In-plane resistivity and an explanation for the characteristic T * in high-T c cuprates George A. Levin and Khandker F. Quader Department of Physics, Kent State University, Kent, Ohio 44242 ~Received 9 August 2000! We offer an explanation for the observed crossover temperature T * in in-plane resistivity r ab of biplanar high-T c cuprates. The key to our picture is the existence of nondegenerate and degenerate carriers possessing different quasiparticle relaxation rates. In the underdoped regime the change of slope d r ab / dT at T * results from the thermal activation of nondegenerate carriers. In the overdoped regime, the nondegenerate carriers tend to become degenerate and a second small Fermi energy emerges, resulting in a change to T 2 behavior in r ab at low T. We compare our results with data on several compounds. We also find an approximate scaling in conductivity. I. INTRODUCTION The unusual normal-state properties of the high-T c cu- prate superconductors are yet to be well understood. The evolution of the normal state with doping is marked by some extraordinary universal features. It is recognized that an un- derstanding of these unusual features is important to the un- derstanding of superconductivity at high temperatures. The temperature ~T! dependence of properties, such as in-plane resistivity r ab ( T ), Hall coefficient R H ( T ), electronic spin susceptibility x s ( T ), entropy S ( T , x ), optical conductivity, etc. in the underdoped cuprates reveal a characteristic tem- perature T * , that demarcates high-T and low-T behavior. In this region, T * decreases with increased doping ~in anticor- relation with the critical temperature T c !, and almost van- ishes at or near optimal doping defined by maximum of T c . In the overdoped region, the T dependence of r ab approaches the usual Fermi-liquid ~FL! behavior with some subtle dif- ferences. Examples of materials which show these types of behavior are Bi 2 Sr 2 CaCu 2 O 8 1d , 1 Bi 2 Sr 2 Ca 1 2x Y x Cu 2 O 2 , 2 Y 1 2x Pr x Ba 2 Cu 3 O 7 2d , 3 Tl 2 Ba 2 CuO 6 1d , 4 etc. Here we focus on the temperature and doping depen- dences of in-plane resistivity r ab ( T , x ). At a given tempera- ture, r ab decreases monotonically with an increasing number of holes. In the underdoped cuprates, 1–3,5 the slope d r ab / dT changes appreciably within a narrow range of temperature around T * ( x ). At T .T * , r ab changes approximately lin- early with temperature, but not as rapidly as it does for T ,T * . The crossover temperature T * ( x ) decreases with in- creasing number of holes, and at optimal doping r ab exhibits a linear-T behavior down to T c , with no apparent change of slope ( d r ab / dT 5const). In overdoped samples, r ab exhibits 4,6 a T 2 behavior at low temperatures, and possibly some power higher than linear T at higher temperatures. The evolution of r ab ( T , x ) correlates with that of x s ( T , x ) and the electronic entropy S ( T , x ). 7,8 With decreasing tem- perature x s ( T , x ) decreases in the underdoped and increases in the overdoped samples. The crossover temperature mark- ing the transition between high and low-T behavior exhibits the same anticorrelation with T c as it does in resistivity. A detailed discussion of experimental results on susceptibility and electronic entropy within the context of the same model that is presented here is given in Ref. 9. Over the past years, several theoretical approaches have been taken to address these issues and the nature of the crossover of the physical properties of the cuprates at T * . In attempts to explain transport properties, and in particular, the linear-T in-plane resistivity ~at optimal doping!, theories based on extensions of Fermi-liquid theory ~FLT!, 10,11 as well as, on non-FL concepts 12 have been proposed. The con- cepts of ‘‘spin gap’’ 13 or ‘‘pseudogap’’ 14 have been pro- posed to explain observed features in r ab ( T , x ) and x s ( T , x ) in the underdoped regime. A central aspect of many of the more FL-like theories is the existence of anomalous electron relaxation rates in the cuprates. Some of these theories have at their core the concepts of ‘‘hot spot’’ 11,15 or ‘‘cold spots’’. 16 While these theories have had varying degrees of success, it has proved to be difficult to explain the evolution of the properties across the full range of doping ~underdoped to optimal to overdoped!. In this paper, we present an alternate, and rather different approach to understanding the origin of the crossover tem- perature T * in the cuprates. We argue that calculations based upon the basic features of our model proposed earlier, 17 sup- port the point of view that the crossover temperature T * is a signature of an underlying small energy scale in the under- doped regime. This decreases with increased carrier density. Beyond optimal doping, our model, in a natural fashion, sug- gests that the system evolves into a Fermi liquid. The small energy scale in the underdoped regime goes over to one that is then related to the small Fermi energy of the second, emerging Fermi surface in the overdoped regime. Here we concentrate on in-plane resistivity, and compare our results with data on a number of materials in the underdoped, opti- mal, and overdoped regimes. The key aspects of our model are ~a! the presence of nondegenerate ~and hence really non-FL like!, as well as degenerate ~FL like! carriers ~denoted as h and j compo- nents, respectively!; ~b! the nondegenerate and degenerate carriers possess different relaxation rates; viz. t h 21 } T 2 , and t j 21 } T . 17 Thus the model is quite different from the models based on two degenerate bands, 18 which we believe cannot reproduce the properties of cuprates being considered here. The idea that two relaxation rates affect transport properties of the cuprates was proposed earlier on by Anderson. 19 A PHYSICAL REVIEW B 1 NOVEMBER 2000-I VOLUME 62, NUMBER 17 PRB 62 0163-1829/2000/62~17!/11879~9!/$15.00 11 879 ©2000 The American Physical Society