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Tabakalar arası çiftlenme ve kuantum faz dalgalanmalarının Josephson-bağlı tabakalı süperiletkenlerin fiziksel özelliklerine etkisi

Tabakalar arası çiftlenme ve kuantum faz dalgalanmalarının Josephson-bağlı tabakalı süperiletkenlerin fiziksel özelliklerine etkisi

##### Dosyalar

##### Tarih

1998

##### Yazarlar

Hüner, Metin

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Tabakalar Arası Çiftlenme ve Kuantum Faz Dalgalanmalarının Josephson-bağlı Tabakalı Süperiletkenlerin Fiziksel Özelliklerine Etkisi Özet Yüksek sıcaklık süperiletkenlerinin yüksek yapısal anizotropiye sahip olmaları yanında, üst kritik manyetik alanın değeri, koherans uzunluğu, manyetik alanın sızma uzunluğu gibi fiziksel büyüklükleri de kristal yönlenime göre büyük fark lılıklar göstermektedir. Ayrıca kritik sıcaklıkları geleneksel süperiletkenlere göre oldukça yüksektir. Bu malzemelerin ortak özelliklerinden bir tanesi de, birim hücrelerinde bir veya daha çok bakır oksit tabakaya sahip olmalarıdır. Elektronlar bu tabakalar boyunca rahatça hareket edebilirken, tabakalara dik yönde ancak zayıf bir tünelleme yapabilirler. Bu yüzden bu malzemeler, bir biriyle Josephson bağlı süperiletken tabakaların art arda dizilmesiyle oluşmuş bir yapı şeklinde modellenebilirler. Tabakalar arası Josephson bağı olan bir sistemin Hamiltonyeninde, tabaka içi çiftlenme yanında, tabakalar arasında olabilecek çeşitli çiftlenme mekanizmaları göz önüne alınmıştır. Daha sonra Gorkov-Nambu mikroskobik yaklaşımıyla sistemin düzen parametrelerinin sağ ladığı denklemler bulunduktan sonra, serbest enerji fonksiyoneli elde edilmiştir. Tabakalar arası çiftlenim sonucu kritik sıcaklıkla, tabaka içi ve arası çiftlerin Josephson bağlanma enerjileri artmış ve Lifschitz değişmezinin katsayısının band doluluğuna olan bağlılığı azalmıştır. Tabakalara dik ve paralel yöndeki üst kritik manyetik alan hesaplanmış ve bunların, kritik alanın kritik sıcaklık civarında gözlenen yukarı doğru kavis ile uyumlu olduğu görülmüştür. Orta lama alan türü bu hesaplardan sonra sistemdeki kuantum faz dalgalanmalarının kritik sıcaklığa nasıl bir etkisi olacağı incelenmiştir. Kritik sıcaklığın ifade sinde bulunan faz-faz korelatörü kendi kendisiyle tutarlı harmonik yaklaşım la hesaplanmıştır. Tabakaların yüklenme miktarı arttıkça kritik sıcaklık düş mektedir. Ayrıca sıfır sıcaklıkta, enine sıkılığın yüklenmeye olan bağlılığının incelenmesi sonucu, tabakalar arası Josephson bağlı sistemlerde süperiletken- normal metal tekrar geçişinin olmadığı sonucuna ulaşılmıştır.

The Effect of Interlayer Pairing and Quantum Phase Fluctuations on the Physical Properties of Josephson-coupled Layered Superconductors Summary After the discovery of superconductivity in mercury at 4 K by Kammerling Onnes in 1911, the search for new superconducting materials led to a slow in crease in the highest known transition temperature Tc over the decades, reach ing a saturation at 23 K with the discovery of the superconductivity of Nb3Ge by Gavaler. After 13 more years, the path to radically higher transition tem peratures was opened by the discovery in 1986 of superconductivity at ~ 35ÜC in "LBCO" (a mixed oxide of lanthanum, barium and copper) by Bednorz and Müller, for which they were awarded the Nobel prize in 1987. The discovery was surprising and exciting, not simply because of the large in crease in Tc, but also because it revealed that the oxides formed an unexpected new class of superconducting materials. Another big jump to Tc ~ 90K fol lowed quickly, with the discovery of the "123" class of materials exemplified by YıBa2Cu307_fi ("YBCO"). In this structure the yttrium atom can be replaced by many other rare earth elements, e.g., La, Nd, Sm, Eu, Gd, Ho, Er and Lu, with similarly high Tc. Shortly after, still higher Tc values were found in the "BSCCO" system (mixed oxides of bismuth, strontium, calcium and copper) and the "TBCCO" system (mixed oxides of thallium, barium, calcium and copper). In all of these systems, copper oxide planes form a common structural element, which is thought to dominate the superconducting properties. Depending on the choice of stoichiometry, the crystallographic unit cell contains varying num bers of Cu02 planes. In addition, the 123 compounds contain CuO chains, which are thought to serve largely as reservoires to control the electron density in the planes. The exact Tc depends on these particulars and the highest Tc achieved in the "YBCO", "BSCCO" and "TBCCO" systems are 93,110 and 130 K, respectively. It is generally argued that the same microscopic mechanism is operating in all the new oxide superconductors. However there is no general consensus yet on what this mechanism is. Most ideas emphasize the importance of quasi- two-dimensional Cu02 layers and doping. The main controversy arises with regard to the origin of the attraction of pairs. In this thesis we will assume that the potentials are attractive. vii The description of superconductivity in an anisotropic material can be given by a simple generalization of the Ginzburg-Landau equations with an anisotropic effective mass: a large one along the c-axis and a small one in the ab-plane. However, this model is appropriate only in compounds with small anisotropy and it becomes invalid in the limit of very large anisotropy. Indeed, in layered compounds, the electron density and the superconducting order parameter are varying in the direction perpendicular to the layers. If coherence length £ is small the order parameter becomes inhomogeneous. It is large within the CuO2 layers but small between them. This situation is equivalent to that of a weak link. Thus, in this approximation, the layered superconductors can be considered as an array of superconducting layers coupled by Josephson inter action. Such a model is referred to as the Lawrence-Doniach model. As it has been observed by many experiments the structural anisotropy has a strong influence on the macroscopic properties of a system. Measurements of the energy gap, A, of high-Tc superconductors show significant anisotropy in the magnitude of A, i.e. the value of A is found to depend on the crystal orientation,. Such measurements raise a question on the symmetry of the superconducting energy gap. Earlier theories of the layered superconductors have proposed the spherically symmetric wave function of the Cooper pairs inside each layer, as it is in the conventional BCS theory, and there is no component of the order parameter along the c-axis which is directed perpen dicularly to the superconducting layers. Even weak interlayer tunneling of the electron pairs prevents phase fluctuations from destroying the long-range order. In particular, an additional attractive interlayer interaction of particles could give rise to the energy gap anisotropy. The strong structural anisotropy of high- Tc superconductors permits us to char acterize the electron motion mainly bandlike inside of each layers. Whereas the electron motion in the perpendicular to layers direction is described in tight- binding approximation. The single electron band anisotropy is characterized by the small parameter <1, We will start from a general effective Hamiltonian for a layered superconductor. Then we will use Gorkov Nambu microscopic theory to obtain the expression for the free energy functional. The dominant pairing interaction in the layered systems is expected to be the intralayer interaction. However, the possible interlayer pairing interac tions should strongly affect on the physical properties of superconductors. The most important interlayer interaction term (with coupling constant of VI) is that when two particles located on the nearest-neighboring layers remain on these layers after scattering. Such an interlayer pairing in addition to an in tralayer attractive interaction has been proposed by many authors to study particularly Tc enhancement in layered superconductors. There may be also exchange type (with coupling constant of V 1) interlayer interaction in the sys tem. Although the origin of interlayer coupling is not specified, the possible mechanism of pairing may be the polarization of the dielectric spaced between viii the layers. Notice that Vi and V\ type interlayer interactions may in principle exist even though the single electron tunneling integral t1 to the neighboring layer vanishes. The another type of interlayer interaction which represents the scattering of two particles from one layer to the nearest-neighboring one by pair (with coupling constant of Vo), has been included in Hamiltonian by Tesanovic. Although this term should be proportional to t1, as an origin of this interaction a strong interband scattering mechanism has been suggested that sets up the off-diagonal long range order (ODLRO) in the system if t± - > 0. Apart from the interlayer interactions considered above, Bulaevskii and Zyskin took into account another type interlayer interaction (with coupling constant of V01) which was shown to give a considerable contribution to the gap anisotropy. This term characterizes the interaction of two incident particles on the same layer with scattering one of the particle to the nearest-neighboring layer, also the interaction of two particles located on the nearest-neighboring layers with scattering of both particles into one layer. In this thesis we include into the Hamiltonian an intralayer attractive inter action as well as interlayer interactions. Our aim is to obtain the expression for the free energy functional, taking into account an intralayer and all pos sible interlayer pairing potentials, and to study the critical temperature and the magnetic properties of a layered system. The Lawrence-Doniach type free energy functional with two order parameters has been obtained in a previous paper when the Hamiltonian of a system includes V0 interlayer pairing inter action and only Vi term of the interlayer interaction. The problem is compli cated by taking into account the additional interlayer pairing interactions Vo, V\ and V01 introducing of which multiplies the number of order parameters. We obtain the Lawrence-Doniach type free energy functional for Josephson coupled layered superconductors by using Gorkov-Nambu microscopic theory. The number of order parameters is reduced by introducing two order parame ters Aoo and Ao1- Although Aoo is defined by Vo, Vo and V0i type interaction terms and Ao1 is defined by Vi, Vi and Vbi type terms, we call Aoo and Ao1 intra- and interlayer order parameters, respectively. Introducing two critical temperatures TCo and TCl which correspond to intra- and interlayer local pairings we show that additional interlayer interactions enhance the values of TCo and TCl, so that they are determined by the pairing constants Vb, Vo and Vi, Vi, correspondingly. The existence of two order parameters in a system gives rise to the appearance of new terms such as the Lifschitz invariant in the free energy functional. The Lifschitz invariant characterizes the mixing of two order parameters in the system. The coefficient under the Lifschitz invariant was found to depend strongly on the filling factor of the two-dimensional (2d) electron band and it vanishes for half-filling case. However, the additional interlayer term V 01 changes the coefficient of the mixing term so that it does not depend on the 2d band fining factor (at least its dependence on the filling factor is negligibly weak) and the value of this coefficient considerably increases, i.e. mixing of two local pairing occurs always if V01 = 0 irrespective of 2d electron band filledness. ix The Josephson coupling between the superconducting layers is realized under the condition of t± < kBTc < £*¦, where Tc is the critical temperature eval uated by the mean field theory and eF is the Fermi energy of the electrons inside the layers. In this case the essential term in the free energy functional is the Josephson coupling term which characterizes the cooper pairs tunneling from one plane to the nearest neighboring one. The coupling energies E\ and E]_ which correspond to the intra- and interlayer pairs tunnelings, respectively, were found to differ by 2, i.e. E± = 2E]_ = 55^. The additional interlayer pair interactions Vo and V01 in the Hamiltonian change the values of the Josephson energies E1_ and E]_ so that E1 increases considerably due to Vo term, while relatively small increase shows E\ energy due to V01 term. Also, the correction to E\ depends on the filling factor and this correction vanishes in the middle of the 2d electron band. The transition temperature to the superconducting state, Tc, is calculated for Josephson coupled layered superconductors. Tc is expressed by the local pair ing temperatures TCo and TCl, as well as by mixing of two order parameters. Since TCo and TCl and the coefficient of the Lifschitz invariant increase due to the additional interlayer interactions VI, Vo and Vol, Tc is also enhanced. We also discuss the influence of the interlayer interactions on the upper critical magnetic field of the layered superconductors. Our results are consistant with the experimental data which yields a positive curvature of the upper critical field near Tc. The results obtained from the free energy functional are of mean field type. It is expected that fluctuations would change physical properties of layered superconductors. There is no superconducting phase transition in a two-di mensional system. The existence of strong fluctuations of order parameter's phase destroys the long range order in a single superconducting layer. The destruction effects of the long wavelength fluctuations become weaker in the Josephson coupled quasi- 2d SC's due to the tunneling of the Cooper pairs from one layer to another. Therefore, the long range order in the system sets in at a non-zero temperature. In this thesis we study also the charging effects on the critical temperature Tc in the Josephson-coupled layered superconductors. The self-consistent mean field method is applied to obtain the equation of Tc. This equation expresses Tc by the phase-phase correlator (cos £(()) cos(p(r)) for the phase (p(r) of the order parameter A(r) = |A(r)| exp [i<^(r)]. The phase-phase correlator is cal culated by using the self-consistent harmonic approximation (SCHA) which permits to study the dependence of Tc on the charging effect for large values of the layer charging. The knowledge of the phase-phase correlator for phases on the different layers permits us to investigate the transverse stiffness of the layered superconductors. Study of the dependence of the transverse stiffness on the charging energy at T = 0 shows that long range phase coherence is destroyed only for sufficiently strong values of the layer charging. The criterion for superconducting-normal (2) metal phase transition and reentrant transition to occur is t± > kBTc, where t± is the one-electron tunneling integral for nearest-neighboring layers and Tc is the superconducting transition temperature for a single layer estimated by mean field theory. However, the condition of t1 > kBTc contradicts the existence of Josephson coupling between the layers.

The Effect of Interlayer Pairing and Quantum Phase Fluctuations on the Physical Properties of Josephson-coupled Layered Superconductors Summary After the discovery of superconductivity in mercury at 4 K by Kammerling Onnes in 1911, the search for new superconducting materials led to a slow in crease in the highest known transition temperature Tc over the decades, reach ing a saturation at 23 K with the discovery of the superconductivity of Nb3Ge by Gavaler. After 13 more years, the path to radically higher transition tem peratures was opened by the discovery in 1986 of superconductivity at ~ 35ÜC in "LBCO" (a mixed oxide of lanthanum, barium and copper) by Bednorz and Müller, for which they were awarded the Nobel prize in 1987. The discovery was surprising and exciting, not simply because of the large in crease in Tc, but also because it revealed that the oxides formed an unexpected new class of superconducting materials. Another big jump to Tc ~ 90K fol lowed quickly, with the discovery of the "123" class of materials exemplified by YıBa2Cu307_fi ("YBCO"). In this structure the yttrium atom can be replaced by many other rare earth elements, e.g., La, Nd, Sm, Eu, Gd, Ho, Er and Lu, with similarly high Tc. Shortly after, still higher Tc values were found in the "BSCCO" system (mixed oxides of bismuth, strontium, calcium and copper) and the "TBCCO" system (mixed oxides of thallium, barium, calcium and copper). In all of these systems, copper oxide planes form a common structural element, which is thought to dominate the superconducting properties. Depending on the choice of stoichiometry, the crystallographic unit cell contains varying num bers of Cu02 planes. In addition, the 123 compounds contain CuO chains, which are thought to serve largely as reservoires to control the electron density in the planes. The exact Tc depends on these particulars and the highest Tc achieved in the "YBCO", "BSCCO" and "TBCCO" systems are 93,110 and 130 K, respectively. It is generally argued that the same microscopic mechanism is operating in all the new oxide superconductors. However there is no general consensus yet on what this mechanism is. Most ideas emphasize the importance of quasi- two-dimensional Cu02 layers and doping. The main controversy arises with regard to the origin of the attraction of pairs. In this thesis we will assume that the potentials are attractive. vii The description of superconductivity in an anisotropic material can be given by a simple generalization of the Ginzburg-Landau equations with an anisotropic effective mass: a large one along the c-axis and a small one in the ab-plane. However, this model is appropriate only in compounds with small anisotropy and it becomes invalid in the limit of very large anisotropy. Indeed, in layered compounds, the electron density and the superconducting order parameter are varying in the direction perpendicular to the layers. If coherence length £ is small the order parameter becomes inhomogeneous. It is large within the CuO2 layers but small between them. This situation is equivalent to that of a weak link. Thus, in this approximation, the layered superconductors can be considered as an array of superconducting layers coupled by Josephson inter action. Such a model is referred to as the Lawrence-Doniach model. As it has been observed by many experiments the structural anisotropy has a strong influence on the macroscopic properties of a system. Measurements of the energy gap, A, of high-Tc superconductors show significant anisotropy in the magnitude of A, i.e. the value of A is found to depend on the crystal orientation,. Such measurements raise a question on the symmetry of the superconducting energy gap. Earlier theories of the layered superconductors have proposed the spherically symmetric wave function of the Cooper pairs inside each layer, as it is in the conventional BCS theory, and there is no component of the order parameter along the c-axis which is directed perpen dicularly to the superconducting layers. Even weak interlayer tunneling of the electron pairs prevents phase fluctuations from destroying the long-range order. In particular, an additional attractive interlayer interaction of particles could give rise to the energy gap anisotropy. The strong structural anisotropy of high- Tc superconductors permits us to char acterize the electron motion mainly bandlike inside of each layers. Whereas the electron motion in the perpendicular to layers direction is described in tight- binding approximation. The single electron band anisotropy is characterized by the small parameter <1, We will start from a general effective Hamiltonian for a layered superconductor. Then we will use Gorkov Nambu microscopic theory to obtain the expression for the free energy functional. The dominant pairing interaction in the layered systems is expected to be the intralayer interaction. However, the possible interlayer pairing interac tions should strongly affect on the physical properties of superconductors. The most important interlayer interaction term (with coupling constant of VI) is that when two particles located on the nearest-neighboring layers remain on these layers after scattering. Such an interlayer pairing in addition to an in tralayer attractive interaction has been proposed by many authors to study particularly Tc enhancement in layered superconductors. There may be also exchange type (with coupling constant of V 1) interlayer interaction in the sys tem. Although the origin of interlayer coupling is not specified, the possible mechanism of pairing may be the polarization of the dielectric spaced between viii the layers. Notice that Vi and V\ type interlayer interactions may in principle exist even though the single electron tunneling integral t1 to the neighboring layer vanishes. The another type of interlayer interaction which represents the scattering of two particles from one layer to the nearest-neighboring one by pair (with coupling constant of Vo), has been included in Hamiltonian by Tesanovic. Although this term should be proportional to t1, as an origin of this interaction a strong interband scattering mechanism has been suggested that sets up the off-diagonal long range order (ODLRO) in the system if t± - > 0. Apart from the interlayer interactions considered above, Bulaevskii and Zyskin took into account another type interlayer interaction (with coupling constant of V01) which was shown to give a considerable contribution to the gap anisotropy. This term characterizes the interaction of two incident particles on the same layer with scattering one of the particle to the nearest-neighboring layer, also the interaction of two particles located on the nearest-neighboring layers with scattering of both particles into one layer. In this thesis we include into the Hamiltonian an intralayer attractive inter action as well as interlayer interactions. Our aim is to obtain the expression for the free energy functional, taking into account an intralayer and all pos sible interlayer pairing potentials, and to study the critical temperature and the magnetic properties of a layered system. The Lawrence-Doniach type free energy functional with two order parameters has been obtained in a previous paper when the Hamiltonian of a system includes V0 interlayer pairing inter action and only Vi term of the interlayer interaction. The problem is compli cated by taking into account the additional interlayer pairing interactions Vo, V\ and V01 introducing of which multiplies the number of order parameters. We obtain the Lawrence-Doniach type free energy functional for Josephson coupled layered superconductors by using Gorkov-Nambu microscopic theory. The number of order parameters is reduced by introducing two order parame ters Aoo and Ao1- Although Aoo is defined by Vo, Vo and V0i type interaction terms and Ao1 is defined by Vi, Vi and Vbi type terms, we call Aoo and Ao1 intra- and interlayer order parameters, respectively. Introducing two critical temperatures TCo and TCl which correspond to intra- and interlayer local pairings we show that additional interlayer interactions enhance the values of TCo and TCl, so that they are determined by the pairing constants Vb, Vo and Vi, Vi, correspondingly. The existence of two order parameters in a system gives rise to the appearance of new terms such as the Lifschitz invariant in the free energy functional. The Lifschitz invariant characterizes the mixing of two order parameters in the system. The coefficient under the Lifschitz invariant was found to depend strongly on the filling factor of the two-dimensional (2d) electron band and it vanishes for half-filling case. However, the additional interlayer term V 01 changes the coefficient of the mixing term so that it does not depend on the 2d band fining factor (at least its dependence on the filling factor is negligibly weak) and the value of this coefficient considerably increases, i.e. mixing of two local pairing occurs always if V01 = 0 irrespective of 2d electron band filledness. ix The Josephson coupling between the superconducting layers is realized under the condition of t± < kBTc < £*¦, where Tc is the critical temperature eval uated by the mean field theory and eF is the Fermi energy of the electrons inside the layers. In this case the essential term in the free energy functional is the Josephson coupling term which characterizes the cooper pairs tunneling from one plane to the nearest neighboring one. The coupling energies E\ and E]_ which correspond to the intra- and interlayer pairs tunnelings, respectively, were found to differ by 2, i.e. E± = 2E]_ = 55^. The additional interlayer pair interactions Vo and V01 in the Hamiltonian change the values of the Josephson energies E1_ and E]_ so that E1 increases considerably due to Vo term, while relatively small increase shows E\ energy due to V01 term. Also, the correction to E\ depends on the filling factor and this correction vanishes in the middle of the 2d electron band. The transition temperature to the superconducting state, Tc, is calculated for Josephson coupled layered superconductors. Tc is expressed by the local pair ing temperatures TCo and TCl, as well as by mixing of two order parameters. Since TCo and TCl and the coefficient of the Lifschitz invariant increase due to the additional interlayer interactions VI, Vo and Vol, Tc is also enhanced. We also discuss the influence of the interlayer interactions on the upper critical magnetic field of the layered superconductors. Our results are consistant with the experimental data which yields a positive curvature of the upper critical field near Tc. The results obtained from the free energy functional are of mean field type. It is expected that fluctuations would change physical properties of layered superconductors. There is no superconducting phase transition in a two-di mensional system. The existence of strong fluctuations of order parameter's phase destroys the long range order in a single superconducting layer. The destruction effects of the long wavelength fluctuations become weaker in the Josephson coupled quasi- 2d SC's due to the tunneling of the Cooper pairs from one layer to another. Therefore, the long range order in the system sets in at a non-zero temperature. In this thesis we study also the charging effects on the critical temperature Tc in the Josephson-coupled layered superconductors. The self-consistent mean field method is applied to obtain the equation of Tc. This equation expresses Tc by the phase-phase correlator (cos £(()) cos(p(r)) for the phase (p(r) of the order parameter A(r) = |A(r)| exp [i<^(r)]. The phase-phase correlator is cal culated by using the self-consistent harmonic approximation (SCHA) which permits to study the dependence of Tc on the charging effect for large values of the layer charging. The knowledge of the phase-phase correlator for phases on the different layers permits us to investigate the transverse stiffness of the layered superconductors. Study of the dependence of the transverse stiffness on the charging energy at T = 0 shows that long range phase coherence is destroyed only for sufficiently strong values of the layer charging. The criterion for superconducting-normal (2) metal phase transition and reentrant transition to occur is t± > kBTc, where t± is the one-electron tunneling integral for nearest-neighboring layers and Tc is the superconducting transition temperature for a single layer estimated by mean field theory. However, the condition of t1 > kBTc contradicts the existence of Josephson coupling between the layers.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1998

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1998

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1998

##### Anahtar kelimeler

Fiziksel özellikler,
Süper iletkenler,
Physical properties,
Superconductors