperature one related to the NFL behavioris characterized by the slow growth. As a result, we canidentify the energy scales near QCP, discovered in Ref.[7]: the thermodynamic characteristics exhibit the kinks(crossover points from the fast to slow growth at elevatedtemperatures) which separate the low temperature LFLscale and high temperature one related to NFL regime. N o r m a li z ed S , y , M , C CeCoIn , C B=8T B=9T He, S x=8.0nm -2 x=8.25nm -2 x=8.5nm -2 x=8.75nm -2 -2 CeRu Si , y (y) B=0.94mTYbRh Si , MT T=0.5KCePd Rh , C B=0.5T B=1.0T B=3.0T CeCoIn , C/T B=8T B=9T He , M 8.00 nm -2 -2 -2 CeRu Si , B=0.94mTHe , S/T 8.00 nm -2 -2 -2 -2 -2 CePd Rh , C/T B=0.5 T B=1 T B=3 T N o r m a li z ed m a ss Normalized temperature
FIG. 3: The left panel. The normalized effective mass M ∗ N versus the normalized temperature y = T /T M . The depen-dence M ∗ N ( y ) is extracted from measurements of S ( T ) /T andmagnetization M on 2D He [25]), from ac susceptibility χ AC ( T ) collected on CeRu Si [24] and from C ( T ) /T col-lected on CePd − x Rh x [26]. The data are collected for differ-ent densities and magnetic fields shown in the left bottom cor-ner. The solid curve traces the universal behavior of the nor-malized effective mass determined by Eq. (5). Parameters c and c are adjusted for χ N ( T N , B ) at B = 0 .
94 mT. The rightpanel. The normalized specific heat C ( y ) of CePd − x Rh x atdifferent magnetic fields B , normalized entropy S ( y ) of Heat different number densities x , and the normalized yχ ( y ) at B = 0 .
94 mT versus normalized temperature y are shown.The upright triangles depict the normalized ‘average’ magne-tization M + Bχ collected on YbRu Si [7]. The kink (shownby the arrow) in all the data is clearly seen in the transi-tion region y ≥
1. The solid curve represents yM ∗ N ( y ) withparameters c and c adjusted for magnetic susceptibility ofCeRu Si at B = 0 .
94 mT.
SCALING BEHAVIOR OF THEMAGNETORESISTANCE
By definition, MR is given by ρ mr ( B, T ) = ρ ( B, T ) − ρ (0 , T ) ρ (0 , T ) , (9)We apply Eq. (9) to study MR of strongly correlated elec-tron liquid versus temperature T as a function of mag-netic field B . The resistivity ρ ( B, T ) is ρ ( B, T ) = ρ + ∆ ρ ( B, T ) + ∆ ρ L ( B, T ) , (10)where ρ is a residual resistance, ∆ ρ = c AT , c is aconstant, A is a coefficient determining the temperaturedependence of the resistivity ρ = ρ + AT . The classicalcontribution ∆ ρ L ( B, T ) to MR due to orbital motion ofcarriers induced by the Lorentz force obeys the Kohler’srule [3]. We note that ∆ ρ L ( B ) ≪ ρ (0 , T ) as it is assumedin the weak-field approximation. To calculate A , we usethe quantities γ = C/T ∝ M ∗ and/or χ ∝ M ∗ as well asemploy the fact that Kadowaki-Woods ratio K = A/γ ∝ A/χ = const [17]. As a result, we obtain A ∝ ( M ∗ ) [17, 18, 19], so that ∆ ρ ( B, T ) = c ( M ∗ ( B, T )) T and c isa constant. Suppose that the temperature is not very low,so that ρ ≤ ∆ ρ ( B = 0 , T ), and B ≥ B c . Substituting(10) into (9), we find that [28] ρ mr ≃ ∆ ρ L ρ + cT ( M ∗ ( B, T )) − ( M ∗ (0 , T )) ρ (0 , T ) . (11)Consider the qualitative behavior of MR described byEq. (11) as a function of B at a certain temperature T = T . In weak magnetic fields, when T ≥ T / and thesystem exhibits NFL regime (see Fig. 1), the main con-tribution to MR is made by the term ∆ ρ L ( B ), becausethe effective mass is independent of the applied magneticfield. Hence, | M ∗ ( B, T ) − M ∗ (0 , T ) | /M ∗ (0 , T ) ≪ ρ L ( B ). As a result,MR is an increasing function of B . When B becomes sohigh that T M ( B ) ∼ µ B ( B − B c ) ∼ T , the difference( M ∗ ( B, T ) − M ∗ (0 , T )) becomes negative and MR as afunction of B reaches its maximal value at T M ( B ) ∼ T when the kink occurs, see the right panel of Fig. 3. Atfurther increase of magnetic field, when T M ( B ) > T , theeffective mass M ∗ ( B, T ) becomes a decreasing functionof B , as follows from Eq. (7). As B increases,( M ∗ ( B, T ) − M ∗ (0 , T )) M ∗ (0 , T ) → − , (12)and the magnetoresistance, being a decreasing functionof B , is negative.Now we study the behavior of MR as a function of T at fixed value B of magnetic field. At low tempera-tures T ≪ T M ( B ), it follows from Eqs. (5) and (7) that M ∗ ( B , T ) /M ∗ (0 , T ) ≪
1, and it is seen from Eq. (12)that ρ mr ( B , T ) ∼ −
1, because ∆ ρ L ( B , T ) /ρ (0 , T ) ≪
1. We note that B must be relatively high to guar-antee that M ∗ ( B , T ) /M ∗ (0 , T ) ≪
1. As the tem-perature increases, MR increases, remaining negative.At T ≃ T M ( B ), MR is approximately zero, because ρ ( B , T ) ≃ ρ (0 , T ) at this point. This allows us to con-clude that the change of the temperature dependence ofresistivity ρ ( B , T ) from quadratic to linear manifests it-self in the transition from negative to positive MR. Onecan also say that the transition takes place when the kinkoccurs (as shown by the arrow in the right panel of Fig.3) and the system goes from the LFL behavior to theNFL one. At T ≥ T M ( B ), the leading contribution toMR is made by ∆ ρ L ( B , T ) and MR reaches its maxi-mum. At T M ( B ) ≪ T , MR is a decreasing function ofthe temperature, because | M ∗ ( B, T ) − M ∗ (0 , T ) | M ∗ (0 , T ) ≪ , (13)and ρ mr ( B , T ) ≪
1. Both transitions (from positive tonegative MR with increasing B at fixed temperature T and from negative to positive MR with increasing T at B=16T B=12T B=10T B=8T B=6T
CeCoIn (B,T)/[ (B=0,T)R M (B)] N o r m a li z ed m agne t o r e s i s t an c e Normalized temperature
FIG. 4: The normalized magnetoresistance R ρN ( y ) given byEq. (15) versus normalized temperature y = T /T Rm . R ρN ( y )was extracted from MR shown in Fig. 6 and collected onCeCoIn at fixed magnetic fields B [1] listed in the right uppercorner. The starred line represents our calculations based onEqs. (5) and (15) with the parameters extracted from ac susceptibility of CeRu Si (see the caption to Fig. 2). Thesolid line displays our calculations based on Eqs. (6) and(15); only one parameter was used to fit the data, while theother were extracted from the ac susceptibility measured onCeRu Si . fixed B value) have been detected in measurements ofthe resistivity of CeCoIn in a magnetic field [1].Let us turn to quantitative analysis of MR. As it wasmentioned above, we can safely assume that the classicalcontribution ∆ ρ L ( B, T ) to MR is small as compared to∆ ρ ( B, T ). Omission of ∆ ρ L ( B, T ) allows us to makeour analysis and results transparent and simple since thebehavior of ∆ ρ L ( B , T ) is not known in the case of HFmetals. Consider the ratio R ρ = ρ ( B, T ) /ρ (0 , T ) andassume for a while that the residual resistance ρ is smallin comparison with the temperature dependent terms.Taking into account Eq. (10) and ρ (0 , T ) ∝ T , we obtainfrom Eq. (11) R ρ = ρ mr + 1 = ρ ( B, T ) ρ (0 , T ) ∝ T ( M ∗ ( B, T )) . (14)It follows from Eqs. (5) and (14) that the ratio R ρ reachesits maximal value R ρM at some temperature T Rm ∼ T M .If the ratio is measured in units of its maximal value R ρM and T is measured in units of T Rm ∼ T M then it is seenfrom Eqs. (5), (6) and (14) that the normalized MR R ρN ( y ) = R ρ ( B, T ) R ρM ( B ) ≃ y ( M ∗ N ( y )) (15)becomes a universal function of the only variable y = T /T Rm . To verify Eq. (15), we use MR obtained in mea-surements on CeCoIn , see Fig. 1(b) of Ref. [1]. The re-sults of the normalization procedure of MR are reportedin Fig. 4. It is clearly seen that the data collapse intothe same curve, indicating that the normalized magne-toresistance R ρN well obeys the scaling behavior given byEq. (15). This scaling behavior obtained directly fromthe experimental facts is a vivid evidence that MR be-havior is predominantly governed by the effective mass M ∗ ( B, T ).Now we are in position to calculate R ρN ( y ) given byEq. (15). Using Eq. (5) to parameterize M ∗ N ( y ), weextract parameters c and c from measurements of themagnetic ac susceptibility χ on CeRu Si [24] and applyEq. (15) to calculate the normalized ratio. It is seenthat the calculations shown by the starred line in Fig.4 start to deviate from experimental points at elevatedtemperatures. To improve the coincidence, we employEq. (6) which describes the behavior of the effective massat elevated temperatures in accord with Eq. (4) andensures that at these temperatures the resistance behavesas ρ ( T ) ∝ T . In Fig. 4, the fit of R ρN ( y ) by Eq. (6)is shown by the solid line. Constant c is taken as afitting parameter, while the other were extracted from ac susceptibility of CeRu Si as described in the captionto Fig. 2. T R m [ K ] R m a x B[T]
FIG. 5: The peak temperatures T Rm (squares) and the peakvalues R max (triangles) versus magnetic field B extracted frommeasurements of MR [1]. The solid lines represent our calcu-lations based on Eqs. (16) and (17). Before discussing the magnetoresistance ρ mr ( B, T )given by Eq. (9), we consider the magnetic field depen-dencies of both the MR peak value R max ( B ) and corre-sponding peak temperature T Rm ( B ). It is possible to useEq. (14) which relates the position and value of the peakwith the function M ∗ ( B, T ). Since T Rm ∝ µ B B , B entersEq. (14) only as tuning parameter of QCP, as both ∆ ρ L and ρ were omitted. At B → B c and T ≤ T Rm ( B ), thisomission is not correct since ∆ ρ L and ρ become com-parable with ∆ ρ ( B, T ). Therefore, both R max ( B ) and T Rm ( B ) are not characterized by any critical field, beinga continuous function at the quantum critical filed B c , incontrast to M ∗ ( B, T ) which peak value diverges and thepeak temperature tends to zero at B c as it follows fromEqs. (7) and (8). Thus, we have to take into account∆ ρ L ( B, T ) and ρ which prevent T Rm ( B ) from vanishingand make R max ( B ) finite at B → B c . As a result, we have to replace B c by some effective field B eff < B c and take B eff as a parameter which imitates the con-tributions coming from both ∆ ρ L ( B, T ) and ρ . Uponmodifying Eq. (14) by taking into account ∆ ρ L ( B, T )and ρ , we obtain T Rm ( B ) ≃ b ( B − B eff ) , (16) R max ( B ) ≃ b ( B − B eff ) − / − b ( B − B eff ) − + 1 . (17)Here b , b , b and B eff are fitting parameters. It ispertinent to note that while deriving Eq. (17) we useEq. (16) with substitution ( B − B eff ) for T . Then,Eqs. (16) and (17) are not valid at B . B c . In Fig.5, we show the field dependence of both T Rm and R max ,extracted from measurements of MR [1]. It is seen thatboth T Rm and R max are well described by Eqs. (16) and(17) with B eff =3.8 T. We note that this value of B eff isin good agreement with observations obtained from the B − T phase diagram of CeCoIn , see the position of theMR maximum shown by the filled circles in Fig. 3 of Ref.[1]. B=16T B=12T B=10T B=8T B=6T [ ( T , B )-( T , B = ) ]/ ( T , B = ) T[K]
FIG. 6: MR versus temperature T as a function of magneticfield B . The experimental data on MR were collected onCeCoIn at fixed magnetic field B [1] shown in the right bot-tom corner of the Figure. The solid lines represent our calcu-lations, Eq. (5) is used to fit the effective mass entering Eq.(15). To calculate ρ mr ( B, T ), we apply Eq. (15) to describeits universal behavior, Eq. (5) for the effective mass alongwith Eqs. (16) and (17) for MR parameters. Figure 6shows the calculated MR versus temperature as a func-tion of magnetic field B together with the experimentalpoints from Ref. [1]. We recall that the contributionscoming from ∆ ρ L ( B, T ) and ρ were omitted. As seenfrom Fig. 6, our description of experiment is pretty good. SUMMARY
Our comprehensive theoretical study of MR shows thatit is (similar to other thermodynamic characteristics likemagnetic susceptibility, specific heat etc) governed by thescaling behavior of the quasiparticle effective mass. Thecrossover from negative to positive MR occurs at elevatedtemperatures and fixed magnetic fields when the systemtransits from the LFL behavior to NFL one and can bewell captured by this scaling behavior. This behaviorpermits to identify the energy scales near QCP, discov-ered in Ref. [7]. Namely, the thermodynamic charac-teristics (like specific heat, magnetization etc) consist ofthe low temperature LFL scale characterized by the fastgrowth and the high temperature one related to the NFLbehavior and characterized by the slow growth. Thesescales are separated by the kinks in the transition region.Obtained theoretical results are in good agreement withexperimental facts and allow us to reveal for the firsttime a new scaling behavior of both magnetoresistanceand kinks separating the different energy scales.
ACKNOWLEDGEMENTS
This work was supported in part by the grants: RFBRNo. 09-02-00056, DOE and NSF No. DMR-0705328, andthe Hebrew University Intramural Funds. ∗∗