• 沒有找到結果。

不同種類的金屬/絕緣體/超導體的結之穿隧

N/A
N/A
Protected

Academic year: 2021

Share "不同種類的金屬/絕緣體/超導體的結之穿隧"

Copied!
64
0
0

加載中.... (立即查看全文)

全文

(1)Tunneling in Different Kinds of Metal/Insulator/Superconductor Junctions. by. Pok-Man Chiu. A thesis submitted in conformity with the requirements for the Degree of Master of Science Graduate Department of Physics National Taiwan Normal University. c Copyright by Pok-Man Chiu 2008 °.

(2) ABSTRACT This thesis applies the theory of tunneling to study different kinds of metal/insulator/ superconductor (N/I/S) junctions. Chapter 1 gives a brief review of the BCS theory. Chapter 2 mentions some basic properties of high-temperature superconductors (HTSC). Possible coexistence of antiferromagnetic (AF) order and the superconducting order in HTSC is emphasized. In chapter 3, theories of tunneling are presented, namely the Blonder-Tinkham-Klapwijk model approach and the tunneling Hamiltonian approach. In chapter 4, we summarize recent experimental and theoretical works on different kinds of N/I/S junctions. Chapter 5 is based on one of my recent paper to be published. We extend the theory of point-contact spectroscopy [Phys. Rev. B 76, 220504(R) (2007). This paper argued that the splitting of zero-bias conductance peak (ZBCP) in electron-doped cuprate superconductor point-contact spectroscopy is due to the coexistence of AF and d-wave superconducting orders.] to study the ferromagnetic metal/electron-doped cuprate superconductor (FM/EDSC) junctions. In addition to the AF order, effects of spin polarization, Fermi-wave vector mismatch (FWM) between the FM and EDSC regions, and effective barrier are also considered. They play a crucial role in determining the spin polarization value. It is shown that there exits the midgap surface state (MSS) contribution to the ZBCP in the junction and Andreev reflections are largely modified due to the exchange field of ferromagnetic metal. A more accurate formula is proposed for determining the spin polarization value in combination with the conductance in point-contact experimental data. Finally in Chapter 6, a brief conclusion and future prospects are given.. ii.

(3) ACKNOWLEDGEMENTS I wish to express my deep gratitude to my supervisor Prof. Wen-Chin Wu for having initiated me into the field of theory of superconductivity, and for his guidance and generous support during my years as a graduate student. Without his encouragement and confidence in me, the research in this thesis could never have been done. I am also thankful my senior, C. C. Huang in our theoretical condensed-matter group for his helping to solve numerous computer-related problems, introducing me the Matlab and teaching me the LaTex. Of course, I am grateful to my father, Kin-Kuen Chiu, mother, Kwan-Fun Chu and sister, Siu-Kit Chiu for their understanding and love. Without their love, my research power would never have kept (literally).. iii.

(4) Contents 1 BCS Theory. 1. 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Cooper Pairs and the Origin of the Attractive Interaction . . . . . . . .. 2. 1.3. BCS Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2 Some Basic Properties of Hight-Temperature Superconductors 2.1. Structure of Hight-Temperature Superconductors . . . . . . . . . . . .. 2.2. Phase Diagram of Hight-Temperature Superconductors and Antiferro-. 7 7. magnetic Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Tunneling Theory. 8 13. 3.1. Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 3.2. Semiconductor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 3.3. Formula of Tunneling Current . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.4. Andreev Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 3.5. Blonder-Tinkham-Klapwijk Model Approach . . . . . . . . . . . . . . .. 17. 3.6. Tunneling Hamiltonian Approach . . . . . . . . . . . . . . . . . . . . .. 22. 4 Tunneling in Different Kinds of Metal/Insulator/Superconductor Junctions. 25. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 4.2. N/I/sSC Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 4.3. FM/I/sSC Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 4.4. N/I/HTSC Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 5 A Detailed Study of FM/I/dSC Junction 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iv. 37 37.

(5) 5.2. Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 5.3. Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 5.3.1. Midgap Surface States . . . . . . . . . . . . . . . . . . . . . . .. 45. 5.3.2. Effects of Fermi-wave-vector Mismatch . . . . . . . . . . . . . .. 46. 5.3.3. Effects of Spin Polarization . . . . . . . . . . . . . . . . . . . .. 48. 5.3.4. Effects of Effective Barrier . . . . . . . . . . . . . . . . . . . . .. 50. 5.3.5. A General Formula for Determining the Spin Polarization . . . .. 54. 6 Conclusions. 55. v.

(6) Chapter 1 BCS Theory 1.1. Introduction. Since Onnes first observed the superconducting phenomenon on mercury in 1911, physicist did not know much about the origin of superconductivity for quite a long time. Until in 1957, after forty-six years searching, a complete theory of superconductivity (called BCS theory) was founded by Bardeen, Cooper, and Schrieffer [1]. BCS theory has a great success in conventional superconductor. The prediction of their theory had a good agreement with different types of experiment. Unfortunately, BCS theory was not so successful in high-temperature superconductor (HTSC) which was found in 1986. Its critical temperature can be order of 100K, while BCS theory predicts that the critical temperature can go only up to about 30 to 40K. Twenty two years after the discovery of HTSC, there is still no successful theory of HTSC. But we have some useful models and theory such as RVB model, Hubbard model, t-J model, and SO(5) theory. They made us understand the HTSC better. Although BCS theory has its drawback on HTSC, there is a consensus that the concept of Cooper pair is still valid. In this chapter, we briefly review the BCS theory. A large part of the context was based on the famous book written by Tinkham – Introduction to Superconductivity [2].. 1.

(7) 2. 1.2. Cooper Pairs and the Origin of the Attractive Interaction. Cooper pair is a key concept of superconductivity that was first introduced by Cooper [3] in 1956. He showed that the Fermi sea of electrons is unstable against the formation of bound pairs, regardless of how weak the interaction is and only if it is attractive. In a microscopic view, it is a consequence of the Fermi statistics and the existence of the Fermi sea background. To see how Cooper pairs form, we consider a simple model of two electrons added to a Fermi see at T = 0, with the assumption that the two extra electrons interact with each other but not with those in the sea, except via the exclusion principle. By the general argument of Bloch that lowest-energy state should have zero total momentum, we expect the two-particle wave function to be of the form ψ0 (r1 , r2 ) =. X. gk eik·r1 e−ik·r2 .. (1.1). k. Taking into account the antisymmetry of the total wave function with exchange of the two electrons, it reduces to the form ψ0 (r1 − r2 ) =. X. gk cos k · (r1 − r2 )(α1 β2 − β1 α2 ),. (1.2). k>kF. where (α1 β2 − β1 α2 ) is the singlet spin function. By inserting it into the Schr¨odinger equation, one obtains (E − 2εk )gk =. X. Vkk0 gk0 ,. (1.3). k0 >kF. where Vkk0 =. 1 Ω. R. 0. V (r)ei(k −k)·r dr is the matrix element of the interaction potential. If. a set of gk satisfying (1.3) with E < 2EF (EF is the Fermi energy) can be found, a bound-pair state then exists. Assuming that Vkk0 = −V for EF ≤ εk < EF + h ¯ ωc (ωc is a characteristic energy cutoff) and vanishes otherwise, Eq. (1.3) can be reduced to Z EF +¯hωc 1 2EF − E + 2¯hωc 1 dε = N (0) = N (0) ln . V 2ε − E 2 2EF − E EF. (1.4). By using the weak-coupling approximation which is valid for N (0)V ¿ 1, the preceding equation can be written as E ≈ 2EF − 2¯hωc e−2/N (0)V .. (1.5).

(8) 3 It means that the formation of an electron pair on the Fermi surface can lower the energy of the system. This is the distinction between normal and superconducting state. Nevertheless it poses a fundamental question: what causes the attractive interaction between electrons? By now, we know the answer for conventional superconductors. Thanks to BCS theory, the complete Hamiltonian [1] for the electrons may be expressed in the form H=. X kσ. 2¯hωq |Mq |2 c∗ (k0 − q, σ 0 )c(k0 , σ)c∗ (k − q, σ)c(k, σ) εk nkσ + Hc + ,(1.6) (εk − εk+q )2 − (¯hωq )2 k,k0 ,σ,σ 0 ,q X. where the second term is the screened Coulomb interaction and the third term is the phonon interaction, which comes from virtual exchange of phonons between the electrons. Mq is the matrix element of phonon-electron interaction and is related to the phonon velocity, υq , by |Mq |2 = |υq |2 (¯ h/2ωq ).. (1.7). Here, Mq varies with isotopic mass in the same way that ωq does. The form of the phonon interaction shows that it is attractive for excitation energies |εk − εk−q | < h ¯ ωq . In the classic paper of BCS [1], their criterion for superconductivity is that the attractive phonon interaction dominates the Coulomb interaction for those matrix elements which are of importance in the superconducting wave function: *. 2 |Mq |2 4πe2 −V = − + 2 h ¯ ωq q. +. < 0.. (1.8). avg. The most important transitions are those for which |εk − εk−q | ∼ kTc ¿ h ¯ ωq . It is the origin of the attractive interaction between electrons near the Fermi surface. A more simple version of understanding the appearance of attractive interaction can be derived by “jellium” model. Pines was the first who attempts to test the theoretical criterion for superconductivity systematically throughout the periodic table. We just quote the result as shown in the book of de Gennes [4]. The jellium model in a certain approximation leads to V (q) =. ωq2 4πe2 4πe2 + , Ω(q 2 + ks2 ) Ω(q 2 + ks2 ) ω 2 − ωq2. (1.9). where the first term is the screened Coulomb repulsion, whereas the second term is the phonon-mediated interaction, which is attractive for ω < ωq but fails to give an attractive interaction for ω = 0..

(9) 4 It is important to recognize that the BCS pairing model requires only an attractive interaction giving a matrix element that can be approximated as −V over a range of energies near the Fermi surface.. 1.3. BCS Ground State. BCS ground state is the soul of superconductivity. Starting from it, all the physical quantity can be calculated and to everyone’s big surprise, many predictions of BCS theory have a very good fitting with experiment. Schrieffer was the first to construct the ground state wave function of superconductivity. His basic idea, as quoted from his Nobel lecture, is “the instantaneous occupancy of the pair state should be essentially uncorrelated with the occupancies of the other pair states at that instant. Rather, only the average occupancies of the pair state are related”. On this basis, he wrote down the trial ground state wave function as Ψ0 =. Y. (uk + vk b∗k ) |0i ,. (1.10). k. where |uk |2 + |vk |2 = 1. |vk |2 is the probability of the pair being occupied, whereas the probability that it is unoccupied is |uk |2 . bk = c−k↓ ck↑ and b∗k = c∗k↑ c∗−k↓ are the creation and annihilation operators for pairs. These satisfy the commutation relations [bk , b∗k0 ]− = (1 − nk↑ − n−k↓ )δkk0 (1.11). [bk , bk0 ]− = 0 [bk , bk0 ]+ = 2bk bk0 (1 − δkk0 ).. Note that from the above commutation relations, the pair operators have both boson and fermion characters! From the pairing Hamiltonian (or reduced Hamiltonian) together with the BCS ground-state wave function, one can determine uk and vk by the variational method. Here, we derive the two coefficients in details since they have important applications in junction tunneling. Pairing Hamiltonian can be written as Hp =. X kσ. εk nkσ +. X. Vkl c∗k↑ c∗−k↓ c−l↓ cl↑ ,. (1.12). kl. which corresponds to the BCS ground-state wave function. Here, we omit many other terms involving electrons not paired [such as (k ↑, −k ↑) one] because these terms have.

(10) 5 zero expectation value in the BCS ground-state wave function. (However, they may be important in other applications.) We first minimize the expectation value of Hp − µN δ hψG | Hp − µN |ψG i = 0.. (1.13). After substituting the BCS ground-state wave function into hψG | Hp − µN |ψG i, one obtains hψG | Hp − µN |ψG i = 2. X. ξk |vk |2 +. k. X. Vkl uk vk ul vl .. (1.14). kl. Eq. (1.14) will be minimized subject to the constraint of |uk |2 + |vk |2 = 1. It is convenient to set uk = sin θk and vk = cos θk . The right-hand side of Eq. (1.14) then becomes X. ξk (1 + cos 2θk ) +. k. 1X Vkl sin 2θk sin 2θl . 4 kl. (1.15). After differentiation with θk , one obtains X ∂ hψG | Hp − µN |ψG i = 0 = −2ξk sin 2θk + Vkl cos 2θk sin 2θl . ∂θk l. (1.16). An alternative form of Eq. (1.16) is P. tan 2θk =. l. Vkl sin 2θl .. 2ξk. (1.17). Defining that ∆k = − q. Ek =. X. Vkl ul vl = −. l. 1X Vkl sin 2θl 2 l. ∆2k + ξk2 ,. (1.18). Ek is thus the excitation energy of a quasiparticle, while ∆k is the energy gap. With the above definitions, Eq. (1.17) becomes tan 2θk = −. ∆k . ξk. (1.19). Combining uk = sin θk , vk = cos θk , and the above equation, one obtains 2uk vk = sin 2θk =. ∆k Ek. (1.20). and vk2 − u2k = cos 2θk = −. ξk . Ek. (1.21).

(11) 6 Note that the choice of signs for the sine and cosine gives the occupation number vk2 → 0 as ξk → ∞, as is required for a reasonable solution. By some algebra, one can write the gap equation as ∆k = −. 1X ∆l q Vkl . 2 l ∆2 + ξ 2 l. (1.22). l. Note that the trivial solution is ∆k = 0. The most simplest nontrivial solution is ∆k = ∆ for |ξk | < h ¯ ωc and ∆k = 0 for |ξk | > h ¯ ωc by means of Cooper’s pairing potential model, i.e., Vkl = −V if |ξl |, |ξk | < h ¯ ωc ; Vkl = 0 if otherwise. Consequently, the gap equation becomes 1=. V X 1 . 2 k Ek. (1.23). Due to BCS theory, k changes smoothly near Fermi surface, so we can replace the summation by an integration from −¯hωc to h ¯ ωc . Gap equation thus becomes Z ¯hωc 1 dξ h ¯ ωc √ 2 = = sinh−1 . 2 N (0)V ∆ ∆ +ξ 0. (1.24). Taking the weak-coupling limit N (0)V ¿ 1, we obtain a more elegant form ∆ ≈ 2¯hωc exp[−1/N (0)V ].. (1.25). The last step is to determine the coefficients of BCS ground state wave function. From Eq. (1.21) and the normalization condition, the athirst results are 1 ξk vk2 = (1 − ) 2 Ek. (1.26). 1 ξk u2k = (1 + ). 2 Ek. (1.27). and. Note that it is very surprising, after plotting vk2 versus electron energy measured from the chemical potential, the curve of vk2 at T = 0 is almost the same as the Fermi function at T = Tc !.

(12) Chapter 2 Some Basic Properties of Hight-Temperature Superconductors 2.1. Structure of Hight-Temperature Superconductors. High-temperature superconductors (HTSC) or cuprate superconductors are known as doped Mott insulators and can be divided into electron-type and hole-type. They have some different properties due to existing different kinds of order such as antiferromagnetism (AF), competing orders for example spin and charge-density waves, and/or superconducting order in them. Fig. 2.1 shows the perovskite crystalline structures of two types of HTSC. Because of the strong interaction between electrons in HTSC, “conventional” BCS theory fails to describe the behavior of these “unconventional” superconductors. Nevertheless, it is believed that the concept of Cooper pair remains valid! Many people attempt to find the pairing mechanism. For example, Emery argued that paring is mediated by a strong coupling to the local spin configurations on Cu sites [5]. Fig. 2.2 shows the crucial characteristic of HTSC, i.e., two dimensional CuO2 planes.. 7.

(13) 8. Figure 2.1: The figure shows the perovskite crystalline structures of two types of HTSC (after Yeh, 2007).. 2.2. Phase Diagram of Hight-Temperature Superconductors and Antiferromagnetic Order. As the schematic phase diagram of HTSC shown in Fig. 2.3, the antiferromagnetism and superconductivity may be coexisting for underdoped hole-type HTSC and optimallyand under-doped electron-type HTSC. The phase diagram was drawn according to experimental data. In theoretical studies, many paper also reported the coexistance of antiferromagnetism and superconductivity [7, 8, 9] and the quantum phase transition between antiferromagnetic Mott insulator and d-wave superconductor in two dimension [10]. It is valuable to briefly mention some theoretical works for the interplay between antiferromagnetism and superconductivity. To our best knowledge, most works were based on the Hubbard model. For example, Inui et al. [7] start from the single-band Hubbard Hamiltonian H = −t. X i,m. c∗iσ ci+m,σ + U. X i. ni↑ ni↓ + V. X. (ni − 1)(ni+m − 1),. (2.1). i,m. where t is the nearest-neighbor hopping matrix element, and U and V are the onsite and intersite Coulomb repulsions, respectively. They found that, as a function of t/U and band filling, the phase diagram exhibits a phase which is simultaneously superconducting and antiferromagnetic (see Fig. 2.4). Assaad et al. [10] gave a new view on how Mott insulator becomes superconducting..

(14) 9. Figure 2.2: Schematic of CuO2 plane which is the crucial element for Mott insulator and HTSC. Speckled shading indicates oxygen pσ orbitals coupled to Cu dx2 −y2 orbitals that leads to superexchange in the insulating phase and carrier motion in the doped metallic phase (after Orenstein and Millis [6]).. They also based on the half-filled Hubbard model and wrote the Hamiltonian as HU = −. X tX 1 1 K~i + U (n~i ↑ − )(n~i ↓ − ), 2 ~i 2 2 ~i. (2.2). where the hopping kinetic energy is K~i =. X ~ δ,σ. (c~∗iσ c~i+~δ,σ +c~∗i+~δ,σ c~i,σ ). (2.3). with ~δ = (±ˆ ax , ±ˆ ay ) (ˆ ax ,ˆ ay are lattice constants). In particular, they have added a new interaction term HW = −W. X ~i. K~i2. (2.4). where W is positive. They found that at half filling and a constant value of the Hubbard repulsion, increasing the strength of the interaction W drives the system from an antiferromagnetic Mott insulator to a d-wave superconductor. Another model is the SO(5) theory which is capable of classifying the complex phases between Mott insulator and HTSC. SO(5) theory gave a whole new phase diagram of HTSC which is shown in Fig. 2.5. In the last part of this section, I briefly review motivation and main idea of the SO(5) theory. SO(5) theory is motivated by a confluence of the phenomenological top-down approach with the microscopic bottom-up approach. From phenomenological top-down.

(15) 10. Figure 2.3: A schematic phase diagram of HTSC. AFM denotes the antiferromagnetic phase, SC denotes the superconducting phase, and CO denotes the competing order. TN and Tc are the N´eel and superconducting transition temperatures, repectively; T ∗ and TP G stand for the low-energy and high-energy pseudogap crossover temperatures, repectively (after Yeh, 2007).. Figure 2.4: A T = 0 and V = 0 phase diagram of t/U vs. δ. s denotes the s-wave superconductivity, d denotes the d-wave superconductivity, AF stands for antiferromagnetism, and FL stands for paramagnetic Fermi liquid (taken from Inui et al. [7])..

(16) 11. Figure 2.5: A theoretical and typical global phase diagram of the extended SO(5) model in the parameter space of chemical potential and the ratio of boson hopping energy over interaction energy. A1, A2, and A3 denote the quantum phase transitions of class A which can be approached by varying the hopping energy, for example, by applying a pressure and magnetic field at constant doping; B1, B2, and B3 denote the quantum phase transitions of class B which can be realized by changing the chemical potential or doping. The vertical dash-dotted line denotes a boundary in the overdoped region (taken from Demler et al. [11]).. approach, SO(5) theory is philosophically inspired by the Landau-Ginzburg theory, a highly successful phenomenological theory in which one first makes observations of the phase diagram, then introduces one order parameter for each broken symmetry phase and constructs a free-energy functional by expanding in terms of different order parameters. A central macroscopic hypothesis of the SO(5) theory is that the ground state and the dynamics of collective excitations in various phases of the HTSC can be described in terms of the spatial and temporal variations of the superspin which can.

(17) 12 be written as       na =     . Re∆ Nx Ny Nz.       ,    . (2.5). Im∆ where Nx ,Ny , and Nz are the three-dimensional AF order parameter, while Re∆ and Im∆ are the two-dimensional SC order parameter. It does include a homogeneous state in which AF and SC phases coexist microscopically that I will be studying in this thesis. Also, it includes states with spin and charge-density-wave orders, such as stripe phases, checkerboards, and AF vortex cores, which can be obtained from spatial modulations of the superspin. From bottom-up approach, the basic microscopic hypothesis of the SO(5) theory is that AF and SC states arise from the same interaction with a common energy scale of J. This common energy scale justifies the treatment of antiferromagnetism and superconductivity on an equal footing and is also the origin of an approximate SO(5) symmetry between these two phases. The relation between SO(5) theory and microscopic models can be summarized by the following block diagram:. Figure 2.6: The relation between SO(5) theory and microscopic models. (taken from Demler et al. [11]).. It is worth mentioning that using SO(5) theory to study FM/I/HTSC or FM/AF/ HTSC junctions remains open..

(18) Chapter 3 Tunneling Theory 3.1. Density of States. Quasi-particle excitations can be simply described as fermions created by the γk∗ , which are in one-to-one correspondence with the c∗k of the normal particle. Here we assume that it only changes the density of states on both sides of the tunneling junction and the number of states in any energy interval dE is not changed. Under this assumption, one can obtain a balancing equation of number of states Ns (E)dE = Nn (ξ)dξ.. (3.1). Here E 2 = ∆2 + ξ 2 and Ns (E) and Nn (ξ) are the superconducting and normal density of states respectively. Since the most important states in tunneling are those near the Fermi level, one can take Nn (ξ) = N (0). For E > ∆, this leads to Ns (E) dξ E = =√ 2 N (0) dE E − ∆2. (3.2). and for E < ∆, Ns (E)/N (0) = 0. The above is for an s-wave superconductor. For a nodal d-wave superconductor, an additional angular average is needed in the right-hand side of the above equation.. 3.2. Semiconductor Model. The Semiconductor model is a simple picture which describes the processes of tunneling. In this model, as illustrated in Fig. 3.1, the normal metal is represented as a continuous distribution of independent-particle energy states with density N (0), including energies 13.

(19) 14. Figure 3.1: Semiconductor model of three type junctions and their conductance spectroscopies. Left curves: density of states plotted horizontally vs. energy vertically. Shading denotes states occupied by electrons. Right curves: tunneling current vs. bias voltage. (a) N/N tunneling at T 6= 0. The current reveals Ohm’s law. (b) N/S tunneling at T 6= 0. When bias voltage above the gap, electrons can tunnel from the right (normal metal) into empty states on the left (superconductor). The current does not obey the Ohm’s law and nonlinear behavior is revealed because of the gap in superconductor. (c) S/S tunneling at T 6= 0. The current shows a more complex behavior because there are two gaps in the junction (taken from Giaever and Megerle [12]).. below as well as above the Fermi level. The superconductor is represented by an ordinary semiconductor with a density of independent-particle energy states by adding its reflection on the negative-energy side of the chemical potential [see Fig. 3.1(b)]. One sees that the states totally unoccupied above the gap for the superconductor, thus electron tunneling to superconductor side is possible..

(20) 15. 3.3. Formula of Tunneling Current. Within the independent-particle approximation, the tunneling current from metal or superconductor 1 to metal or superconductor 2 can be written as I1→2 = A. Z ∞ −∞. |T |2 N1 (E)f (E)N2 (E + eV )[1 − f (E + eV )]dE,. (3.3). where T is the tunneling-matrix element, V is the applied voltage, and N (E) is the normal or superconducting density states. Subtracting the reverse current, one obtains the net current I = A |T |. 2. Z ∞ −∞. N1 (E)N2 (E + eV )[f (E) − f (E + eV )]dE,. (3.4). where we assume tunneling-matrix element T as a constant. Eq. (3.4) is a general current formula for different kinds of junctions. When both sides of the junctions are normal metal or in the normal state, current formula is Inn = A |T |2 N1 (0)N2 (0). Z ∞ −∞. [f (E) − f (E + eV )]dE. = A |T |2 N1 (0)N2 (0)eV ≡ Gnn V.. (3.5). For N/I/SC junction case, the form becomes 2. Ins = A |T | N1 (0) =. Z ∞ −∞. N2s (E)[f (E) − f (E + eV )]dE. Gnn Z ∞ N2s (E) [f (E) − f (E + eV )]dE. e −∞ N2 (0). (3.6). From Eq. (3.6), one obtains the corresponding conductance formula Gns. Z ∞ N2s (E) ∂f (E + eV ) dIns = Gnn [ ]dE. = dV ∂(eV ) ∞ N2 (0). (3.7). When T → 0, it will reduce to a more simple form Gns (T → 0) =. dIns N2s (e |V |) (T → 0) = Gnn . dV N2 (0). (3.8). Note that one can utilize Eq. (3.7) and measure the conductance to determine the density of states of quasiparticle. In the limit kT ¿ ∆, it can be reduced to µ. Gns (V → 0) 2π∆ = Gnn kT. ¶1/2. e−∆/kT .. (3.9).

(21) 16. Figure 3.2: The process of Andreev reflection in a N/S junction at the interface (after Wikipedia, 2008).. 3.4. Andreev Reflection. Andreev reflection reveals the emerging phenomenon of a superconductor. In the mathematical view, it is a simplest solution of Bogoliubov equation, i.e., when x → −∞, one can put ∆ = 0 and thus the reflecting wave function . ψrefl = a . 0 1. .  iq − x. e. + b. 1 0.  +  e−iq x .. (3.10). Andreev reflection behavior is not easily understood in a natural way. People usually interpret the Andreev reflection as a reflection of a hole. The more physical interpretation, however, is the following. Andreev reflection is a process that ”excites” a electron moving in the opposite direction of the hole. Wolf’s book [13] has a more microscopic description. We follow his statement that because a freely propagating quasiparcitle state at E < ∆ in the superconductor is energetically impossible, it will be exponentially decayed. But reflection of the k > kF electronlike excitation to the degenerate k < kF holelike excitation can occur. This process occurs as the original electron, under the influrnce of the anomalous or pairing potential at the interface, that joins a second electron (leaving a hole in normal metal side), with the resultant pair propagating to the superconductor side (see Fig. 3.2). It should be mentioned that there are also other types of Andreev reflections such as crossed Andreev reflection and specular Andreev reflection. These occur in superconductor-related’s junctions because of different physics in those junctions..

(22) 17. 3.5. Blonder-Tinkham-Klapwijk Model Approach. Bogoliubov-de Gennes (BdG) equations are a easy tool to study tunneling in superconductorrelated junctions due to different symmetry of pairing. The orginal form of BdG equations [4] was written as εu(r) = [He (r) + U (r)]u(r) + ∆(r)v(r) εv(r) = −[He∗ (r) + U (r)]v(r) + ∆∗ (r)u(r), where He (r) =. 1 (−i¯ h∇ 2m. −. eA 2 ) c. (3.11). − EF and U (r) is the arbitrary external potential.. There are two methods to obtain the BdG equations such as those given in de Gennes’ famous book [4] and his review paper [14]. Blonder et al. [15] were the first to systematically apply BdG equations to investigate normal/superconducting micro-constriction contacts. Their work motivate many further investigations. Thus it is valuable to review their paper in more details. In their paper, they create some meaningful physical quantity such as excess current, charge imbalance, and quasiparticle/supercurrent conversion. As widely known, BdG equations comes from Bogoliubov transformation, BCS ground state wave-function, and its corresponding paring Hamiltonian. The solution of BdG equations is related to BCS ground state wave-function’s coefficients. The usual Bogoliubov transformation was written as † γk0 = uk c†k↑ − vk c−k↓ ,. (3.12). where uk and vk are the BCS coherence factors which are given by u2k = 1 − vk2 = (1 + εk /Ek )/2.. (3.13). Note that γk† is an operator that creates quasiparticle excitations of the two spin directions from the superconducting ground state and its corresponding energy is Ek = r q. ∆2k + ε2k . The relation between Ek and k can be shown as h ¯ k± =. 2m(εF ±. q. Ek2 − ∆2k ).. As shown in Fig. 3.3, there exist four k’s which give rise to a same Ek . Considering the semiconductor model appropriate for a superconductor, one can extend Fig. 3.3 to include the quasiparticle branches to the lower half energy plane. One thus obtains the picture of Fig. 3.4. Based on Fig. 3.5, we now consider A(E), B(E), C(E), and D(E) as the four probabilities of tunneling at a N/S junction which can be determined by matching.

(23) 18. Figure 3.3: The dispersion of excitation energy Ek with k along the x axis (after Blonder, Tinkham, and Klapwijk, 1982).. Figure 3.4: Semiconductor model of Fig. 3.3 that extends the quasiparticle branches to the lower half plane (after Blonder, Tinkham, and Klapwijk, 1982).. the slope and the value of the wave function across the interface. More specifically, A(E) is the probability of Andreev reflection as a hole on the other side of the Fermi surface, while B(E) is the probability of normal reflection. C(E) and D(E) are the probabilities for the transmission of a electron and a hole into the superconducting side, respectively. The conservation of probability requires that A(E) + B(E) + C(E) + D(E) = 1.. (3.14). Note that for |E| < ∆, there are no transmitted quasiparticles. Based on the tunneling probability consideration and assuming that the scattering at the interface is ballistic, Blonder et al. obtained a current formula of N/S junctions.

(24) 19. Figure 3.5: Schematic diagram of the possible tunneling processes at the N/S interface. Open circles denote holes, closed circles denote electrons, and the arrows points denote the direction of the group velocity (after Blonder, Tinkham, and Klapwijk, 1982).. which differs from Eq. (3.6), i.e., IN S = 2N (0)evF A. Z ∞ −∞. [f0 (E − eV ) − f0 (E)][1 + A(E) − B(E)]dE.. (3.15). Here A is an effective-neck cross-sectional area and N (0) refers to the one-spin density of states at ²F . If both sides of the interface are normal metal, Eq. (3.15) reduces to IN N =. 2N (0)e2 vF A V V ≡ 2 1+Z RN. (3.16). Figs. 3.6 and 3.7 are their main results. Fig. 3.6 shows that the current running through the N/S junction can be greater or lesser than the current through the N/N junction at T = 0. Fig. 3.7 shows the conductance which can be experimentally measured directly. The most interesting result is the Andreev reflection that leads to a double-time value of the conductance in the normal state. Besides, Blonder et al. proposed a measurable physical quantity, the excess current Iexc , which is defined as Iexc ≡ (IN S − IN N )|eVÀ∆ =. Z ∞ 1 [A(E) − B(E) + B(∞)]dE. (3.17) eRN [1 − B(∞)] 0. However people have paid little attention to the excess current that requires measurements at high bias voltages. Fig. 3.8 shows the normalized excess current as a function of barrier. In addition, they analyzed the charge imbalance generation at the interface. At first, they defined the quasiparticle charge as qk = u2k + vk2 = ±1/Ns (Ek ). (3.18).

(25) 20. Figure 3.6: Current vs. voltage for various barrier strength Z at T = 0 (after Blonder, Tinkham, and Klapwijk 1982).. Figure 3.7: Conductance vs. voltage for various barrier strength Z at T = 0. Clearly, the peaks all locate at the gap value (after Blonder, Tinkham, and Klapwijk, 1982)..

(26) 21. Figure 3.8: Excess current (in units of ∆/eRN ) as a function of barrier strength Z (after Blonder, Tinkham, and Klapwijk, 1982).. and the total quasiparticle charge Q∗ =. X. qk fk .. (3.19). k. In order to define the rate of generation of quasiparticle charge at the interface, they distinguish quasiparticles in the two channels, k + and k − , and also take into account their fractional charge. A formula thus follows IN∗ S = 2N (0)evF A. Z ∞ −∞. [f0 (E − eV ) − f0 (E)][C(E) − D(E)]Ns−1 (E)dE. (3.20). and the ratio F ∗ ≡ IN∗ S /IN S gives the charge imbalance generated as a fraction of the injected current. We recall that IN S was given in (3.15) which, by means of the conservation law, can be rewritten as IN S = 2N (0)evF A. Z ∞ −∞. [f0 (E − eV ) − f0 (E)][2A(E) + C(E) + D(E)]dE.. (3.21). Due to different natures in normal side and superconducting side, there must exist current conversion process near the interface. The most important point is that there are solutions of the Bogoliubov equations even for |E| < ∆, i.e., the Andreev reflection wave function related to Andreev reflection. However they are evanescent waves which decay in a distance √ h ¯ vF /2 ∆2 + E 2 ∼ ξ(T ).. (3.22).

(27) 22. Figure 3.9: Schematic diagram of conversion of normal current to supercurrent at N/S interface. The evanescent waves due to Andreev reflection die out over a distance of order ξ, while the injected quasiparticle charge imbalance relaxes over a charge diffusion length ΛQ∗ (after Blonder, Tinkham, and Klapwijk, 1982).. Thus Andreev current is carried for ∼ ξ as a quasiparticle current before decaying into a pair current, while quasiparticle charge imbalance diffuse to a more long length ΛQ∗ =. q. DτQ∗ , where D and τQ∗ are the diffusion constant and charge imbalance. relaxation time. Fig. 3.9 shows the current conversion systematically.. 3.6. Tunneling Hamiltonian Approach. In this section, we introduce another approach for the study of junction tunneling. It is the tunneling Hamiltonian which Cohen et al. [16] proposed in 1962 and became universally adopted for the discussion of tunneling. Their idea was that the Hamiltonian is divided into three parts: H = HL + HR + HT ,. (3.23). where the tunneling part HT =. X. † † ∗ Ckσ ). Cpσ (Tkp Ckσ Cpσ + Tkp. (3.24). kpσ. The term HL (HR ) is the Hamiltonian for particles on the left (right) side of the tunneling junction that satisfy [HL , HR ] = 0. The tunneling matrix element Tkp describes.

(28) 23 the particles transferred through an insulating junction. This transfer rate is assumed to depend only on the wave vectors on the two sides k and p and not on other variables, such as the energy or spin of the particles. Note that the tunneling Hamiltonian is believed to be an improper formalism only when the applied voltages are large, say 1eV. Following the derivation in Mahan’s book [17], we utilize the Heisenberg equation of motion on NL , i.e., N˙ L = i[H, NL ] = i[HT , NL ] = i. X. † ∗ † (Tkp Ckσ Cpσ − Tkp Cpσ Ckσ ). (3.25). kpσ. and the total tunneling current is then given by D. E. I(t) = −e N˙ L (t) .. (3.26). By transforming to the interaction representation and taking the first order of the S-matrix, one obtains, I(t) = −ei. Z t. D. −∞. E. dt0 [N˙ L (t), HT (t0 )] ,. (3.27). where 0 0. 0 0. HT (t0 ) = eiH t HT e−iH t. (3.28). 0 0 0 0 N˙ L (t) = eiH t N˙ L e−iH t .. (3.29). and. Here, we set H = HL + HR + HT ≡ H 0 + HT . Because chemical potentials are not the same on the two sides of the system, one should define the Hamiltonian with respect to the chemical potentials, i.e., KR = HR − µR NR , KL = HL − µL NL , and K 0 = KR + KL . Thus the time development of Ckσ operator is governed by Ckσ (t) = eiKR t Ckσ e−iKR t and Cpσ operator by Cpσ (t) = eiKL t Cpσ e−iKL t . After some algebra, one obtains a general formula for the total current I(t) = IS (t) + IJ (t),. (3.30). where IS (t) = e. Z ∞ −∞. 0. D. E. 0. D. E. dt0 Θ(t − t0 ){eieV (t −t) [A(t), A† (t0 )] − eieV (t−t ) [A† (t), A(t0 )] } (3.31).

(29) 24 is the quasiparticle current with A(t) ≡. P kpσ. IJ (t) = e. Z ∞ −∞. † Tkp Ckσ (t)Cpσ (t) and. 0. 0. D. E. dt0 Θ(t − t0 ){e−ieV (t +t) h[A(t), A(t0 )]i − eieV (t+t ) [A† (t), A† (t0 )] } (3.32). is the pair current or supercurrent due to the Josephson effect. It was very surprised that supercurrent is a natural outcome of the tunneling Hamiltonian! The quasiparticle current can be expressed by the spectral functions [17], i.e., IS (t) = 2e. X. |Tkp |. kp. 2. Z. dξ AL (p, ξ + eV )AR (k, ξ)[f (ξ) − f (ξ + eV )]. 2π. (3.33). Substituting the spectral functions for the normal and superconducting sides of the junction into it, then GN S eV = Θ(eV − ∆) q = ρ(eV ), GN N (eV )2 − ∆2. (3.34). where ρ(eV ) is the density of states of quasiparticle or normalized conductance which Giaever first measured this quantity! Note that from tunneling Hamiltonian method, one can obtain a normalized conductance formula which in other words, is the density of states of quasiparticle! Finally, it’s worth mentioning that there also exists another method, namely the Green’s function approach [18, 19, 20, 21, 22]. In this thesis, we shall not touch this approach however..

(30) Chapter 4 Tunneling in Different Kinds of Metal/Insulator/Superconductor Junctions 4.1. Introduction. In this chapter, we apply the theory of tunneling to investigate metal/insulator/ superconductor (N/I/S) junctions. Since there are many kinds of superconductor-related junctions, it is convenient to introduce some short hands for them. In this thesis, we are only concerned of four types of N/I/S junctions which are normal metal/insulator/swave superconductor (N/I/sSC), ferromagnetic metal/insulator/s-wave superconductor (FM/I/sSC), metal/insulator/d-wave superconductor (N/I/dSC), and ferromagnetic metal/insulator/d-wave superconductor (FM/I/dSC). However, as far as HTSC of d-wave pairing symmetry is concerned, they are also divided into hole-doped superconductor (HDSC) and electron-doped superconductor (EDSC). Here, normal metal means no impurity (for simplicity) and no ferromagnetism in it. Due to the diversity of electron tunneling spectroscopy both in theories and in experiments, we neglect some interesting effects such as multiple gaps, effects of magnetic field, magnetic impurities, pressure effects, interactions with electromagnetic radiation and superconducting fluctuations completely. For a detailed discussion on these effects, we recommend Wolf’s book [13].. 25.

(31) 26. Figure 4.1: Tunneling current vs. bias voltage for a N/I/sSC junction at different temperatures – the lower T the lower curves (after Giaever, 1960).. 4.2. N/I/sSC Junctions. N/I/CSC junction tunneling was a powerful tool that can probe the density of states and the superconducting gap structure. This technique was pioneered by Giaever [23] who first measured the tunneling current in Al/AlO/Pb sandwiches junction under different temperature. In his experiment, he applied magnetic field to adjust the state transition. His data are shown in Fig. 4.1. Fig. 4.2 plots the slope, dI/dV , or the conductance at T = 1.6K and H = 0, where Pb is in the superconducting state. To understand his measurement, Giaever assumed that the conductance must measure the density of states. Later, his assumption was confirmed by the theoretical works of Bardeen [24] and Cohen et al. [16]..

(32) 27. Figure 4.2: The conductance dI/dV vs. bias voltage. The curve is made from the slope of curve 5 relative to the slope of curve 1 in Fig. 4.1 (after Giaever, 1960).. 4.3. FM/I/sSC Junctions. Using point contact technique to measure the spin polarization in ferromagnetic metal/ conventional superconductor (FM/CS) junctions was pioneeringly done by Soulen et al. [25] and Upadhyay et al. [26] in 1998. Their works showed that determining the spin polarization at Fermi surface is essentially not an easy task. That leads to some definitions of spin polarization such as “tunneling polarization” proposed by Tedrow and Meservey [27] and “point-contact polarization” proposed by Soulen et al. [25]. One year later, Zhu et al. [28, 29] and Kashiwaya et al. [30] have utilized the ideas to study the spin-polarized quasiparticle transport in ferromagnet/d-wave superconductor junctions. Zhu et al. [28, 29] predicted that conductance resonances occur in a normal-metal-ferromagnet/d-wave superconductor junction and in a following paper, they have also studied the junctions by solving the Bogoliubov-de Gennes (BdG) equations within an extended Hubbard model which included the proximity effect, the spin-flip interfacial scattering at interface, and the local magnetic moment. They have reported that the proximity induced order parameter oscillation in the ferromagnetic region. In contrast, Kashiwaya et al. [30] emphasized the spin current and spin filtering effects at the magnetic interface. In the works of Zutic and Valls [31, 32], they first considered the effect of Fermi-wave vector mismatch (FWM). They have pointed out that if one neglects FWM, the effect of spin polarization invariably leads to the suppression of Andreev reflection (AR). Among many other junction studies, Dong et al. [33] studied a little different junction which forms a four layer sandwich, i.e.,.

(33) 28. Figure 4.3: Schematic of the process for Pc = 0 where the Andreev reflection is unhindered by a spin minority population at EF . The solid circles denote electrons and open circles denote holes. The black and ashy area are the occupied states, and white area are the unoccupied states (after Soulen et al., 1998).. FM/I/d + is/d-wave junctions by taking into account the roughness of the interfacial barrier and broken time-reversal symmetry states. After the pioneering works of Soulen et al. and Upadhyay et al., several experimental groups [34, 35, 36, 37, 38, 39, 40, 41] followed. Normal and ferromagnetic metal/conventional superconductor or s-wave superconductor (FM/s-wave SC) junctions have been intensely studied experimentally and theoretical modelings (BlonderTinkham-Klapwijk (BTK) formula [15] and its extension) had a good fitting with the conductance data measured. Recently Linder and Sudbø [42] presented a theoretical study of FM/s-wave SC junction that investigated the possibility of induced triplet pairing state in the ferromagnetic metal side. They also used the BTK approach but allowed for arbitrary magnetization strength and direction in the ferromagnet, spinactive barrier, Fermi-vector mismatch, and different effective masses in the two side of the junction. As is well known, there is no retroreflection process when an exchange.

(34) 29 field is present. However, they pointed out that retroreflection can occur under some conditions [42]. In this section, we briefly review Soulen et al. [25] and Upadhyay et al.’s [26] experimental work since they are the first using Andreev reflection to measure spin polarization. The spin degrees of freedom emerge in FM/I/sSC junctions that results in the modification of the semiconductor model as follows (see Fig. 4.3). Based on the point contact technique, a definition of spin polarization or contact polarization is proposed: Pc =. N↑ (EF )vF ↑ − N↓ (EF )vF ↓ , N↑ (EF )vF ↑ + N↓ (EF )vF ↓. (4.1). where vF σ is the Fermi velocity for spin-σ particles. Because Iσ ∝ Nσ (EF )vF σ , this leads to Pc =. I↑ − I↓ . I↑ + I↓. (4.2). They further relate the contact polarization to the normalized conductance. Let I = I↑ + I↓ = 2I↓ + (I↑ − I↓ ) ≡ Iupol + Ipol ,. (4.3). where Iupol is the unpolarized current or supercurrent, while Ipol is the polarized or quasiparticle current. Differentiating both sides of Eq. (4.3) with respect to V , one obtains d d d I(V, T, Pc , Z) = Iupol (V, T, Pc , Z) + Ipol (V, T, Pc , Z). dV dV dV. (4.4). As suggested by Soulen et al. [25], it can be viewed that Iupol carries no net spin polarization (Pc ) and obeys the conventional BTK theory, while Ipol carries all of Pc , for simplicity (by ignoring the nonlinear effect), one can instead use the following phenomenological formula d d d I(V, T, Pc , Z) ≡ (1 − Pc ) Iupol (V, T, Z) + Pc Ipol (V, T, Z). dV dV dV. (4.5). Eq. (4.5) is the formula commonly used in the study of spin-polarized junction tunneling. If the interfacial scattering is minimal, Z ≈ 0, and when eV ¿ ∆ and kT ¿ ∆, there is no normal reflection and Andreev reflection coefficient becomes unity. That is 1 d Iupol = 2 GN dV. (4.6).

(35) 30. Figure 4.4: Conductance for several spin-polarized metals showing the suppression of Andreev reflection with increasing spin polarization Pc (after Soulen et al., 1998).. and d Ipol = 0. dV. (4.7). 1 d I(V → 0, T → 0, PC , Z = 0) = 2(1 − PC ), GN dV. (4.8). Therefore. or in a more common form G(0)/GN = 2(1 − Pc ).. (4.9). Soulen et al. [25] utilized this formula and combined with the experimental conductance data to determine the contact polarization. Fig. 4.4 gives their measured result. In Table 4.1, Soulen et al. [25] summarized the contact polarization of several FM metals and have compared the results with those of Tedrow and Meservey [27]. The work of Upadhyay et al. [26] is somewhat different from that of Soulen et al. [25]. It should be mentioned that Upadhyay et al. in fact measured the normalized Andreev reflection conductance. Fig. 4.5 shows schematically their experimental setup. In their work, they defined a dimensionless function g(V ) ≡. GS (V ) − GN (V ) . GN (0). (4.10).

(36) 31. Table 4.1: Summary of experimental results of Andreev reflections for determining the Pc at EF of several FM metals (after Soulen et al., 1998). N is the number of point contact adjustments. Each adjustment represents a distinct point contact junction and an independent determination of Pc . Columns 5 and 6 represent a comparison between the PT reported in Ref. [27] and the value of Pc reported in Ref. [25].. It is a normalized Andreev reflection conductance or a measure of the Andreev reflection probability. Fig. 4.6 was their experimental results showing that the spin polarization caused the zero-bias peak split. Table 4.2 summarizes the data of Upadhyay et al. [26]. In view of Tables 4.1 and 4.2, it seems that the value of spin polarization was not consistent and may be dependent of experiment setups and samples.. Figure 4.5: Schematic of a Pb-Co nanocontact (after Upadhyay et al., 1998)..

(37) 32. Figure 4.6: (a) g(V ) for two Pb-Co samples with normal state resistances 15.5 Ω (◦) and 12.1 Ω (+) at 1.41 K; (b) g(V ) for a Pb-Ni device with normal state resistance of 7.3 Ω at 2.5 K. The solid lines are a fit of modified BTK model to the data (after Upadhyay et al., 1998).. Table 4.2: Spin polarization and transmission coefficients of direct FM/I/S interface currents as measured by Andreev reflection, and in comparison with previous data obtained by Soulen et al. [25]. Note that the definitions of P (tunneling) and P (Andreev) are different (after Upadhyay et al., 1998)..

(38) 33. 4.4. N/I/HTSC Junctions. As Qazilbash et al. mentioned in their paper [43], the pairing symmetry is predominantly d-wave in hole-doped copper oxide high temperature superconductor that are supported by many different types of experiment. However, in the electron-doped HTSC, the point contact spectroscopy (PCS) and tunneling data have been more controversial. For example, the absence of a zero-bias conductance peak in the tunneling spectra is argued to be against the d-wave pairing symmetry, while other types of experiments support a d-wave scenario. Therefore, in this thesis and the paper to be published, we present a model to study this problem. Since the PCS in the hole-doped HTSC is well established, we will focus on the electron-doped side of the HTSC. In the following we summarize some recent experimental and theoretical works of PCS on the electron-doped HTSC. Fig. 4.7 shows experimental PCS results of underdoped Pr2−x Cex CuO4 (PCCO). No ZBCP splitting is found and the result is interpreted to be in support of d-wave pairing in underdoped PCCO. It is useful to compare the results with those of hole-doped YBa2 Cu3 O7−δ (YBCO) (see Fig. 4.8). In contrast to the results in Fig. 4.7, PCS data on electron-doped HTSC in Figs. 4.9 and 4.10 are shown to exhibit ZBCP splitting. Moreover, the phenomenon of ZBCP splitting is strongly doping dependent. We will discuss this phenomenon in more details in next chapter. On the theoretical side, Figs. 4.11 and 4.12 show a theoretical fitting using a generalized BTK model [43] and four-component generalized BTK model [44]..

(39) 34. Figure 4.7: Conductance of underdoped PCCO at T = 1.43 K with the variation of magnetic field applied parallel to the c axis. RN for this junction is 9.8Ω. The spectra are shifted for clarity. Inset (a): the solid line is the fit of the theoretical calculation (after Qazilbash et al., 2003).. Figure 4.8: Conductance of optimally doped YBCO taken at T =4.23 K in a magnetic field applied parallel to the c axis of the film. RN at high-voltage bias is 49Ω (after Qazilbash et al., 2003)..

(40) 35. Figure 4.9: Normalized conductance of optimally doped PCCO for different junction resistances at T =1.43 K (after Qazilbash et al., 2003).. Figure 4.10: Normalized conductance of overdoped PCCO for different junction resistances at T =1.43 K (after Qazilbash et al., 2003)..

(41) 36. Figure 4.11: (a) A fit of the low-resistance (RN = 2.6 Ω) conductance (circles) for overdoped PCCO to the generalized BTK model with an isotropic gap (solid line). (b) A fit of the same data to the d + is pairing symmetry model (after Qazilbash et al., 2003).. Figure 4.12: Fitting to the low-resistance conductance data of electron-doped cuprates by using four-component generalized BTK model (after Liu and Wu, 2007)..

(42) Chapter 5 A Detailed Study of FM/I/dSC Junction 5.1. Introduction. Following the idea of Soulen et al., if one replaces the conventional superconductor by the high-temperature or d-wave superconductor into the junction, it will exist some novel phenomena due to its d-wave pairing symmetry, complex band structure and magnetic properties, and other order parameters such as AF order. Of equal interest, it is strongly suggested that antiferromagnetic (AF) order may coexist with the dwave superconducting order in the electron-doped cuprate superconductor (EDSC), especially in the underdoped and optimally-doped regimes [44]. In this chapter, we shall explore the possible novel phenomena in the FM/EDSC junction case, taking into account the interplay between antiferromagnetic order and spin polarization. The ideas and models developed in NM/CS junctions will be applied to current FM/EDSC junctions case. This chapter is organized as follows. In Sec.5.2, the basic formulation is given. We set up the condition of the junction and generalize the Bogoliubov-de Gennes equations to include AF order parameter. As the formal process, we utilize WKBJ approximation to obtain the more simple Andreev-like equations, solving them and determining the four spin-dependence reflection coefficients. Sec. 5.3 is our main results and discussions. In Sec. 5.3.1, the surface bound state condition was derived. In Sec. 5.3.2, the effect of FWM was studied. In Secs. 5.3.3 and 5.3.4, we discuss the effect of spin-polarization and the generalized effective barrier, respectively. In Sec. 5.3.5, we propose a more. 37.

(43) 38 general formula to determine the spin-polarization using the experimental zero-bias conductance data.. 5.2. Formalism. Our formulation is given based on the following assumptions. We consider a point contact or planar FM/I/EDSC junction where the superconductor overlayer is coated with a clean, size-quantized, normal-metal overlayer of thickness d, that is much shorter than the mean free path l of normal electrons. The interface is assumed to be perfectly flat and infinitely large. Considering l → ∞ limit, the discontinuity of all parameters at the interface can be neglected, except for the SC order parameter to which the proximity effect is ignored [45]. When both SC and AF orders exist, quasiparticle (QP) excitations of an inhomogeneous superconductor can have a coupled electronhole character associated with the coupled k and k + Q [Q = (π, π)] subspaces. QP states are thus governed by the generalized BdG equations [4, 44] ˆ σ u1σ (x) + Eu1σ (x) = H Z. Ev1¯σ (x) =. dy∆(s, r)v1¯σ (y) + Φu2σ (x). ˆ σ v1¯σ (x) + Φv2¯σ (x) dy∆∗ (s, r)u1σ (y) − H. ˆ σ u2σ (x) − Eu2σ (x) = H Z. Ev2¯σ (x) = −. Z. Z. dy∆(s, r)v2¯σ (y) + Φu1σ (x). ˆ σ v2¯σ (x) + Φv1¯σ (x), dy∆∗ (s, r)u2σ (y) − H (5.1). ˆ σ ≡ −¯ where s ≡ x − y, r ≡ (x + y)/2, H h2 ∇2 /2m − EFF,S − σJ with J the exchange energy and σ = 1 (−1) for up (down) spin, σ ¯ = −σ, and Φ is the AF order parameter. In the FM region, we define EFF ≡ h ¯ 2 qF2 /2m = (¯h2 qF2 ↑ /2m + h ¯ 2 qF2 ↓ /2m)/2 as the ¯ 2 kF2 /2m, spin averaged value. It differs from the value in the superconductor, EFS = h to which a FWM is assumed between the FM and EDSC regions for generality [32]. In (5.1), the two-component wave functions u1 and v1 are considered related to k subspaces, while u2 and v2 are related to k + Q subspaces. Comparing to the first and second lines of Eq. (5.1), minus sign associated with the ∆(s, r) terms in the third and fourth lines occurs because for d-wave SC gap, ∆(k + Q) = −∆(k), with ∆(k) the Fourier transform of the Cooper pair order parameter in the relative coordinate [45, 46]. It is emphasized that Andreev reflection are largely modified due to the exchange.

(44) 39 field of ferromagnetic metal when electron is not normally incident into the EDSC region. Owing to the momentum conserved parallel to the interface, the Snell’s law [47, 32, 30] limits the Andreev reflection angle, i.e., incident angle is typically not equal to the Andreev reflection angle except when J = 0 or for normal incidence. The Snell’s law requires that qF σ sin θN σ = qF σ¯ sin θA¯σ = kF sin θSσ ,. (5.2). where θN σ , θA¯σ , and θSσ are the angle of normal reflection, angle of Andreev reflection, and angle of transmission into the SC respectively. Assuming that there is no FWM, ranges of the six scattering angles will be limited to 0 < θN ↑ < sin−1 (kF /qF ↑ ) ≡ θc2 , 0 < θA↑ < sin−1 (qF ↓ /qF ↑ ) ≡ θc1 , and 0 < θS↓ < sin−1 (qF ↓ /kF ); θA↓ and θS↑ can be any angles. AF-Andreev reflection angle and AF-Normal reflection angle are θAσ + π and 2π−θN σ respectively. We emphasize that Snell’s law also allows sin−1 (qF ↓ /qF ↑ ) < θA↑ < sin−1 (kF /qF ↑ ). Within this range, qF ¯↑ becomes. q. qF2 ↓ − qF2 ↑ sin2 θN ↓ (purely imaginary). [32, 30]. It affects the conductance largely if the Andreev reflection angle for spin down electron is in this range. Although Andreev reflection for spin down electron as a propagating wave is impossible, it can still transmit into the superconductor side. For clearly presenting the main physical idea, the Fermi surface (FS) will be approximated by a square (see Fig. 5.1). Under the WKBJ approximation [48, 49, 50, 51, 15, 45, 52, 46], . u1σ. . . eikF ·r u˜1σ.       ikF ·r  v1¯σ   e v˜1¯σ  =    ikF Q ·r  u2σ   e u˜2σ   . v2¯σ.     ,   . (5.3). eikF Q ·r v˜2¯σ. one obtains a set of Andreev equations in the x direction, ˆF )˜ E u˜1σ (x) = Hσ u˜1σ (x) + ∆(k v1¯σ (x) + Φ˜ u2σ (x) ˆF )˜ E˜ v1¯σ (x) = ∆∗ (k u1σ (x) − Hσ v˜1¯σ (x) + Φ˜ v2¯σ (x) (5.4) ˆF Q )˜ E u˜2σ (x) = Φ˜ u1σ (x) − Hσ u˜2σ (x) + ∆(k v2¯σ (x) ˆF Q )˜ E˜ v2¯σ (x) = Φ˜ v1¯σ (x) + ∆∗ (k u2σ (x) + Hσ v˜2¯σ (x), 2. where Hσ = − i¯hmkF. d dx. − σJ. Solving the above system of first-order differential equa-. tions, one obtains four eigenvectors which form the wave function in the superconductor.

(45) 40. No rm. AF N. or m. al r. al r. e fle c. ef le c. i Inc. ti on. Insulat or. Normalmetal. Superconductor. c El e. ke Q nl i tr o. Hol e li. ti on. P. ke Q P. ( π, π). t den. e dr e An. A. ion le c t f e vr. e nd r F -A. ev. ec r efl. tion. (-π , π ). θ Q. kF. kF θ Q. kx (π-,. π). (- π,- π). Figure 5.1: Schematic plot showing all possible reflection and transmission processes for an electron of particular spin incident into an FM/EDSC junction, where an AF order exists in the SC side. For convenience for a d-wave superconductor, kx axis is chosen to be along the [110] direction. The right-bottom inset shows the incident wave vector kF = (kF , ky , kz ) and its corresponding AF wave vector kF + Q ≡ kF Q = (−kF , ky , kz ) due to an AF coupling. Both vectors are tied to the Fermi surface, which is approximated by a square (thick line)..

(46) 41 region [32], i.e., . . ∆0. . . E+.           E−   ∆0     ψSσ (x) = c1σ   + c2σ    0   Φ    . Φ . . E−. .     ik+ x  e   . 0 . −∆0.          E+  −∆0      + c c + 4σ    3σ   0  Φ      .     −ik− x  e   . (5.5). Φ. 0. For simplicity, we will set k + = k − = kF cos θS [15]. Here E± ≡ E ± ε with ε = √ E 2 − ∆2 − Φ2 and ciσ are the coefficients of corresponding waves. One does not need to normalize the quasiparticle amplitude because, as pointed out by Blonder et al., it will only complicate the calculation. If we set Φ = 0, J = 0, and normalize the quasiparticle amplitude, it will reduce to the case for a typical N/I/S junction [15, 52, 46]. Because we consider that there is an AF order in the EDSC side, an incident electron from FM side will have four possible reflections [44]. The wave function [15, 46, 30] in the ferromagnetic metal side (x < 0) for spin σ and an injection angle θN can be written as     ΨN σ (x) =    . eiqF σ cos θN x + RN σ e−iqF σ cos θN x RA¯σ eiqF σ¯ cos θA¯σ x AF iqF σ cos θN x RN σe.     ,   . (5.6). AF −iqF σ¯ cos θA¯ σx RA¯ σe. where the above four coefficients can be determined by the boundary conditions: ψN σ (x) |x=0− = ψSσ (x) |x=0+ , dψN σ (x) 2mH dψSσ (x) |x=0+ − |x=0− . 2 ψSσ (x) |x=0+ = dx dx h ¯. (5.7). The barrier potential is assumed to take a delta function, V (x) = Hδ (x). According to the paper of Kashiwaya et al. [30], one should have two types of conductance in a FM, namely the charge and spin conductances. After some algebra, we obtain the normalized tunneling charge conductance for up spin ¯ ¯. ¯2 ¯. ¯ ¯. ¯2 ¯. AF AF Cq↑ = 1 − |RN ↑ |2 + a |RA↓ |2 + ¯RN ↑ ¯ − a ¯RA↓ ¯ .. (5.8).

(47) 42 and for down spin ¯ ¯. ¯2 ¯. ¯ ¯. ¯2 ¯. AF AF Cq↓ = 1 − |RN ↓ |2 + |RA↑ |2 + ¯RN ↓ ¯ − ¯RA↑ ¯ .. (5.9). Here the factor a ≡ L↓ λ2↓ /L↑ λ1↑ (see later). Note that factor a does not appear in the down spin channel. Similarly, the spin conductance for spin-up channel is obtained to be ¯ ¯. ¯2 ¯. ¯ ¯. ¯2 ¯. ¯2 ¯. ¯ ¯. AF AF Cs↑ = 1 − |RN ↑ |2 − a |RA↓ |2 + ¯RN ↑ ¯ − a ¯RA↓ ¯ ,. (5.10). while for spin-down channel ¯ ¯. ¯2 ¯. AF AF Cs↓ = 1 − |RN ↓ |2 − |RA↑ |2 + ¯RN ↓ ¯ − ¯RA↑ ¯ .. (5.11). Comparing to the results of charge conductance, due to the spin imbalance induced by the exchange field, different sign of RAσ terms occurs in the spin conductances. The four reflection coefficients are obtained to be RN σ = − − RA¯σ = + AF RN σ = AF RA¯ = σ. (E − ε)(1 − Lσ λ1σ + 2iZθ )B (1 + Lσ λ1σ + 2iZθ )D ∆(1 + Lσ¯ λ2¯σ + 2iZθ )A (1 + Lσ λ1σ + 2iZθ )D 1 − Lσ λ1σ − 2iZθ 1 + Lσ λ1σ + 2iZθ ∆(1 + Lσ λ1σ − 2iZθ )B (1 + Lσ¯ λ2¯σ − 2iZθ )D (E − ε)(1 − Lσ¯ λ2¯σ − 2iZθ )A (1 + Lσ¯ λ2¯σ − 2iZθ )D ΦB D ΦA , D. (5.12). where A = 2∆Lσ λ1σ [1 − Lσ Lσ¯ λ1σ λ2¯σ + 4Zθ2 + 2iZθ (Lσ λ1σ + Lσ¯ λ2¯σ )] B = 2Lσ λ1σ [2Lσ¯ λ2¯σ E + ε(1 + L2σ¯ λ22¯σ )] D = ∆2 [(1 − Lσ Lσ¯ λ1σ λ2¯σ + 4Zθ2 )2 + 4Zθ2 (Lσ λ1σ + Lσ¯ λ2¯σ )2 ] + [2Lσ λ1σ E + 4εZθ2 + ε(1 + L2σ λ21σ )] × [2Lσ¯ λ2¯σ E + 4εZθ2 + ε(1 + L2σ¯ λ22¯σ )].. (5.13).

(48) 43 Here we have defined Zθ = Z/ cos θSσ with Z = mH/¯h2 kF , λ1σ = cos θN / cos θSσ , λ2¯σ = cos θA¯σ / cos θSσ , and Lσ = note that the coefficients. AF RN σ. and. q. qF /kF − σ(qF /kF )(J/EF N ). It is interesting to. AF RA¯ σ. in (5.12) are proportional to the AF order Φ,. AF as is expected. Later we will show that RN σ dominates in the total spectra in the case. of large Φ and as a consequence, unusually interesting results occur. The normalized total charge conductance can be written as Gq (E) = Gq↑ (E) + Gq↓ (E),. (5.14). where Gqσ (E) =. 1 Zβ dθN cos θN Cqσ (E, θN )Pσ GN q α. (5.15). with GN q =. Z π/2 −π/2. dθN cos θN [CN ↑ P↑ + CN ↓ P↓ ].. (5.16). In contrast, the normalized total spin conductance is given by Gs (E) = Gs↑ (E) − Gs↓ (E),. (5.17). where Gsσ (E) =. 1 Zβ dθN cos θN Csσ (E, θN )Pσ GN s α. (5.18). with GN s =. Z π/2 −π/2. dθN cos θN [CN ↑ P↑ − CN ↓ P↓ ].. (5.19). In addition, the normalized total charge current Iq can be given by Iq (E) = Iq↑ (E) + Iq↓ (E). (5.20). where Iqσ (E) =. 1 Z IN q. β α. dθN cos θN Cqσ (E, θN )Pσ kF. (5.21). with IN q =. Z π/2 −π/2. dθN cos θN [CN ↑ P↑ qF ↑ + CN ↓ P↓ qF ↓ ]. (5.22).

(49) 44 3 G (. ). G (. ). Normalized Conductance. q. s. G (. ). G (. ). q. 2. s. 1. 0. 0. 1. 2. Normalized Energy E/. Figure 5.2: Normalized charge conductance Gq and spin conductance Gs for Φ = 0.5 and Φ = 0 with Z = 1, X = 0.5, L0 = 1, and θN = 0. and the normalized total spin current Is Is (E) = Is↑ (E) − Is↓ (E). (5.23). where Isσ (E) =. 1 Z IN s. β. α. dθN cos θN Csσ (E, θN )Pσ kF. (5.24). with IN s =. Z π/2 −π/2. dθN cos θN [CN ↑ P↑ qF ↑ − CN ↓ P↓ qF ↓ ].. (5.25). Here CN σ (θN ) =. 4λ1 Lσ |1 + λ1 Lσ + 2iZ|2. (5.26). is the conductance of the junction in the normal state. We introduce a factor Pσ = (EFF + σJ)/2EFF which can be interpreted as the probability of spin σ incident electron related to the exchange energy [30, 32, 42]. The values of α and β are restricted by Snell’s law or experimental setup. It should divide the incident angle to two range, i.e., 0 < |θN | < θc1 and θc1 < |θN | < θc2 when one calculates the conductance. Fig. 5.2 shows a normalized conductance spectra Gq and Gs for Φ = 0.5 and Φ = 0 with Z = 1, X = 0.5, m = 1 and θN = 0 for a d-wave superconductor..

(50) 45. 5.3 5.3.1. Results and Discussions Midgap Surface States. In this section, we derive the condition of midgap surface states in FM/EDSC case. It’s an extension of Hu’s [45] and Liu and Wu’s [44] results. To find the surface bound state on the ferromagnetic metal overlayer, we take the following assumption  . . u˜lσ. .  = e−γσ x . v˜lσ. . uˆlσ. ,. vˆlσ. (5.27) q. ˆF )|2 + Φ2 . Consequently |∆(k. where γσ is the attenuation constant for |E(qF σ )| < Eq. (5.4) becomes . uˆ1σ. . . h0.        v   ∆0 ˆ 1¯ σ   E =     u   Φ ˆ 2σ   . vˆ2¯σ. 0. ∆0. Φ. −h0. 0. 0. −h0. Φ. −∆0. . 0. uˆ1σ. .      v  ˆ   1¯σ     u −∆0  ˆ    2σ . Φ. h0. (5.28). vˆ2¯σ. for the superconducting overlayer (x > 0). Here h0 = ε0σ −σJ, ε0σ = i¯h2 m−1 γσ qF cos θN , ˆ The wave-vector components parallel to the interface are conserved and ∆0 = ∆0 (k). for all possible processes.. q. Solving Eq. (5.28), one obtains eigenvalues E = ± ∆2 + Φ2 + ε0σ 2 − σJ, where + (−) corresponds to the electron- (hole-) like QP excitation. Since ε0σ (−qF cos θN ) = −ε0σ (qF cos θN ) and ∆(−qF cos θN ) = −∆(qF cos θN ), states for kx = k + and −k + are actually degenerate. Thus for kx = k + , one can have two degenerate eigenstates for electron-like QP excitation, while for kx = −k + , one can have another two degenerate eigenstates for electron-like QP excitation. Superposition of these four eigenstates thus gives a formal wave function for the superconductor overlayer . . ∆0.       E−  ψSσ (x) = c1σ     0  . . . E+.       ∆0  + c2σ      Φ  . Φ . . E−. .    −γ x ik+ x  e σ e   . 0 . −∆0.           −∆0   E+     + c3σ   + c4σ    Φ   0    . 0. .     −γ x −ik− x  e σ e .   . Φ (5.29).

(51) 46 For simplicity, we set k + = k − = kF cos θS [15]. Here E± ≡ E±ε0σ , ε0σ =. q. (E + σJ)2 − ∆2 − Φ2 ,. and ci are the coefficients of corresponding waves. The formal wave function for the ferromagnetic-metal overlayer can be obtained to be: . . eik1σ x ∆0.    −ik1σ x   e  E−  ψN σ (x) = c1σ     0  . . . eik1σ x E+.    −ik1σ x   e  ∆0  + c2σ   −ik1σ x   e  Φ  . eik1σ x Φ . . e−ik1σ x E−.       −eik1σ x ∆0  + c  3σ   eik1σ x Φ   . .    ik+ x  e   . 0 .        + c4σ       . −e−ik1σ x ∆0 eik1σ x E+ 0 −ik1σ x. e. 0. .     −ik− x  e .   . Φ (5.30). In the above wave function, we have neglected the factor eik⊥ ·r⊥ which does not affect the final result. Due to the free boundary at x = −d, it requires that ψN σ (x = −d) = 0, thus we obtain the eigencondition for the surface bound states: e−2ik1σ d E+ + e2ik1σ d E− = 2Φ.. (5.31). The above is a generalization to our previous result without spin dependence. If J = Φ = 0 is set, the result is reduced to Hu’s case [45], i.e., 4ik1 d. e. ε0 + E . =− 0 ε −E. (5.32). Since there exists an AF order, ZBCP will be split. This splitting, however, is in addition to (and physically different from) the splitting caused by spin polarization. In Sec. III.D, we will discuss the ZBCP splitting in more details. Beyond the quasiclassical approximation, a more accurate calculation for the surface bound-state energies in dx2 −y2 -wave and other unconventional cuprate superconductors was reported by Walker et al. [53].. 5.3.2. Effects of Fermi-wave-vector Mismatch. As reported in several theoretical studies [32, 42], conductance is strongly modified by the effect of Fermi-wave-vector mismatch (FWM). In our case, due to the presence of.

(52) 47 Normalized Charge Conductance G. q. 2.0. L =0.2 0. L =0.5 0. 1.5. L =1 0. L =1.2 0. 1.0. 0.5. 0.0 0.0. 0.5. 1.0. 1.5. 2.0. Normalized Energy E/. Figure 5.3: Effect of Fermi-wave-vector mismatch on normalized charge conductance spectra Gq for various wave-vector mismatch value L0 with Z = 0, Φ = 0.5, and X = 0.5. ˇ c et the AF order, the conductance patterns are somewhat different from those of Zuti´ al. [32] and Linder et al. [42]. To see clearly how the effect affects, all the figures are plotted for normal incidence (θN = 0) unless mentioned otherwise. In this subsection, we introduce a parameter L0 ≡ qF /kF to account for the effect of FWM. We consider the cases for both L0 lesser and greater than unity. In view of Fig. 5.3 for a dx2 −y2 -wave superconductor, one sees a strong effect of FWM when L0 < 1. The normalized charge conductance curves are almost identical when L0 ≥ 1. In our previous paper [44], it is shown that the zero-bias conductance peak (ZBCP) of a dx2 −y2 -wave superconductor can be split by the AF order. If there is no spin-active barrier [30, 42], external magnetic field, and spin polarization, the splitting of ZBCP may be caused by an AF order, or can be a natural result of a d+is-wave symmetry gap. As a comparison, Fig. 5.4 shows that there occurs a strong ZBCP in case of Φ = X = 0. One sees that FWM (or the value of L0 ) has a strong effect on the strength of the ZBCP. Once again, the curves start to overlap when L0 is greater than one. Fig. 5.5 shows that the ZBCP of a dx2 −y2 -wave superconductor can be split by the AF order and by the spin polarization. When L0 is less than unity, FWM can lower.

參考文獻

相關文件

Sometimes called integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.. Note that integer programming

6 《中論·觀因緣品》,《佛藏要籍選刊》第 9 冊,上海古籍出版社 1994 年版,第 1

† Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan.. It is also important to note that we obtain an inequality with exactly the

Population: the form of the distribution is assumed known, but the parameter(s) which determines the distribution is unknown.. Sample: Draw a set of random sample from the

(a) A special school for children with hearing impairment may appoint 1 additional non-graduate resource teacher in its primary section to provide remedial teaching support to

◦ 金屬介電層 (inter-metal dielectric, IMD) 是介於兩 個金屬層中間,就像兩個導電的金屬或是兩條鄰 近的金屬線之間的絕緣薄膜,並以階梯覆蓋 (step

Then, it is easy to see that there are 9 problems for which the iterative numbers of the algorithm using ψ α,θ,p in the case of θ = 1 and p = 3 are less than the one of the

Key words: theory of the nature of the mind, the Buddha nature, one who possesses a gotra, non-resultant activity which is neither positive nor negative and is able