Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor

The finite-temperature properties of a two-dimensional d-wave superconductor with the Lifshitz disorder, introduced by dopants, are studied. The doping dependence of the mean-field critical temperature Tc MF and of the superconducting critical temperature Tc defined by the Berezinskii—Kosterlitz...

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Datum:	2006
Hauptverfasser:	Loktev, V.M., Turkowski, V.M.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2006
Schriftenreihe:	Физика низких температур
Schlagworte:	К 100-летию со дня рождения Б.Г. Лазарева
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spelling	irk-123456789-1203452017-06-12T03:04:59Z Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor Loktev, V.M. Turkowski, V.M. К 100-летию со дня рождения Б.Г. Лазарева The finite-temperature properties of a two-dimensional d-wave superconductor with the Lifshitz disorder, introduced by dopants, are studied. The doping dependence of the mean-field critical temperature Tc MF and of the superconducting critical temperature Tc defined by the Berezinskii—Kosterlitz—Thouless transition are calculated at different values of coupling, dopant potential, and intermediate boson energy. It is shown that superconductivity tends to disappear with increasing doping when the dopant potential is large enough, though the metallic properties of the system are preserved. 2006 Article Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor / V.M. Loktev, V.M. Turkowski // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1055–1064. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 74.20.Rp, 74.25.Dw, 74.40.+k, 74.62.Dh, 74.72.–h http://dspace.nbuv.gov.ua/handle/123456789/120345 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	К 100-летию со дня рождения Б.Г. Лазарева К 100-летию со дня рождения Б.Г. Лазарева
spellingShingle	К 100-летию со дня рождения Б.Г. Лазарева К 100-летию со дня рождения Б.Г. Лазарева Loktev, V.M. Turkowski, V.M. Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor Физика низких температур
description	The finite-temperature properties of a two-dimensional d-wave superconductor with the Lifshitz disorder, introduced by dopants, are studied. The doping dependence of the mean-field critical temperature Tc MF and of the superconducting critical temperature Tc defined by the Berezinskii—Kosterlitz—Thouless transition are calculated at different values of coupling, dopant potential, and intermediate boson energy. It is shown that superconductivity tends to disappear with increasing doping when the dopant potential is large enough, though the metallic properties of the system are preserved.
format	Article
author	Loktev, V.M. Turkowski, V.M.
author_facet	Loktev, V.M. Turkowski, V.M.
author_sort	Loktev, V.M.
title	Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor
title_short	Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor
title_full	Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor
title_fullStr	Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor
title_full_unstemmed	Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor
title_sort	temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2006
topic_facet	К 100-летию со дня рождения Б.Г. Лазарева
url	http://dspace.nbuv.gov.ua/handle/123456789/120345
citation_txt	Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor / V.M. Loktev, V.M. Turkowski // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1055–1064. — Бібліогр.: 34 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT loktevvm temperaturecarrierconcentrationphasediagramofatwodimensionaldopeddwavesuperconductor AT turkowskivm temperaturecarrierconcentrationphasediagramofatwodimensionaldopeddwavesuperconductor
first_indexed	2025-07-08T17:42:10Z
last_indexed	2025-07-08T17:42:10Z
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fulltext	Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9, p. 1055–1064 Temperature–carrier-concentration phase diagram of a two-dimensional doped d -wave superconductor V.M. Loktev Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine 14-b, Metrologicheskaya Str. Kiev, 03143, Ukraine E-mail: vloktev@bitp.kiev.ua V.M. Turkowski Department of Physics, Georgetown University, Washington, D.C. 20057, USA E-mail: turk@physics.georgetown.edu Received March 1, 2006 The finite-temperature properties of a two-dimensional d-wave superconductor with the Lifshitz disorder, introduced by dopants, are studied. The doping dependence of the mean-field critical temperature Tc MF and of the superconducting critical temperature Tc defined by the Berezinskii—Kosterlitz—Thouless transition are calculated at different values of coupling, dopant potential, and intermediate boson energy. It is shown that superconductivity tends to disappear with increasing doping when the dopant potential is large enough, though the metallic properties of the system are preserved. PACS: 74.20.Rp, 74.25.Dw, 74.40.+k, 74.62.Dh, 74.72.–h Keywords: two-dimensional d-wave superconductor. 1. Introduction It is known that the high-temperature superconduc- tors (HTSCs) in the non-doped regime are antiferromagnetic insulators (or semiconductors, more precisely), similarly to many other transition metal oxides, with dielectric gap � 2 eV (see, for example reviews [1–4]). Free carriers (electrons or holes) can be introduced into the conduction band by changing temperature or pressure or by injecting some donor or acceptor impurities. The last possibility is the most ef- fective way to transform the layered copper oxide in- sulators into metals, which exhibit the high-tempera- ture superconductivity. Since, in principle, different valence dopants can be introduced into the system in any proportion, the physical properties of HTSC com- pounds are studied as a function of dopant concentra- tion x, or equivalently as a function of carrier concen- tration c. It should be noted, however, that YBa2Cu4O8 is probably the unique HTSC cuprate compound with the metallic phase in its initial stoichiometric state. This doesn’t allow one to study the effects of doping in this compound. Despite the fact that the quantities x and c can be considered equal (or proportional to each other, as in the cuprates with many layers), the effects of carriers and dopants on the material properties are quite differ- ent. Changing of the carrier density leads to a rather smooth evolution of the metallic, transport, and optical properties of HTSCs. At the same time, randomly dis- tributed dopants [5] together with chaotic electric and deformation fields, which are always generated by them, play a role of carrier scatterers, or centers of lo- calization. As a result, even clean HTSCs should be considered as «bad metals» due to the condition c x c� � dop . The mean free path l and the Fermi mo- mentum kF in these compounds are related as k lF � 1, and the number of the impurity centers (including ex- ternal ones) cannot be smaller than the carrier number, contrary to the cases of the conventional metals and su- perconductors. In fact, as it follows from the experi- ments, the doping initially leads to destruction of the long-range magnetic order in the dielectric phase. © V.M. Loktev and V.M. Turkowski, 2006 Then, the system gets properties of a doped semicon- ductor with increasing x (or c), and it becomes a metal when c is larger than the mobility threshold value cmet . The effect of impurities on the properties of usual (low-temperature) s-wave superconductors was estab- lished a long time ago. Nonmagnetic impurities with small concentration have practically no effect on the superconducting properties (in particular, they hardly change the critical temperature Tc [9]), and their role is constrained to make the superconducting gap more isotropic. In contrast, magnetic impurities result in a fast suppression of Tc, when their concentration is in- creasing [10]. Impurities are external objects in both cases, and their concentration is not connected with charge carrier concentration. It is important that the superconducting condensate in ordinary superconduc- tors is formed and exists without any doping. This difference is principal, since practically in all cases the doping is necessary for metalization of the cuprate materials. Therefore the role of the doping is «constructive». On the other hand, at densities equal to the carrier density, the presence of the dopants can cause suppression of superconductivity, similarly to external magnetic impurities in usual superconduc- tors. Moreover, since both magnetic and nonmagnetic external impurities change Tc in the HTSCs signifi- cantly (see [11–16] and also review [4]), this question becomes even more interesting. A zero-temperature self-consistent study of this two-fold role of the dop- ants in the case of a two-dimensional (2D) system with the s-wave pairing demonstrates [17] that super- conductivity exists only in small concentration region c c cmet � � sup , where csup is the concentration value above which superconductivity is suppressed [17]. This quantity is defined by the condition of equality of the inverse lifetime of a pair �pair �1 and supercon- ducting gap � s : � s �pair � �. The value of �pair is defined by rate of the scattering of pairs on local inhomogeneites, observed in HTSCs [18]. The zero-temperature superconducting properties of a 2D model with the doping induced by the Lifshitz disorder in the case of a d-wave pairing were studied in [19,20]. The corresponding finite temperature problem is complicated by the fact that a two-dimensional sys- tem with a continuous symmetry of the order parameter can’t have a long-range order at finite temperatures. Due to strong order parameter fluctuations only the phase with algebraically correlated order parameter is possible in this case, even when the system has no im- purities. The critical temperature of this transition is usually called the Berezinskii–Kosterlitz–Thouless (BKT) transition temperature (for review, see [21]), but we shall identify it with Tc below. Therefore, the temperature–carrier-density phase diagram of the 2D doped superconducting metals consists of three quali- tatively different regions: i) T T cc MF� ( ), when the superconductivity is ab- sent (normal phase); ii) T c T T cc c MF( ) ( )� � , when the order parameter already exists, but its phase correlations rapidly (ex- ponentially) decay with distance. This phase is often interpreted as the pseudogap phase of HTSCs; iii) T T cc� ( ), when the system demonstrates a quasi-long-range superconducting order with an alge- braically correlated order parameter. This phase diagram was studied in cases with dif- ferent kinds of electron attraction. For example, it was considered in the case of a model with the local at- traction in [22] (see also a recent paper [23]), and with a phonon-exchange attraction in ( [24,25]) (for review see [21]). It is important to understand how the disorder affects the critical temperature Tc in the 2D system at different values of doping. Below we make an attempt to study self-consis- tently the superconducting temperature-carrier den- sity phase diagram of the system with a boson-medi- ated attraction in the presence of the Lifshitz disorder, which is introduced by doping into the initially insu- lating system. It is a great pleasure and honor for us to contribute this special issue of Low Temperature Physics in mem- ory of Prof. B.G. Lazarev, an outstanding low-temper- ature and solid state physicist, whose works led to a deeper understanding of the physics of normal and superconducting metals, especially mechanisms of the behavior of these systems in the presence of strong ex- ternal fields and pressure. 2. The main equations The Hamiltonian of the 2D fermion system with a boson-exchange interaction and in the presence of a disorder can be written as: H t a a( ) ( ) ( ) , , †� � � � � �� nm n m n m � � � � � ��W a a 2 � � �� n n n † , ( ) ( ) � ��1 2 a a V a an n m m nm m n † , , †( ) ( ) ( ) ( ) � � � �� V N a aL n n n� � � † , , (1) where a an n� �� † ( ), ( ) are creation and annihilation operators of fermion with spin � � � �, on site n; � is the imaginary time; tnm is the nearest neighbor hop- ping operator. The dispersion relation of the free elec- 1056 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 V.M. Loktev and V.M. Turkowski trons on the square lattice can be chosen in the simplest form: � �k � � � �4 2t t k a k ax y[cos( ) cos( )] with the bandwidth W t� 8 , where t is the nearest neighbor hopping parameter; � is the chemical poten- tial. The interaction potential Vnm in (1) corresponds to the potential Vkq in the momentum space. We shall consider the d-wave pairing case, which can be generated by the potential V Vkq k q� � � , where � k � � � �[cos( ) cos( )] ( )k a k ax y D� � �2 k 2 . The theta-func- tion guarantees that the quasiparticle with the ener- gies in the «BCS» belt �D around the Fermi level interact with each other (�D is the maximal energy of the intermediate bosons, which lead to the inter- particle attraction); VL is an one-site impurity (here—dopant) potential. It corresponds to a ran- domly distributed shift of the on-site energy, accord- ing to the Lifshitz model of the disorder [26,27]. In order to study properties of the system, it is con- venient to derive the thermodynamic potential. It can be calculated by evaluating the partition function of the system. This function can be written as Z Da Da d a a H� � � � � � � � � � � � � � � �† † , exp ( ) ( ) ( )� � � � � � � � � 0 n n n� ! " # $ . (2) The Hubbard–Stratonovich transformation with bilocal fields % �nm ( ) and % �nm † ( ) can be applied to the nonlinear term in the partition function (2): exp ( ) ( ) ( ) ( )† , †d a a V a a� � � � � � 0 � � � � � � � � � � � � � � � n n m m nm m n � � � �� D D d V a a a% % � % � % � � � � † †exp \| ( )\| ( ) ( ) ( ) 0 2 nm nm nm n m n m nm n m � � � � � � � � � � � � � � �� † † , ( ) ( ) ( )� � % �a . (3) The complex order parameter can be expressed as % � � � �nm nm nm( ) ( ) exp( ( )),� � i where � nm ( )� is the modulus and � �nm ( ) is the phase of the order parame- ter. We assume that � nm ( )� depends on the relative coordinate of the operators r R Rn m� � only, and � �nm ( ) depends only on the center of mass coordinate R R Rn m� �( )/2: % � � � �nm r R( ) ( , ) exp( ( , )).� � i It is easy to understand this approximation in the fol- lowing way. The Cooper pair dynamics is described by the modulus of the order parameter, and its space sym- metry depends on the relative pair coordinate. On the other hand, the motion of the superconducting con- densate is described by the phase of the order parame- ter, which changes slowly in space, and it can be de- scribed by the center of mass coordinate. In order to evaluate the partition function of the system, it is convenient to introduce the two-compo- nent Nambu spinors & &n n n n n n( ) ( ) ( ) , ( ) ( ( ), ( )† † †� � � � � �� a a a a ) , and to make the following decomposition of the fermion fields: a i /n n n� �� ' � � �( ) ( ) exp( ( ) ),� 2 a i /n n n� �� ' � � �† †( ) ( ) exp( ( ) ).� � 2 Then, in the continuum limit % � � � � � % � % �nm n m n m nm nm † n † † † †( ) ( ) ( ) ( ) ( ) ( ) (a a a a� � � � � ( & )� ( )� ��&m � � �& &n m nm †( )� ( ) ( )� � � % � � � ) )( , ) ( )� (†� � � �*R Rn m n m� 1 , (4) where )n †( )� and )m ( )� are so called neutral Nambu spinor operators: )n n n ( ) ( ) ( )†� ' � ' � � � � � � � � � , )m m m † †( ) ( ( ), ( ))� ' � ' �� , and �� are combinations of the Pauli matrices ��1 and ��2: � (� � )� � �� +1 2 2/ . Substitution of (3) into (2) with approximation (4) allows one to obtain the expression for the parti- Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1057 tion function after integration over the neutral Nambu spinors: Z D D� �� , �� - �exp( ( , )), where the thermodynamic potential , �( , )� has the following form: - � � � � , � � ( , ) ( , ) ( ) .� �� d d r V G 0 2 2 1r r Tr ln The Green function for the Nambu spinors Gnm n m( , ) ( ) ( ) ( )†� � � � � � �1 2 1 2 1 2� � � �& & (5) has the standard structure: G� �� 1 1G ., whereG �1 is the part of the inverse Green’s function that doesn’t depend on the phase of the order parameter: G G� � � � � / 0 �1 1 1 2 1 1 1 2nm n mR R( , ) , \| \| ,� � � � � � � � �1 1 � � � � ��nm ( )[ � ( )]1 2 31 4t � � � � � ��1 1 � � � � � �nm a n mR R( )� � ( , )1 2 3 1 1 2t � � .L( , , , )� �1 2 R Rn m . The disorder induced self-energy .L can be calculated by making an average over the disorder. We use the fully renormalized group expansion method to evaluate an approximate expression for this function. It has the following structure: .L( , , , )� �1 2 R Rn m � 1nm.L 2 2 �( )� �1 2 , where .L( )� �1 2� in the Matsubara rep- resentation can be approximate by: .L n L N n ncV i i( ) ~ ( )3 4 3 � 5 3� � � � � � � � 2 2 sign (6) and 4 5N / W� 4 ( ) is the normal Fermi density of states in the case �F W/� 2. The effective disorder potential ~VL in (6) is ~ ( )V V / V gL L L� �1 3 , g3 � � �4 � �N / Wln[ ( )] (see Appendix A, and [19], where the original derivation is presented in detail). The self-energy . in the continuum limit is: . .nm n mR R( , ) , \| \| ,� � � �1 2 1 2� / 0 � � 1 1 � �( ) ( )R Rn m� � 21 2 2 � � 6 � � � � i i m � ( , ) ( , � � � � �� 3 1 2 12 41 R Rn R nn � 6 � 6 6 � � � � ( ( , )) ( , ) � � � � �3 1 2 18 2m i mR n R n Rn n n R R , where the effective fermion mass m is connected with the hopping parameter t in the ordinary way: m / a t� 1 2( ). In the limit of small fluctuations of the phase of the order parameter the thermodynamic potential can be expanded in powers of the self-energy .: , � � , �( , ) ( , ) ( ) ( , ),� � � � � � � �� d d r V 0 2 2 1r r Tr ln kinG where the kinetic part of the thermodynamic poten- tial is , � .kin Tr( , ) ( ) .� -� � �1 1 1 n n nG (7) As a rule, it is enough to consider the thermody- namic potential up to the second order in the phase gradients 6�. In this case, only the terms with n � 1 2, contribute to the kinetic part of the thermodynamical potential in (7). The kinetic (6�-dependent) part of the effective potential in this case has the following structure: , �kin ( , ) ( ) ,� �� 6� J d r 2 2 2 (8) where J T T T d k i n i n n( , , ( )) ( ) [� ( ) ]� 5 � 3 �� 1 2 2 2 2 3 3tr eGk� � � �t T d k i i n n n 2 2 22 2( ) [ ( ) ( )], 5 3 3k k k 2tr G G (9) (for details see [29], for example). The neutral fermion Green’s function G in the momentum and Matsubara frequency space has the following form: Gk k k ( ) � � ( ) i in n L n 3 3 � � � 3 � � � � 1 1 3� . . (10) Now, having the expression for the thermodynamic potential, we are able to derive the final set of equa- tions which define the phase diagram of the system. 2.1. The gap equation In order to evaluate the equation for the mean-field critical temperature of the system above which � k � 0 it is necessary to minimize the potential part of the thermodynamic potential with respect to the order pa- rameter � k: 1058 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 V.M. Loktev and V.M. Turkowski 1 1 � , � k 6 � � 0 0 . (11) This equation has the following form in the momen- tum space: � �k k q q k� � �� V d q T i n x n� 5 � � 3 � 2 22( ) [� ( )]tr G . (12) It is seen from this equation that the order parame- ter has the momentum dependence supposed a priori (see above). The momentum integrals have to be performed over the region given by \| \| ,� �k 7 D � � �k � 8 mob. The last condition requires that the en- ergy of the band electrons must be larger than the mo- bility edge energy �mob. The value of this parameter can be estimated as � 4mob met� ( )ln ( )c/ c/cN 2 , where the metalization concentration cmet is: cmet � � \| \|� 4loc N/2, which is defined by the local energy level � 4 4loc � � �( ) (exp[ ] )1 1/ / VN L N (see [19] for details) The gap equation (12) can be transformed to 1 2 1 2 12 2 2 2 2 2 2 2 � � � 99 �� V T d k n / T E / TL: 5 5 � :5 :5( ) [ ( )] ( ) k k.n � � , (13) where E cV /L Nk k k� � � �� : 4 �2 2 21� , ~ , and 99 � � �. .L L n L Ni / cVIm ( )( ) ~sign 3 5 41 2 2 . The summation over the Matsubara frequencies can be formally performed in (13), and the gap equation can be written as 1 4 2 1 2 2 2 2 2 2 � � � 99 � � � � �� iV d k E T i E T L :5 5 � ; :5 :5 ; ( ) k k k. 1 2 2 2 � 99 � � � � � � � � .L T i E T:5 :5 k � � 99 � � � � � � 99 � ; :5 :5 ; :5 :5 1 2 2 2 1 2 2 2 . .L L T i E T T i E T k k� � � � � �, (14) where ; �( )x n n x n � � � � � � � � � � 1 1 1 0 is the di-gamma function, � ;� � ( ) .1 0 577� is Euler’s constant [30]. It is easy to see that equation (14) has the standard form in the clean limit: 1 2 2 2 2 2 2� �V d k E / T E( ) tanh[ ( )] 5 � k k k . 2.2. The number equation The number equation which connects the free car- rier concentration c with the chemical potential � of the system is defined by: 1 0v c � � 6 � � � , � � , (15) where v is volume of the system. The explicit calculations give (see, for example [21]): c T d k i n n n� �� 2 2 3 3 2( ) [ ( )� � ] 5 3 � 13 � � tr exp( )Gk .(16) In detail, c d k T n / T E / TL � � � � 99 �� 2 2 2 2 2 2 22 1 2 2 1( ) [ ( )] ( )5 : 5 � :5 :5 k k.n � � � � � � � � � � . (17) Similarly to the gap equation, the summation over n can be formally performed, and the number equation can be transformed to Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1059 c d k i E T i E T L� � � 99 � � � � �� 2 22 1 2 1 2 2 2 1 ( )5 � :5 ; :5 :5 ;k k k. 2 2 2 � 99 � � � � � � � � � � .L T i E T:5 :5 k � � 99 � � � � � � 99 � ; :5 :5 ; :5 :5 1 2 2 2 1 2 2 2 . .L L T i E T T i E T k k� � � � � � � � � . (18) This equation in the clean limit also acquires the standard form: c d k E E T � � � � � � � � � � �� 2 22 1 2( ) tanh 5 �k k k . The carrier concentration dependence of the mean-field critical temperature Tc MF is defined by Eqs.(14) and (18) at � � 0. 2.3. The Berezinskii–Kosterlitz–Thouless equation The temperature below which the phases of the or- der parameter in a 2D system with continuous symme- try become algebraically correlated is called the criti- cal temperature of the BKT transition, in analogy with the phase transition in the 2D spin XY model. This is the only possible finite temperature phase tran- sition in such systems. In the case of the 2D spin XY model with the Hamiltonian H J/XY � �� ( ) ( ) , 2 2 n n n � � , where 4 is the nearest neighbor vector, the BKT transition critical temperature is: T JBKT � 5 2 . (19) In the superconducting case the equation for such a critical temperature can be obtained immediately after a comparison of the Hamiltonian HXY and Eq.(19) with the kinetic part of the thermodynamic potential (8): T J T Tc c c� 5 � 2 ( , , ( )),� (20) where the function J T Tc c( , , ( ))� � , gap �( )Tc , and the chemical potential � are defined by (9), (14), and (18), respectively. Contrary to the XY-model case, the BKT equation (21) is a complicated equation, since «the stiffness» J depends on temperature, chem- ical potential, and the gap amplitude, which are also functions of temperature. It is possible to show from (9) that the expression for J is: J T T t c t T d k n n L ( , , ( )) ( ) [ � : 5 5 � . � � � � 99 �� 2 2 2 12 2 2 2 2 2k / T E / T n / T E / TL ( )] ( ) [( ( )) ( ) ] :5 :5 :5 :5 2 2 2 2 2 22 1 � � � 99 � k k. 2 . Therefore, the critical temperature equation for the doped metallic system is: T t c t T d k n / T c c n L c � � � � 99 �� 5 : 5 5 :5 4 2 2 2 12 2 2 2 2 ( ) [ ( ) k . ] ( ) ( ) [( ( )) ( ) ( 2 2 2 2 22 1 � � � 99 � E T / T n / T E T / c c L c c k k :5 :5 :5. Tc) ]2 2 . (21) Similarly to the gap and the number equations, this equation can be expressed by means of an integral over di-gamma functions: T t c it d k E T E / c � � � � � � � �� 5 : 5 :5 :54 2 2 1 2 2 2 2 2 2 2 2 2( ) ( k k k T T i E T c L c c)2 1 2 2 2 � � � � � � � � � 999 � �� ; :5 :5 . k � � 99 � �� 99 �; :5 :5 ; :5 1 2 2 2 1 2 2 2 . .L c c L cT i E T T i Ek k :5 ; :5 :5T T i E Tc L c c �� 99 � �� 1 2 2 2 . k . (22) In the clean limit, equation (22) transforms to: T t c t T d k E / T c c c � � � 5 5 54 8 2 1 2 2 2 2 2 2( ) cosh ( ( )) k k . The carrier concentration dependence of the critical temperature can be found by solving the set of self-consistent equations (14), (18), and (22). 1060 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 V.M. Loktev and V.M. Turkowski 3. Solutions The solution of the system of equations (14) and (18) at zero value of the gap � � 0 defines the doping dependence of the mean-field critical temperature Tc MF . For simplicity, we put the mobility edge �mob equal to zero, and the impurity potential ~VL equal to VL. This is correct when c c( )� dop andVL are not very large. The dependencies T cc MF ( ) in the caseVL � 0 at dif- ferent values of the attractionV and �D is presented in Fig. 1. The critical temperature Tc MF is extremely small at low concentrations when the coupling V is small comparing to W (see also [34], for example). Formally, it is caused by presence of the factor � k in the gap equation (14). The presence of this factor de- creases the effective density of states, and hence the effective coupling at low carrier densities, since � k 2 2 � kF is small in this case. It is interesting that the boson frequency �D decreasing leads to a similar phe- nomena. The factor � � �( )D 2 2� k is more important in the d-wave pairing channel comparing with the s-wave pairing case. The same carrier density dependence of the critical temperature Tc can be found by solving system (14), (18) and (22) (Fig. 2). It is interesting that T cBKT � at low carrier densities in the s-wave pairing channel at any V and �D. This indicates the fact that the sys- tem with the s-wave pairing at low carrier densities is in the Bose-condensation regime. The situation is quite different in the d-wave pairing case. The system is in the Bose–Einstein condensation regime only, whenV is larger than some critical value. In fact, as it follows from Fig. 2, T cc � at low carrier densities and rather large V. However, this is true only when �D is Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1061 0.25 0.5 0 0.15 0.3 T cM F / W V/W = 1.0 b 0.25 0.5 c 0 0.25 0.5 T cM F / W V/W = 2.0 c 0.25 0.50 0.05 0.1 T c M F / W V/W = 0.5 a c c Fig. 1. The doping dependence of the mean-field critical tem- perature Tc MF in the case when the dopants are absent at dif- ferent values of the coupling and Debye frequency. The val- ues of the Debye frequency are �D/W � 0125 025. , . and 1.0. Here and in Fig. 2 the critical temperature increases with �D increasing. 0.25 0.5 c 0 0.015 0.03 T c / W V/ W = 0.5a 0.25 0.5 c 0 0.025 0.05 T c / W V/ W = 1.0b 0.25 0.5 c 0 0.025 0.05 T c / W V/W = 2.0 c Fig. 2. The doping dependence of the superconducting crit- ical temperature Tc in the case, when the dopants are ab- sent at different values of the coupling and of the Debye frequency. The values of �D/W are the same as in Fig. 1. of order or larger than W. In the cases of low V and �D, the temperature Tc becomes significantly different from zero at some finite value of c, which increases with V and �D decreasing. In the strong coupling and large �D case, the term proportional to c is the largest one on the right-hand side of equation (22). In this case T tc/c � 5 4 and it is much smaller than Tc MF (Fig. 3). The disorder leads to a significant change of the phase diagram at large carrier densities. Namely, the curves T cc MF ( ) and T cBKT ( ) decrease with the doping increasing whenVL is large enough (Fig. 4). This pro- vides a bell-like shape of the doping dependence of the superconducting critical temperature, similar to the ones found experimentally in cuprates. However, we must stress that it seems problematic to reproduce quantitatively the phase diagram of HTSCs within the framework of this model. Despite the fact that the model has a phase diagram which qualitatively resem- bles that of HTSCs, a more complicated realistic choice of the model parameters must be considered in order to describe experimental results. In particular, the correlation effects, the realistic quasiparticle spec- tra and the boson dispersion relation should be taken into account more precisely. 4. Conclusions To conclude, it has been shown that the dopants in- deed play a twofold role in formation of the d-wave superconducting condensate in 2D doped metals at fi- nite temperatures. On one hand, they are a source of metalization, supplying carriers to the system, but on the other hand, they are the centers of carrier scatter- ing, both in free and bound states. It was shown that, similarly to the zero-temperature case, the supercon- ductivity exists in finite range of carrier densities, pro- vided that the disorder is large enough. The doping dependencies of the mean-field and observed critical temperatures have a bell-like shape in this case. It was not an aim of the paper to describe quantita- tively real HTSC cuprates, therefore, the problem was simplified. A self-consistent study of the dopant role in HTSCs and their influence on magnetic, electric, and superconducting subsystems requires a more spe- cific consideration. We would like to thank Prof. Yu.G. Pogorelov for numerous enlightening discussions. V.M.T. acknowl- edges support from the National Science Foundation under grant number DMR-0210717. Appendix A: Disorder average for the supercon- ducting system In order to find the neutral fermion Green’s func- tion averaged over disorder, let us write the Hamil- tonian for neutral fermions in the momentum space: H � � � �� ' ' ' ' � � �k k k k k k k k, , † , ,[ ]� h.c. � 9�� V N iL exp( ( ) ) , , , , † ,k k n k k n k k 1 � � �' ' , (A.1) where n corresponds to the sites where a dopant is present, and the superconducting gap � k is defined self-consistently by Eqs.(14) and (18). The retarded fermion Nambu Green’s function is defined by (5). This Green’s function is a 2 22 matrix 1062 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 V.M. Loktev and V.M. Turkowski 0.25 0.5 c 0 0.25 0.5 T / W T c MF Tc Fig. 3. The phase diagram at V/W = 2, �D/W = 1 for the case when the dopants are absent. 0.1 0.2 0.3 0.4 0.5 c 0 0.025 0.05 T c / W V L / W = 0 LV / W = 0.25 LV / W = 0.50 b Fig. 4. The doping dependence of the mean-field critical temperature Tc MF and of the critical temperature Tc at V/W /WD� �2 1, � and different values of the disorder potential. 0.25 0.5 c 0 0.25 0.50 0.75 T cM F / W VL / W = 0 VL / W = 0.125a / W = 0.250VL in the Nambu space. It is important to distinguish in the equations of motion, which follow from (A.1), the diagonal and non-diagonal matrices in the Nambu (N) and in the momentum (M) spaces. In particular, we have the following equation for the non-homogeneous system Green’s function in the frequency and momen- tum space: G Gk k k k k k, , ( ) ,( ) ( )� � �� 10 � � 99 � � � � ��G Gk k n k k kk k n, ( ) , ,( ) � exp( ( ) ) ( )0 � �V i , (A.2) where the unperturbed Green’s function is Gk k k k , ( ) ( ) � � 0 3 1 1 0 � � � � � � � � �� i with V VL� ��3, �� j are the Pauli matrices. The M-diagonal one-particle Green’s function is a self-averaged function, which can be found from (A.2) after making average over the disorder [31–33]. The general solution of (A.2) has the following form G Gk k k k k, , ( )( ) {[ ( )] ( )} ,� � �� 0 1 1.L (A.3) where the self-energy part is defined by the group ex- pansion .L cV Vk( ) �( ( ) �)� �� 1 1G [ ( ( ) ( ) ( ))( ( ) ( ))1 1 0 � � � � ��c A A A A Ai 0n n kn 0n n0 0n n0� � � � �e � �1 ...], (A.4) with G( ) ( );,� �� 1 N Gk k k (A.5) A V Vi 0n k n k k k k( ) � ( )( ( ) �),� � �� e G G1 1. In the case, when .Lk( )� is approximated by the first term in (A.4), the self-energy is momentum-inde- pendent: .L cV V( ) �( ( ) �) .� �� 1 1G (A.6) The self-energy can be found self-consistently from the system (A.3), (A.6) and (A.5). It can be shown [19] that in the limit W / W� [ ( )]5� �� 1 and at small values of frequencies \| \|� �� D the self energy is: .L L NcV i( ) ~� 4 � � 5 � � � � � � � � 2 2 , (A.7) where 4N and ~VL are defined after Eq.(6). Equation (A.7) immediately leads to the disorder self-energy in the Matsubara frequency representation Eq. (6). 1. V.M. Loktev, Fiz. Nizk. Temp. 22, 3 (1996) [Low Temp. Phys. 22, 3 (1996)]. 2. M.A. Kastner, R.J. Birgeneau, G. Shirane, and Y. Endoh, Rev. Mod. Phys. 70, 897 (1998). 3. M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). 4. Yu.A. Izyumov, Phys. Usp. 42, 215 (1999). 5. 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Temperature–carrier-concentration phase diagram of a two-dimensional doped d-wave superconductor

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