The Pennsylvania State University The Graduate School PUMPING SPIN, SINGLET PAIRS, AND LIGHT. A Thesis in Physics by Sungjun Kim. c 2007 Sungjun Kim

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1 The Pennsylvania State University The Graduate School PUMPING SPIN, SINGLET PAIRS, AND LIGHT A Thesis in Physics by Sungjun Kim c 2007 Sungjun Kim Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2007

2 The thesis of Sungjun Kim was reviewed and approved by the following: Ari Mizel Assistant Professor of Physics Thesis Advisor, Chair of Committee Jainendra Jain Erwin W. Mueller Professor of Physics Nitin Samarth Professor of Physics Sharon Hammes-Schiffer Professor of Chemistry Jayanth Banavar Professor of Physics Head of the Department of Physics Signatures are on file in the Graduate School.

3 Abstract We apply quantum pumping to generate spin current, the flow of singlet pair, and the flow of light. We consider quantum pumping on two extended lattice models. We develop quantum pumping formula with single particle Green s functions. We find closed form for pumping formula in terms of full Green s functions. We present a prescription for generating pure spin current or spin selective current, based on quantum pumping in a 1-D tight-binding model. Our calculations indicate that some pumping cycles produce the maximum value 2 of pumped spin while others reverse the direction of current as a result of small alterations of the pumping cycle. We find pumping cycles which produce essentially any ratio of spin current to charge current. We propose a method to produce singlet pair flow by quantum pumping. We introduce Hubbard terms as pumping parameters. We extend our formalism to the interacting regime. We develop singlet pair pumping formula with two particle Green s function. We present plots which show how singlet pair flow depends on the size of squared cycle. We study interference effects in the current generated by quantum pumping in two extended lattice models. The first model contains an Aharonov-Bohm loop within a 1-D tight-binding model. It exhibits interference between the two arms of the loop. We also investigate the effect of magnetic field reversal on the pumped current. Our second model is a 1-D tight-binding model with next-nearest-neighbor hopping terms. The resulting band structure can have 4 degenerate Fermi wave vectors ±k 1F and ±k 2F rather than the usual 2 Fermi wave vectors ±k F. It exhibits signatures of interference between these degenerate conduction band states. We propose a way to transport bosonic particle by pumping. We develop pumping formula based on Maxwell s equations. We choose localized time dependent dielectric constants as pumping parameters. By pumping, we generate net energy flow and its quantity can be precisely controlled by selecting pumping cycle. iii

4 Table of Contents List of Figures Acknowledgments vi xi Chapter 1 Introduction GaAs-AlGaAs heterostructure and two dimensional electron gas Quantum dots Electron pumping on a closed dot Electron pumping on an open dot Landauer formula Scattering matrix formalism Quantum pumping and Berry s phase Chapter 2 Pumping Electron Introduction Lippmann-Schwinger Equation Pumping formalism Chapter 3 Pumping Spin Introduction Model Hamiltonian and Pumped Current Spin current without charge current Selective spin pumping Arbitrary combinations of spin current and charge current iv

5 3.6 Conclusion Chapter 4 Pumping Singlet Introduction Pumping Singlet Pair Chapter 5 Pumping on extended lattice Introduction Pumping Through a Loop Geometry Pumping on a chain with next-nearest-neighbor hopping Conclusion Chapter 6 Pumping Light Introduction Formalism Universal form Appendix A Derivation of Eq. (1.25) 79 Appendix B Detailed Calculations for Chapter 2 81 B.1 Proof for identity (2.37) B.2 Detailed derivation for Eq. (2.48) Appendix C Proofs for identities (4.25),(4.26), and (4.27) 84 Appendix D Detailed Calculations for Chapter 5 91 D.1 Derivation of current operator D.2 Derivation of Eq. (5.6) Bibliography 103 v

6 List of Figures 1.1 Conduction band has narrow potential well at the interface between GaAs and AlGaAs. As electrons are confined in the narrow region, 2DEG is formed. This figure is the redrawing of a figure in reference [18] Schematic diagram for a diffusive dot and a ballistic dot: (a) A diffusive dot. Small circles in the dot are impurities. There are many impurities in the dot, electron experiences many elastic scattering events through the dot. (b) A ballistic dot. Since there is no impurities in the dot, electron scatters at the boundaries of the dot. This figure is the redrawing of a figure in reference [22] A quantum dot used by Kouwenhoven et al.[26]: The region enclosed by gates 1, 2, C, and F is a quantum dot. Electrons are transported through the channels between gate 1 and gate F and between gate 2 and gate F. This is the simplified drawing of a figure in reference [26] Pumping on closed dot with two oscillating potential barriers: (a) Two potential barriers are at equilibrium height. Tunnelings through barriers are weak at both sides. (b) By lowering left barrier and raising right barrier, one electron tunnels into the dot from left reservoir. (c) Two barriers return to the equilibrium height. At this stage, the dot has N+1 electrons. (d) By lowering right barrier and raising left barrier, N+1th electron tunnels out from dot to right reservoir. The cycle completes by returning two barriers to the equilibrium height. The complete one cycle is (a) (b) (c) (d) (a). This is redrawing of a figure in reference [26] A quantum dot used by Switkes et al. [12]: A quantum dot is formed by gates A, B, C, and 1, 2. Couplings between dot and reservoirs are determined by gates A, B, and C. Gates 1, 2 are used to deform dot shape by applying ac voltage to the gates. This is simplified drawing of a figure in reference [12] vi

7 1.6 Two reservoir are connected by a wire. Each reservoir is characterized by the chemical potential µ L, µ R Dispersion relation is determined by wire. Left reservoir feeds electrons to left side of band up to µ L and right reservoir feeds electrons to right side of band up to µ R. The gap between µ L and µ R, which is shown as blue bar in the figure, contribute to net current. We ignore side band due to transverse mode Wire has scattering region, transmission probability and reflection probability are determined by the scatterer. For simplicity, we assume T, R are independent of k Two impurities (Gray dots) are located sites l, l on a wire. On-site energies u l, u l characterize those impurities Gray dots located at l and l sites represent impurities on a wire; the impurities are characterized by potential barriers u l,u l. Localized magnetic fields B l,b l are applied on those impurities (a) Two identical square boxes are located symmetrically about the X l = X l line. For spin up, the parameters {X l, X l } begin at (b, a δ) and go counterclockwise. For spin down, the {X l, X l } begin at (a, b) and go clockwise. (b) The spin up and spin down cycles are located on the same rectangular box. For spin up, the cycle begins at (a, b) and goes counterclockwise. For spin down, the cycle begins at (a + δ, b) and goes clockwise (a) Three pairs of square boxes are located symmetrically about the median X l = X l and share the same meeting point. We move those pairs along the X l = X l line. For the three pairs, the box side lengths are 2,4, and 6 respectively. The pairs shift a distance 2d along the X l = X l line. (b) Pumped spin q s vs d at fixed k F = 1.4 and for l = 1: The solid line is for the pair of large boxes of side length 6, the long-dashed line is for the pair of medium boxes of side length 4, and the short-dashed line is for the pair of small boxes of side length 2. The result shows monotonic increase of pumped spin with box size and monotonic decrease of pumped spin with projected distance d vii

8 3.4 Plots of q s vs d. In each plot, the solid line is for a large box of side length 6, the long-dashed line is for a medium box of side length 4, and the short-dashed line is for a small box of side length 2. (a) We move each square box along the line X l = X l, where the position of the lower left corner is (d, d). (b) Plot at k F = 1.4 and for l = 1. For small d, negative pumped spin is found for the smaller two boxes (i.e. spin is pumped in the opposite direction). (c) Plot at k F = 3.1 and for l = 1. The maximum value 2 of pumped spin per cycle occurs on the solid line. The pumped spin q s is independent of box size for sufficiently large d (a) An example of pure spin current. There is no net flow of charge. (b) An example of selective spin current. Only one kind of spin contributes to current, there is no cancellation of charge (a) Pumping cycle for spin-up parameters, beginning at the point (a,b) in the generalized parameter space and proceeding counterclockwise. It encloses a non-zero area, and generates a non-zero spin-up current. (b) Pumping cycle for spin-down parameters, also beginning at (a,b). This cycle encloses no area, so there is no spindown current generated (a) We consider two congruent rectangular boxes symmetrically located within one large rectangular box. By varying δ, we shift the interior edges of the rectangular boxes. The left rectangular box shows the trajectory of spin-up pumping parameters, while the right rectangular box is for spin-down pumping parameters. For the first type of cycle (shown here), both the small rectangles are traversed in a counterclockwise direction. For the second type of cycle (not shown here), spin-up parameters traverse in a counterclockwise direction while spin-down parameters traverse in a clockwise direction. (b) Plot of q s /q c vs δ for first type of cycle at k F = 3.1 and for l = 1. We set (a, b) = (0, 0) and β = γ = 8. δ is varied from 1 to 8. Any q s /q c in the range 0 to 1 can be found. Note that the origin of plot is (1, 0), not (0, 0) Q pumped for U min and U max : We consider square cycle where leftdown corner has (U min, U min ) and right-up corner has (U max, U max ). This plot is for E F = 3.9. We set J = Q pumped for U min and U max : This plot is for F F = Q pumped for U min and U max : This plot is for F F = viii

9 4.4 Q pumped vs E F : We consider square cycle where left-down corner has (0, 0) and right-up corner has (8, 8). This plot is for l = 1 which determines distance between two impurities. We set J = Q pumped vs E F : This plot is for l = Two quantum dots are connected to left and right leads in parallel. They form a closed loop in the central region which is threaded by a magnetic field B. One dot has on-site energy u a and the other has on-site energy u b (a) Pumped charge q vs ϕ: We set J /J = 0.5 and traverse a squareshaped pumping cycle with corners (X 0, X 0) = (0, 0) and (100, 100) as shown in (b). The solid line is for k F = 1.7, and the dashed line is for k F = 2.1. This plot shows the antisymmetry and periodicity of pumped charge as a function of the magnetic field. The solid line curve near the origin shows that the pumped current is highly sensitive to small magnetic fields Pumped charge q vs k F : We set J /J = 0.5 and traverse a squareshaped pumping cycle with corners (X 0, X 0) = (0, 0) and (100, 100). The solid line is for ϕ = π, which corresponds to Φ = 1Φ 2 4 0, and the dashed line is for ϕ = 3π, which corresponds to Φ = 3Φ Since the current is periodic for ϕ and is antisymmetric when the sign of ϕ is reversed, the two curves shown have opposite values of pumped charge at any given k F We consider chain with next-nearest-neighbor hopping. J is the hopping amplitude for nearest neighbor hopping, and J is the hopping amplitude for next-nearest-neighbor hopping. On this chain, we assume only two sites (gray dots) located at l and l have on-site energies u l, u l This is the dispersion relation for J = 1 and J = 1. It has doublewell shape curve. Dashed line is for E k = E F = 1. Dashed line has four cross points with doublewell curve, those give four Fermi wave vectors {k 1F, k 2F, k 1F, k 2F } Pumped charge q vs l (there is exactly one value of q for each choice of l in both (a) and (b)). In both (a) and (b), the Fermi level is E F = 1.5, and the pumping cycle is square shaped with lowerleft corner (0, 0) and upper-right corner (100, 100). The distance between the two time-dependent potentials is 2l. (a) Standard chain with J = 1 and no nnn hopping, J = 0. A regular periodicity is evident. (b) Chain with nnn hopping, J = 1 and J = 1. The dependence of q on l is quite irregular and complicated ix

10 5.7 Pumped charge q vs k 1F : We set J = 1, J = 1 and l = 10, squareshaped pumping cycle has (4, 4) and (100, 100) for left-lower corner and right-upper corner. High peak near k 1F = 1.9 is related to resonance transmission and low peak between k 1F = 1.7 and k 1F = 1.8 indicates destructive interference between two wave vectors Lpumped vs x max 1 : We define γ by k 2 = γk 1, for this plot, we set γ = 2, which means (x max 2a, x max 2b ) = (2x max 1a, 2x max 1b ). We consider square shape cycles where (x min 1a, x min 1b ) = (0, 0) and (x max 1a, x max 1b ) = (x max 1, x max 1 ). Lpumped is defined by L pumped = L pumped /( ɛ ). We 8π define s by s = k 1 (b a), for this plot, we set s = 1. Red curve is the contribution from cycle formed by {x 1a, x 1b }, blue curve is the contribution from cycle formed by {x 2a, x 2b }, the sum of those two curves eventually generates pumped light, which is shown by black curve Lpumped vs x max 1 : We set γ = 3.3 ans s = 1 for this plot Lpumped vs x max 1 : We set γ = 4.15 ans s = 1 for this plot. As we increase x max 1, two contributions from red curve and blue curve cancel each other and no light is transported Lpumped vs s: We set γ = 2 and square cycle has x mim 1 = 0 and x max 1 = 10. As we increase s, as expected, it shows periodic behavior. Since there are two periodic contributions, resulting curve shows narrow peaks Lpumped vs s: We set γ = 1.7 and square cycle has x mim 1 = 0 and x max 1 = 10. Complicated pattern shows periodic behavior x

11 Acknowledgments I would like to begin the acknowledgments by thanking Ari Mizel first. His guidances and patiences lead me to come at this point. Always, I m more focused on numbers and logics between equations, rather than physics itself, he encourages me to look at physics without bounded formula. I believe I learn what doing physics means during my study at Penn State with Ari Mizel. I would like to thank Kunal Das for his many helps when we begin first work. At that time, we are not experts in the area, we spent much time without getting anywhere, with help of his hard work, we can penetrate into the area and we are able to produce some results. I would like to thank Rusko Ruskov, Xingxiang Zhou, and Ryan Woodworth for their advices and encouragements. During group meetings, I learn it s important to discuss physics to make a progress and get some other perspectives. I particularly thanks Xingxiang Zhou, I learn other perspectives of quantum pumping from him. I would like to thank Jainendra Jain, Nitin Samarth, and Sharon Hammes- Schiffer for their useful comments about thesis. From their feedbacks, I believe, this thesis becomes more beneficial and readable to others. Finally, I would like to thank my mother. Her great supports during my study make me reach here. Sometimes she sent Korean foods and clothes, it s always great help for my every day life. I believe, if there is something valuable in my works at Penn State, that is the answer for her every day prayer to God. xi

12 Dedication To my mother and father xii

13 Chapter 1 Introduction The notion of quantum pumping was first introduced by Thouless [1]. He pointed out that when an electron is in a bound state of a potential, the electron can be transported by shifting the potential slowly. This provides a means of charge transport without an applied voltage bias. Later, quantum pumping was considered for electron in a scattering state rather than a bound state [2]. Suppose the electron is propagating in some conduction channel. By periodically changing two parameters that define the shape of the channel, we can make the electron get swallowed down the channel like food down a throat. In this way, again we have a mechanism to generate a direct current under no voltage bias [2]. Physically, quantum pumping can be seen as an energy exchange processes between an electron and an oscillating scatterer (for example, channel walls can constitute the scatterer) [3, 4, 5]. When an electron hits the oscillating scatterer, it gains energy quanta or loses energy quanta. The net result of these processes is a net current. Quantum pumping is widely described by Brouwer s formula [2], which expresses instantaneous pumped charge in terms of scattering matrix and its derivative to parameter. This formula is based on the earlier work [6, 7] which is called the BTP formula. More recently, a quantum pumping formula has been developed that is based directly on the form of scattering states rather than on the scattering matrix [8]. In addition to its appeal as a means of delivering charge without bias, the

14 2 formula for pumped current has been intriguing to theorists because of its similarity to a quantum Berry s phase formula. This similarity between pumped current and Berry s phase was studied in the papers [9, 10, 11]. The first attempt at the experimental realization of quantum pumping was performed in 1999 [12]. In the experiment, the confining potential of an open quantum dot is deformed. A pumped direct current arises in response to the deformation at zero bias. They observed some characteristic behaviors of quantum pump, for instance, sinusoidal dependence of pumped voltage for phase difference π. However, some of their observations about the relation between pumped voltage and magnetic field don t agree with theoretical prediction[13], which raised some arguments about what their experiments really measured [14]. In this thesis, we develop a new form of the quantum pumping formula in terms of the single particle Green s function [15] starting with the scattering state approach of Entin et al. [8]. The result is an exceptionally flexible formalism that we apply to the quantum pumping of spin [15], entangled electron pairs [16], and electrons in two extended lattice models [17]. One model is a one-dimensional lattice with an Aharonov-Bohm ring, the other is one-dimensional lattice with next-nearest-neighbor hopping terms. Rich interference effects are found in these models. Finally, we apply our formalism in a very different direction by turning our attention to bosons from fermions. We use our theory to develop a photonics pumping formula for light energy, based on Maxwell s equations. In the following sections, first we introduce two dimensional electron gas and quantum dot which provide the experimental set-up for quantum pumping. We introduce electron pumping on closed dot which gives some insight about transporting electron by pumping. Then we introduce some details about the quantum pumping experiment done by Switkes et al.[12]. We introduce the scattering matrix formalism that is the widely accepted description of quantum pumping. To do so, it is necessary to first present the Landauer scattering approach for conductance, because the scattering matrix formalism is couched in the framework of the Landauer theory. We confirm that the scattering matrix formalism is reasonable, verifying charge conservation in quantum pumping. Finally we introduce Berry s phase to show the similarity between quantum pumping and Berry s phase.

15 1.1 GaAs-AlGaAs heterostructure and two dimensional electron gas Experimental set up[12] for quantum pumping is based on a GaAs-AlGaAs heterostructure. In a doped GaAs-AlGaAs, a confined two dimensional electron gas (2DEG) is formed at the interface between GaAs and Al x Ga 1 x As[18, 19]. Typical value of x has 0.3. Fermi energy of n-doped AlGaAs is higher than that of GaAs, when two layers have contact, electrons spread out from AlGaAs. At equilibrium, the conduction band has narrow valley at the interface between GaAs and AlGaAs (see Fig. 1.1), which causes sharp peak in electron density at the interface and ending up with 2DEG. 2DEG formed in GaAs-AlGaAs heterostructure has high electron mobility. There are two reasons for that. First reason is the suppressed boundary scattering at the interface. This suppression comes from the fact that the lattice constant of GaAs is close to that of AlGaAs. Second reason for high mobility is the low effective mass in GaAs. The effective mass in GaAs is m = 0.067m e. It is possible to create lateral confinements in 2DEG. Lateral confinements make channels or dots on 2DEG. Most popular way to make channels or dots on 2DEG 3 Figure 1.1. Conduction band has narrow potential well at the interface between GaAs and AlGaAs. As electrons are confined in the narrow region, 2DEG is formed. This figure is the redrawing of a figure in reference [18].

16 4 is to apply a negative voltage to a gate which is put on the top surface of GaAs- AlGaAs heterostructure. Electric field due to a negative voltage repels electrons in the region below the gate, and the shape of gates and the magnitudes of applied voltages determine the shape or geometry of created confined area on 2DEG. Quantum point contact[20, 21] is one example for this kind of lateral confinement. Instead of applying a voltage to a gate, lateral confinement also can be created on 2DEG by removing a thin layer of AlGaAs. One example for this method is multi probe electron waveguides. 1.2 Quantum dots Quantum dots are small conducting devices which confine electrons in a small region[22, 23, 24]. A quantum dot can have up to several thousand electrons, and the area of small region is about µm 2. As already mentioned in previous section, 2DEG formed at the interface of semiconductor heterostructure provides bases for quantum dots. By applying negative voltages to gates, one can make a small region where electrons are confined. This small region is termed quantum dot because electrons on the dot don t have any spacial degree of freedom in macroscopic sense. We used a term small, however, a quantum dot is bigger than atoms, we need to understand what the term small really means. The meaning of small can be clarified in terms of phase coherence length L φ. Phase coherence length is the distance electron runs preserving its phase. Quantum dot is small indicates the size of quantum dot is comparable to or smaller than phase coherence length. Physical meaning of smallness of quantum dot is that electrons moving within quantum dot don t lose their phase informations. Therefore, quantum interference effects can be clearly demonstrated when some physical quantity, for instance, conductance, is measured at quantum dot device. A huge class of devices which have smaller or comparable size to phase coherence length are termed mesoscopic devices. Quantum dot is one of mesoscopic devices. Mesoscopic devices can be classified to two groups in terms of elastic mean free path l which characterizes elastic scattering of electron. Since elastic scattering with impurities preserves phase coherence, even though the system size L is larger

17 5 Figure 1.2. Schematic diagram for a diffusive dot and a ballistic dot: (a) A diffusive dot. Small circles in the dot are impurities. There are many impurities in the dot, electron experiences many elastic scattering events through the dot. (b) A ballistic dot. Since there is no impurities in the dot, electron scatters at the boundaries of the dot. This figure is the redrawing of a figure in reference [22]. than elastic mean free path l, the system is still in mesoscopic regime. The system belonging to this category is termed diffusive (see Fig. 1.2a). If the system size is smaller than elastic mean path, the system is termed ballistic. In ballistic device, elastic scatterings occur at the boundaries of the device (see Fig. 1.2b). Most experimental works on quantum dots are about ballistic quantum dots. One can consider two kinds of quantum dots based on how it couples to leads. When the coupling between quantum dot and leads are strong, the quantum dot is considered an open dot because electron easily moves from lead to dot or from dot to lead. For the weak coupling, there are barriers between dot and leads, electron transports between dot and leads only by tunneling. Since quantum dot is isolated by barriers, the quantum dot coupled this way is termed a closed dot. The closed dot has quantized charge and discrete energy levels. 1.3 Electron pumping on a closed dot Quantized charge on the closed dot allows turnstile device which transports electron one by one[25, 26]. By applying a single rf signal to central isolated region of

18 6 an array of four tunnel junctions, a current plateau I = ef was observed[27]. Two phase-shifted rf signals were applied to generate a current plateau I = ef in three nanoscale tunnel junction geometry[28]. These turnstile devices belong to metal system. In 1991, turnstile device in a semiconductor quantum dot was realized[26]. In this experiment, the gate geometry is fabricated on the surface of a GaAs-AlGaAs heterostructure. 2DEG about 100nm is formed in the interface between GaAs and AlGaAs. A quantum dot is formed by applying -400 mv to gate F, C, 1, and 2 (see Fig. 1.3) since electric fields due to these gates repels electrons at those regions. The resulting quantum dot has a diameter of 0.8 µm. At this gate voltage -400 mv, two narrow channels formed by gate 1 and gate C and by gate 2 and C are pinched off. The channels between gate 1 and gate F and between gate 2 and gate F provides paths between a quantum dot and the rest of wide regions. Therefore, the couplings between a quantum dot and wide regions are controlled by gate 1 voltage, gate 2 voltage, and gate F voltage. When the coupling defined by gate voltage 1 and gate voltage F or gate voltage 2 and gate voltage F is strong, more precisely, when corresponding conductance G 2e 2 /h, the quantum dot is considered as an isolated system and charge effects become crucial. Charging effects roughly represent electron-electron interactions and expressed in terms of charging energy which is the electrostatic energy related to adding one electron to the dot. In this regime, the couplings form tunneling Figure 1.3. A quantum dot used by Kouwenhoven et al.[26]: The region enclosed by gates 1, 2, C, and F is a quantum dot. Electrons are transported through the channels between gate 1 and gate F and between gate 2 and gate F. This is the simplified drawing of a figure in reference [26].

19 7 barriers between dot and reservoirs (wide regions on 2DEG), classically electron cannot pass from dot to reservoir or from reservoir to dot. For the dot in this experiment, the energy difference between discrete states is about 0.03 mev, and charging energy is about 1 mev, so charging energy has dominant effect. Pumping is performed by applying two rf signals with a phase difference π to gate 1 and gate 2. Two signals produce oscillating tunneling barriers. They provide some insight for how oscillating tunneling barriers pump electron one by one per cycle. At the beginning of cycle, the dot has N electrons and both tunneling barriers are in the same heights where tunneling probabilities are small (see Fig. 1.4a). As the left barrier is lowered, one electron can tunnel into the dot because lowered barrier makes tunneling probability large. At the same time, by increasing the right barrier, the entering electron cannot escape to right reservoir because of very low tunneling probability (see Fig. 1.4b). One more electron s tunneling from left reservoir to the dot is prevented by charging energy. To tunnel into the dot, the Figure 1.4. Pumping on closed dot with two oscillating potential barriers: (a) Two potential barriers are at equilibrium height. Tunnelings through barriers are weak at both sides. (b) By lowering left barrier and raising right barrier, one electron tunnels into the dot from left reservoir. (c) Two barriers return to the equilibrium height. At this stage, the dot has N+1 electrons. (d) By lowering right barrier and raising left barrier, N+1th electron tunnels out from dot to right reservoir. The cycle completes by returning two barriers to the equilibrium height. The complete one cycle is (a) (b) (c) (d) (a). This is redrawing of a figure in reference [26].

20 8 electron need energy to overcome charging energy, however, there is no such an energy source, the electron cannot tunnel into the dot. This mechanism is termed Coulomb blockade. Two barriers return to the initial heights as the dot has N + 1 electrons (see Fig. 1.4c). As right barrier is lowered and left barrier is raised simultaneously, the extra electron tunnels out to right reservoir (see Fig. 1.4d). By returning two barriers to the initial heights (see Fig. 1.4a), the cycle completes. During one cycle, one electron is pumped through a dot from left reservoir to right reservoir. Generated current is expressed in terms of frequency f, it has I = ef. By increasing the bias voltage, V = (µ L µ R )/e, the number of charge states between µ L and µ R can have integer n. In this set-up, the cycle pumps n electrons, therefore, current has I = nef. This experiment demonstrates pumping on a closed dot can produce quantized current. 1.4 Electron pumping on an open dot Pumping on an open dot can be termed quantum pumping because it is the interference effect of coherent electron. In an open system, there are no barriers between dot and reservoirs, electron classically passes through the dot. By deforming confining potentials which defines quantum dot, quantum dot experiences shape changes. Electron wave function responds to these shape changes in a way of affecting the interference pattern[12]. The resulting pumped current has some randomness, for instance, the direction of current is not predicted[2]. In case of pumping on a closed dot, the direction of current is determined by the cycle. Pumping on an open quantum dot was attempted by Switkes et al.[12] in In this experiment, quantum dots are defined by electrostatic gates on the top of a GaAs-AlGaAs heterostructure. Negative voltages, 1V, are applied to gates to form the dot (see Fig. 1.5). The dot has a lithographic area, 0.5 µm 2. Couplings between reservoirs and dot are controlled by gates A, B, and C. Voltages on these gates were set so that average conductance through dot has G 2e 2 /h. The periodic shape deformation of dot was performed by applying ac gate voltages to gates 1, 2. For most measurements, pumping frequency has f = 10 MHz, base temperature is T = 330 mk, dot conductance is G 2e 2 /h, and ac voltage is 80 mv peak

21 9 Figure 1.5. A quantum dot used by Switkes et al. [12]: A quantum dot is formed by gates A, B, C, and 1, 2. Couplings between dot and reservoirs are determined by gates A, B, and C. Gates 1, 2 are used to deform dot shape by applying ac voltage to the gates. This is simplified drawing of a figure in reference [12]. to peak. The range of applied magnetic field, B is from 30 to 80 mt. They demonstrated some characteristic behaviors of quantum pump, such as antisymmetry about phase difference φ = π, sinusoidal dependence of φ, and random fluctuations of amplitude as a function of perpendicular magnetic field. They also investigated the relation between pumped voltage (they measured voltage instead of current.) and applied magnetic field. They observed symmetric relation between them, which doesn t agree with theoretical prediction[13]. Theory predicts some definite symmetric relations only if deformed dot keeps certain definite spacial symmetry at every single moments of the cycle. Our work on extended lattice model is partially motivated by this contradiction. 1.5 Landauer formula In Landauer s transport theory[29, 18], when two reservoirs are connected by a wire (see Fig. 1.6), the current flowing in a wire is expressed as I = e = e h den(e)v(e)f L (E) e den(e)v(e)f R (E) (1.1) de[f L (E) f R (E)], (1.2) where N(E) = 1 dk 2π de is the density of states per length, v(e) = 1 de dk is the

22 10 velocity, and f L (E) is the distribution function for left reservoir, f R (E) is for right reservoir. We ignore transverse mode contribution, and we don t include spin degree of freedom here. Main feature of Landauer s formula is the cancellation between density of states and velocity. At the end, the current is independent of details of density of states and velocity for each state k and is solely determined by distribution function of each reservoir. At zero temperature, since a distribution function has simple step function, f L (E) = θ(e µ L ), f R (E) = θ(e µ R ), Eq. (1.2) reduces to I = e h (µ L µ R ), (1.3) where we assume µ L > µ R. Left reservoir feeds electrons up to µ L at the right hand side of band, right reservoir feeds electrons up to µ R at the left hand side of band, and the current is determined by net flow which occurs due to difference between two chemical potentials, µ L and µ R [30] (see Fig. 1.7). When a wire has scattering region in the middle (see Fig. 1.8), some of electrons transmit and some of them reflect, I R = e h T (µ L µ R ), (1.4) I L = e h (1 R)(µ L µ R ), (1.5) where T is transmission probability, R is reflection probability, and I R is the Figure 1.6. Two reservoir are connected by a wire. Each reservoir is characterized by the chemical potential µ L, µ R.

23 11 Figure 1.7. Dispersion relation is determined by wire. Left reservoir feeds electrons to left side of band up to µ L and right reservoir feeds electrons to right side of band up to µ R. The gap between µ L and µ R, which is shown as blue bar in the figure, contribute to net current. We ignore side band due to transverse mode. current flowing from right region of wire into right reservoir, I L is the current from left reservoir to left region of wire. We assume T, R are independent of k for simplicity. Charge conservation I L = I R is guaranteed by the relation T + R = Scattering matrix formalism Scattering matrix relates initial free state to final free state for a system which experiences scattering processes and scattering matrix has all relevant informations about the system[31, 32]. Scattering matrix element S fi is the amplitude for initial free state φ i to be at final free state φ f after scattering events. In terms of scattering operator S, this can be written as φ f = Sφ i. (1.6) Figure 1.8. Wire has scattering region, transmission probability and reflection probability are determined by the scatterer. For simplicity, we assume T, R are independent of k.

24 12 Formal scattering theory is well developed and scattering matrix can be expressed in terms of scattering states S fi = ψ f ψ+ i, (1.7) where scattering states are defined by ψ i = 1 φ i + E i H iɛ V φ i, (1.8) ψ f = 1 φ f + E f H iɛ V φ f. (1.9) In the scattering states, H is full Hamiltonian, V is scattering potential, and ɛ = 0 +. One can prove orthogonality for scattering states as follows. ] ψ + i ψ+ j [ φ = 1 i + φ i V ψ + j E i H iɛ = φ i ψ + j + 1 E i E j iɛ φ i V ψ + j [ ] 1 = φ i φ j + E j H 0 + iɛ V ψ+ j 1 + E i E j iɛ φ i V ψ + j 1 = φ i φ j + E j E i + iɛ φ i V ψ + j + 1 E i E j iɛ φ i V ψ + j [ ] 1 = φ i φ j + E j E i + iɛ + 1 φ i V ψ + j E i E j iɛ = φ i φ j = δ ij. (1.10) In this proof, we used the following alternative expression for scattering state ψ + i = φ i + 1 E i H 0 + iɛ V ψ+ i. (1.11)

25 13 Unitarity of scattering matrix, SS = S S = 1, can be proven by the orthogonality and completeness of scattering states. Completeness of scattering states is 1 = g ψ g ψ g or 1 = g ψ+ g ψ + g. The proof is quite simple. [SS ] fi = g S fg S ig = g ψ f ψ+ g ψ + g ψ i = ψ f ψ i = δ fi. (1.12) Pumping formula was developed based on scattering matrix. Pumped charge is expressed in terms of scattering matrix and its derivative to parameter[2]. We introduce scattering matrix formalism for pumping and show charge conservation for resulting pumped charge by following the notations in the paper[9]. Charge conservation is required so that the evaluated quantity should be a physical quantity. Infinitesimal charge coming into j reservoir is where j = 1, 2 and S is scattering matrix dq j = ie 2π T r(q jdss ), (1.13) ( ) r t S = t r. (1.14) Q 1 is defined by Q 1 = ( ) (1.15) Q 2 is defined by

26 14 ( ) 0 0 Q 2 =. (1.16) 0 1 Eq. (1.13) is called Brouwer s formula[2], which is considered the generalization of so called BTP formula[6]. Eq. (1.13) is widely used to describe quantum pumping. From Eq. (1.13), instantaneous current coming into reservoir 1 is I 1 = dq 1 dt = ie [ S ds 11 2π 11 + S ds 12 ] 12 dt dt = ie [ S 2π 11 ( u a ua S 11 + u b ub S 11 ) + S12( u a ua S 12 + u b ub S 12 ) ] = ie [ ua (S 2π 11 ua S 11 + S12 ua S 12 ) + u b (S11 ub S 11 + S12 ub S 12 ) ] = ie [ ua (r ua r + t ua t ) + u b (r ub r + t ub t ) ], (1.17) 2π where we assume scattering matrix has time-dependence through two parameters {u a, u b }. Similarly, instantaneous current coming into reservoir 2 is given by I 2 = dq 2 dt = ie [ S ds 21 2π 21 + S ds 22 ] 22 dt dt = ie [ S 2π 21 ( u a ua S 21 + u b ub S 21 ) + S22( u a ua S 22 + u b ub S 22 ) ] = ie [ ua (S 2π 21 ua S 21 + S22 ua S 22 ) + u b (S21 ub S 21 + S22 ub S 22 ) ] = ie [ ua (t ua t + r ua r ) + u b (t ub t + r ub r ) ]. (1.18) 2π We consider pumped charge for a closed cycle

27 15 q pump 1 = = ie 2π = ie 2π dti 1 dt [ u a (r ua r + t ua t ) + u b (r ub r + t ub t ) ] [dua (r uar + t uat ) + du b (r r + ub t ub t ) ]. (1.19) By the definition of pumped charge, for reverse cycle, it has the opposite sign value. Eq. (1.19) can be transformed to surface integral by Green s theorem[2] q pump 1 = ie du a du b [ ua (r ub r + t ub t ) ub (r ua r + t ua t ) ], 2π D (1.20) where D indicates surface region enclosed by cycle. From Eq. (1.20), we define field f(u a, u b ) on parameter space f(u a, u b ) = ua (r ub r + t ub t ) ub (r ua r + t ua t ). (1.21) Since we have all informations about the field in parameter space, based on those informations, we make a cycle which gives desired charge per cycle. Quantum pumping has precise control over generated current. Similarly, for pumped charge coming into reservoir 2 q pump 2 = = ie 2π = ie 2π dti 2 [dua (t uat + r uar ) + du b (t t + ub r ub r ) ] D du a du b [ ua (t ub t + r ub r ) ub (t ua t + r ua r ) ]. (1.22) We require charge conservation

28 16 q pump 1 + q pump 2 = 0. (1.23) Otherwise, charge is accumulated in scattering region during the cycle. To prove charge conservation, we consider general 2 2 unitary matrix[9] ( cos θe S = e iγ iα i sin θe iφ ) i sin θe iφ. (1.24) cos θe iα We calculate sum of two infinitesimal charges (see Appendix A for detailed calculation) dq 1 + dq 2 = ie 2π T r[(q 1 + Q 2 )dss ] = ie 2π T r[dss ] = e dγ. (1.25) π For a closed cycle, the integration of Eq. (1.25) is zero, we have q pump 1 + q pump 2 = (dq 1 + dq 2 ) = e dγ π = 0. We have proved charge conservation. 1.7 Quantum pumping and Berry s phase Similarity between pumping formula and Berry s phase was studied in some theoretical works[9, 10, 11]. We introduce Berry s simple, but profound observation[33] about geometrical phase in quantum mechanics.

29 17 We consider time-dependent Hamiltonian H(t) where its time-dependence is only through some parameters. For simplicity, we consider two parameter case. The Hamiltonian can be written as H(t) = H(X(t)) where X = {X 1 (t), X 2 (t)} are two parameters. A quantum state should satisfy Schrödinger equation i t ψ(t) = H(X(t)) ψ(t). (1.26) We can consider the instantaneous eigenstate at any given time for the Hamiltonian H(X) n(x) = E n (X) n(x). (1.27) Now we can ask ourself what the zeroth order time evolution of the eigenstate n(x(0)) is. One can say that the answer could be ψ n (t) = e i R t 0 dt E n(x(t )) n(x(t)). (1.28) This solution is just adding dynamical phase to the instantaneous eigenstate. The problem of this solution is that it doesn t satisfy the Scrödinger equation (1.26). Therefore, this cannot be a solution. What Berry found in the problem is that by putting additional phase in the form of e iγn(t), one can resolve the problem occurring in the solution which only has dynamical phase. This is Berry s zeroth order solution ψ n (t) = e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)). (1.29) The crucial point in this solution is that γ n (t) is not a function of X(t). If γ n (t) depends on time through parameters X(t), for a closed cycle, it should return to its initial value. Since γ n (t) γ n (X(t)), after a closed cycle, it doesn t have to

30 18 return to the initial value. (1.26) To figure out the detailed form of γ n (t), we put Berry s solution (1.29) into Eq. i t [e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)) ] = H(X(t)) [e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)) ], (1.30) i t [e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)) ] = E n (X(t))e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)) γ n (t)e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)) +i e i R t 0 dt E n(x(t )) e iγn(t) t n(x(t)) = E n (X(t))e i R t 0 dt E n(x(t )) e iγn(t) n(x(t)), (1.31) γ n (t) n(x(t)) i t n(x(t)) = 0. (1.32) From Eq. (1.32), we have γ n (t) = i n(x(t)) t n(x(t)) = i n(x(t)) X1 n(x(t)) Ẋ1 + i n(x(t)) X2 n(x(t)) Ẋ2. (1.33) For a closed cycle C, we have γ n (C) = dt γ n (t) = i n(x(t)) X1 n(x(t)) dx 1 + n(x(t)) X2 n(x(t)) dx 2.(1.34) Eq. (1.34) shows the detailed form of Berry s phase. As seen in Eq. (1.34),

31 19 Berry s phase depends on the shape of cycle in parameter space and by repeating the cycle, it accumulates the same value over and over again. Comparing Eq. (1.34) with Eq. (1.19) which is for pumped charge, one can find they have the similar mathematical structures. One may demonstrate the direct relation between Berry s phase and pumped charge by defining Berry s phase in a unique way for scattering states. For scattering states, causality could be crucial to define Berry s phase.

32 Chapter 2 Pumping Electron 2.1 Introduction In this chapter, we start by introducing the Lippmann-Schwinger equation. In quantum pumping, we are interested in solving a scattering problem with timedependent parameters. To treat this time-dependent problem, first we solve a simpler, time-independent scattering problem. The solution to this simpler problem is conveniently expressed by the Lippmann-Schwinger equation. After we have the Lippman-Schwinger solution in hand, we return to the original time-dependent problem. We analyze quantum pumping within the adiabatic approximation, assuming that all time-dependent parameters are slowly varying with time. The slow variation in parameters allows us to regard the Lippmann-Schwinger solution as a zeroth order solution. By finding first order corrections directly from the Schrödinger equation, we develop a pumping formalism to compute how much current the time-dependence produces. 2.2 Lippmann-Schwinger Equation Lippmann-Schwinger Equation[34, 35] gives the solution of scattering state for a particle which experiences elastic scattering processes. For a given 1-D Hamiltonian

33 21 H = H 0 + V, (2.1) we assume H 0 is free Hamiltonian, and V is time-independent scatterer. We consider the eigenstate k of H 0 H 0 k = E k k, (2.2) where E k = k2, we set = 1. 2m We introduce scattering state χ k. First, χ k is the eigenstate of full Hamiltonian H. Second, χ k has the same energy E k which k has for free Hamiltonian H 0. This implies scattering process is elastic, which means no energy change during the process. H χ k = E k χ k. (2.3) We claim χ k = 1 k + V χ k E k H 0 (2.4) = 1 k + V k. E k H (2.5) By apply E k H 0 to both sides of Eq. (2.4), it reproduces Eq. (2.3). Similarly, applying E k H to to both sides of Eq. (2.5), it also reproduces Eq. (2.3). 1 1 However, we have singularities in both operators E k H 0 and E k. To resolve the H singularity problem, we extend the energy E k to complex region, E k E k ± iη. χ ± k = k + 1 E k H 0 ± iη V χ± k (2.6) 1 = k + V k, (2.7) E k H ± iη

34 22 where η = 0 +. Eqs. (2.6), (2.7) are called Lippmann-Schwinger equation, they give the analytic solution for elastic scattering processes. We introduce Green s operators[35, 36, 37] G ± 0 (E k ) = G ± (E k ) = 1 E k H 0 ± iη, (2.8) 1 E k H ± iη. (2.9) G + 0 is retarded free Green s operator, G 0 is advanced free Green s operator. G + is retarded full Green s operator, G is advanced full Green s operator. Retarded Green s operator satisfies causality, we get physical solution from retarded Green s operator. In terms of Green s operators, we have χ ± k = k + G± 0 (E k )V χ ± k (2.10) = k + G ± (E k )V k. (2.11) By iterating scattering state in Eq. (2.10 ) χ ± k = k + G± 0 (E k )V χ ± k = k + G ± 0 (E k )V k + G ± 0 (E k )V G ± 0 (E k )V k +G ± 0 (E k )V G ± 0 (E k )V G ± 0 (E k )V k +. (2.12) By comparing Eq. (2.12) to Eq. (2.11), we have G ± (E k ) = G ± 0 (E k ) + G ± 0 (E k )V G ± 0 (E k ) + G ± 0 (E k )V G ± 0 (E k )V G ± 0 (E k ) + (2.13) Eq. (2.13) can be written in two closed forms which are called Dyson s equation

35 23 G ± (E k ) = G ± 0 (E k ) + G ± (E k )V G ± 0 (E k ) (2.14) = G ± 0 (E k ) + G ± 0 (E k )V G ± (E k ). (2.15) From Eqs. (2.14), (2.15), we evaluate full Green s function with free Green s functions which have the known explicit expressions. We consider real space representation of Eq. (2.11). For simplicity, we consider simple scattering potential, V = α V α α α x χ ± k = x k + x G± (E k )V k = x k + α V α G ± (E k )(x, α) α k, (2.16) where G ± (E k )(x, α) = x G ± (E k ) α. The full Green s functions, G ± (E k )(x, α) have all informations about scattering states. We evaluate G ± (E k )(x, α) from Dyson s equation, Eq. (2.14). G ± (E k )(x, α) = G ± 0 (E k )(x, α) + α V α G ± (E k )(x, α )G ± 0 (E k )(α, α).(2.17) We know explicit expression for free Green s function, for the present free Hamiltonian H 0, G ± 0 (x, x ) = ± me±ik x x ik, by solving algebraic equation, Eq. (2.17), we get full Green s functions G ± (E k )(x, α) and scattering states which we have an interest in. 2.3 Pumping formalism We develop formalism for pumping current in terms of single particle Green s function. We introduce the following Hamiltonian (see Fig. 2.1). H = H 0 + V, (2.18)

36 24 H 0 = J n a n+1a n + a na n+1, (2.19) V = u l n l + u l n l, (2.20) where J is the nearest neighbor hopping amplitude and a n is the electron creation operator at site n, n l = a l a l is the number operator at site l. H 0 is free Hamiltonian which represents homogeneous one-dimensional lattice, V is the impurity potential and the impurities are characterized by on-site energies {u l, u l }, which simulate potential barriers at sites l and l. From Lippmann-Schwinger equation, we have zeroth order scattering state χ p = [1 + G(E p )V ] p, (2.21) where the energy E p = 2J cos p, G(E p ) = 1/(E p H + i0 + ) is retarded full Green s function. We set lattice constant to 1. We consider adiabatic variation of time-dependent parameters, adiabatic condition is satisfied if the variation is slow compared to the dwell time of an electron in the scattering region[3]. We consider first order adiabatic correction[8]. From time-dependent Schrödinger s equation, (i t H) ψ p (t) = 0, (2.22) where we set = 1. We define Figure 2.1. Two impurities (Gray dots) are located sites l, l on a wire. On-site energies u l, u l characterize those impurities.

37 25 ψ p (t) = e iep φ p (t), (2.23) φ p (t) = p + φ p (t). (2.24) Putting Eqs. (2.23), (2.24) to Eq. (2.22), we have (E p H) φ p (t) = V p i t φ p (t). (2.25) In terms of full Green s function, we have integral equation for unknown φ p (t) φ p (t) = GV p ig t φ p (t), (2.26) where we drop energy dependence of G, G = G(E p ). By iterating φ p (t) φ p (t) = GV p ig t φ p (t) = GV p ig t (GV p ig t φ p (t) ). (2.27) Up to first order correction, φ p (t) = GV p ig t (GV p ig t φ p (t) ) GV p ig t (GV p ). (2.28) We have first order correction for φ p (t) φ p (t) = p + GV p ig t (GV p ) = χ p ig t (GV p ). (2.29)

38 26 We derive an useful identity for t (GV p )[8]. state χ p satisfies Since zeroth order scattering (E p H) χ p = 0. (2.30) Taking time derivative to Eq. (2.30) (E p H) χ p V χ p = 0 χ p = G V χ p. (2.31) Since t (GV p ) = χ p, from the identity (2.31), we have φ p (t) = χ p ig t (GV p ) = χ p igg V χ p. (2.32) We express first order correction scattering state in terms of zeroth order scattering state and full Green s function. Instantaneous pumped current is given by j pump (n) = e π dp def (E) π 2π δ(e E p) φ p j n φ p, (2.33) where F (E) is Fermi distribution function and j n is the current operator j n = J i (a n+1a n a na n+1 ). (2.34) We evaluate the integration over p in Eq. (2.33) first. π π dp 2π δ(e E p) φ p j n φ p

39 27 where n = a n 0. π dp = 2J Im π 2π δ(e E p) n φ p n + 1 φ p, (2.35) We put Eq. (2.32) to Eq. (2.35) π 2J Im 2J Im π π π dp 2π δ(e E p) n φ p n + 1 φ p dp 2π δ(e E p)[ n χ p n + 1 χ p +i n χ p n + 1 G 2 V χp i n + 1 χ p n G 2 V χp ]. (2.36) To evaluate Eq. (2.36), we find identity (see Appendix B.1 for proof) π π dp 2π δ(e E p) n χ p m χ p = 1 2πi [G(n, m) G (n, m)], (2.37) where G(n, m) = n G m. From the identity (2.37), the first term in Eq. (2.36) is gone since it is a real number. This is the static scattering case where there should be no current at no bias. With the identity (2.37), we can replace scattering states with Green s functions. π 2J Im dp π = 2J Im π u m m=±l 2π δ(e E p)[i n χ p n + 1 G χ p i n + 1 χ p n G χ p ] π dp 2π δ(e E p)[i n + 1 G 2 m n χ p m χ p i n G 2 m m χ p n + 1 χ p ] = J π Im [ u m n + 1 G 2 m [G(n, m) G (n, m)] m=±l n G 2 m [G(m, n + 1) G (m, n + 1)] ] (2.38) In Eq. (2.38), we need to deal with G 2. We use the following identity.

40 28 G(E k )(n + 1, m) = e ik G(E k )(n, m) for n m. (2.39) We evaluate Eq. (2.38) for n, which means the observation point is far away from the scattering center and is located on the right hand side of the scatterer. J π Im m=±l u m [ n + 1 G 2 m [G(n, m) G (n, m)] n G 2 m [G(m, n + 1) G (m, n + 1)] ] = J π Im [ u m e ik n G 2 m [G(n, m) G (n, m)] m=±l n G 2 m [e ik G(n, m) e ik G (n, m)] ] = J π Im [ u m n + 1 G 2 m G(n, m) + n G 2 m G (n + 1, m) ] m=±l = J π Im m=±l u m E [G(n, m)g (n + 1, m)], (2.40) where E = E k, and at the last step of calculation, we have used the identity n G 2 m = E G(n, m). This identity can be easily shown by taking energy derivative to G 1 G = 1. G 1 G = (E H)G = 1 G + G 1 E G = 0 G 2 = E G. (2.41) We put the result (2.40) to the original Eq. (2.33) which is the formula for instantaneous pumped current. j pumped (n) = ej π Im m=±l u m def (E) E [G(n, m)g (n + 1, m)].

41 29 (2.42) We perform the integration in Eq. (2.42) at zero temperature where F (E) = θ(e F E). j pumped (n) = ej 2 π Im m=±l Ẋ m G(n, m)g (n + 1, m) E=EF for n.(2.43) where we introduce dimensionless pumping parameters {X l, X l }, which are defined as X l = u l /J, X l = u l /J. We find the expression for instantaneous pumped current in terms of retarded full Green s functions. To evaluate the instantaneous pumped current, we need to compute full Green s functions. The full Green s function can be expressed in terms of free Green s function by Dyson s equation[35], G(E) = G 0 (E) + G(E)V G 0 (E), where G 0 (E) = 1/(E H 0 + i0 + ). From Dyson s Equation G(n, m) = f( m)g 0 (n, m) + h( m)g 0 (n, m), (2.44) where f(m) = 1 JX mg 0 (0, 0), Z (2.45) h(m) = JX mg 0 (m, m), Z (2.46) Z = 1 J[X m + X m ]G 0 (0, 0) + J 2 X m X m G 2 0(0, 0) J 2 X m X m G 2 0(m, m). (2.47) We evaluate Eq. (2.43) with Eq. (2.44) and the explicit expression for free Green s function, G 0 (E k )(n, m) = eik n m 2iJ sin k (see details in Appendix B.2).

42 30 j pumped = ej π m=±l Ẋ m [1 2 [f( m)f ( m) + h( m)h ( m)]im[g 0 (0, 0)] +Re[f( m)h ( m)]im[g 0 (m, m)] sign(m)im[f( m)h ( m)]re[g 0 (m, m)] ] E=EF (2.48) We evaluate pumped charge for a closed cycle by integrating instantaneous current with respect to time. q pumped = dt j pumped (2.49) = ej [1 dx m π 2 [f( m)f ( m) + h( m)h ( m)]im[g 0 (0, 0)] m=±l +Re[f( m)h ( m)]im[g 0 (m, m)] sign(m)im[f( m)h ( m)]re[g 0 (m, m)] ] E=EF (2.50) As shown in Eq. (2.50), pumped charge for a closed is determined by shape of pumping in parameter space and is independent of how the shape is formed, this implies that, for sinusoidal pumping parameters, pumped charge is independent of frequency of pumping parameters. Also one can observe mathematical structure of Eq. (2.50) is similar to that of Berry s phase.

43 Chapter 3 Pumping Spin 3.1 Introduction The field of spintronics offers a vision of electronics that utilizes carrier spin in addition to carrier charge [38]. The rich potential of carrier spin for applications ranges from non-volatile devices to quantum computation. In order to realize this potential, however, it is essential to develop effective tools for the manipulation and transport of spin. Adiabatic quantum pumping provides such a tool. As a result of cycling two or more physical parameters, a direct current is generated[2]. This method can deliver precise currents and requires no voltage bias. Recently, it has been shown that adiabatic quantum pumping in the presence of a magnetic field can also generate a spin current[39, 40, 41, 42, 43]. For some fortuitous choices of experimental parameters, it has even been possible to generate a spin current without any charge current, which is termed a pure spin current [40, 41, 43]. This is a promising development, and one wonders if it is possible to establish complete control over both the amount of spin and the amount of charge pumped per cycle. A device for generating a pure spin current with improved control appears in Ref. [42], in which Zeeman energy is chosen as one of the adiabatic pumping parameters. There is no need to make a fortuitous choice of parameters in this device when the minimum and maximum Zeeman energies involved in the pumping cycle are equal in magnitude but opposite in sign, a pure spin current arises. However, if the maximum value of Zeeman energy is not equal and opposite the

44 32 minimum value, various combinations of spin current and charge current arise in a way that is not easily controlled. In this chapter, we introduce a flexible approach to adiabatic pumping in which essentially any composition of spin/charge current can be chosen as desired. The scheme utilizes generalized pumping parameters each of which depends on more than one physical parameter. With generalized pumping parameters, many different physical processes map to the same path in pumping parameter space. The result is greatly improved control over the pumping products. For instance, a pure spin current can be generated in the following way. In adiabatic quantum pumping, carriers are transported with each cycling of the pumping parameters. By reversing the direction of the pumping cycle trajectory, one reverses the direction of the current flow. With generalized parameters, it is possible to make the spin-up pumping parameters traverse the exact same trajectory as the spin-down pumping parameters, but in the opposite direction. The result is that spin-up carriers get pumped in one direction and spin-down carriers in the other direction, leading to zero net transport of charge but non-zero spin current. This technique can be applied to any desired cycle in parameter space, so that the amount of spin pumped per cycle can be set by judicious choice of the trajectory in parameter space. Generalized pumping parameters also enable selective spin pumping wherein the current consists only of spin-up carriers or only of spin-down carriers. One spin s parameters traverse a degenerate cycle that pumps no charge, while the other spin s parameters traverse a productive cycle. This selective spin pumping is a valuable tool; by combining and repeating spin selective pumping, it is possible to generate any rational proportion of spin current to charge current. Finally, in addition to pure spin pumping and spin selective pumping, we consider a family of cycles which produce arbitrary ratios of spin current to charge current after exactly one cycle. Unlike the schemes mentioned above, the correct cycle in this case cannot be fixed in a deterministic way; trial and error is necessary. However, we present an argument that an appropriate cycle generally exists. Moreover given a cycle that produces a given ratio of spin current to charge current, we also show how to traverse a cycle that produces the inverse ratio. This chapter is based on our published paper[15].

45 Model Hamiltonian and Pumped Current We consider transport through a one-dimensional channel of sites, schematically depicted in Fig The following is our model Hamiltonian H = H 0 + V 1 + V 2, H 0 = J n,σ a n+1σa nσ + a nσa n+1σ, V 1 = σ V 2 = σ u l n lσ + u l n lσ, σe Z l n lσ σe Z l n lσ. (3.1) In the Hamiltonian, J is the nearest neighbor hopping amplitude, a nσ is the electron creation operator at site n for spin σ and n lσ = a lσ a lσ is the number operator at site l for spin σ. The first term H 0 is the Hamiltonian of carriers in a homogeneous channel and V 1 is the impurity potential. The impurities are characterized by on-site energies {u l, u l }, which simulate potential barriers at sites l and l. The V 2 term contains the Zeeman spin energies E l Z = gµ B B l, El Z = gµ B B l for carriers in the localized external magnetic fields at sites l and l. We assume that all four experimental parameters in Fig. 3.1 can be tuned precisely; naturally, this would be challenging to realize in the laboratory. We set spin σ equal to 1 instead of 1/2. We assume that the four quantities {u l, u l, E l Z, EZ l } are slowly varying time-dependent parameters. We define generalized spin-dependent Figure 3.1. Gray dots located at l and l sites represent impurities on a wire; the impurities are characterized by potential barriers u l,u l. Localized magnetic fields B l,b l are applied on those impurities.

46 34 pumping parameters {X lσ, X lσ }. X lσ = (u l σe Z l)/j X lσ = (u l σe Z l )/J (3.2) in terms of which the Hamiltonian becomes H = H 0 + V, H 0 = J n,σ a n+1σa nσ + a nσa n+1σ, V = J σ X lσ n lσ + X lσ n lσ. (3.3) For mathematical convenience the following are set to unity = e = latticespacing = 1. By defining the spin-dependent parameters {X lσ, X lσ }, we make explicit the fact that spin-up carriers and spin-down carriers are separately controllable. We apply the procedures developed in chapter 1 to evaluate spin current. In this chapter, scattering state has spin index. Instantaneous scattering states χ pσ of the Hamiltonian are obtained by ignoring the time-dependence of the pumping parameters and using the Lippmann-Schwinger equation[35] χ pσ = [1 + G(E p )V ] c pσ 0, (3.4) where c pσ is the carrier creation operator for energy E p = 2J cos p and spin σ, and G(E p ) = 1/(E p H + i0 + ) is the retarded full Green s function. The timevariation of the potential is adiabatic if it is slow compared to the dwell time of a carrier in the scattering region [3]. The time-dependence is then restored to first order by adiabatic corrections [8] φ pσ = χ pσ i G(E p ) χ pσ. (3.5) In terms of these first order scattering states (3.5), the instantaneous pumped current associated with spin σ is

47 35 j σ (n) = [ π dp de F (E) 2J Im π 2π δ(e E p) nσ φ pσ n + 1σ φ pσ ], (3.6) where Im indicates imaginary part, F (E) is the Fermi distribution function, and nσ = a nσ 0. First the integral over p is evaluated π dp 2J Im π 2J 2 Im π Ẋ mσ m=±l 2π δ(e E p) nσ φ pσ n + 1σ φ pσ (3.7) π dp 2π δ(e E p) [ i n + 1σ G 2 (E p ) mσ nσ χ pσ mσ χ pσ i nσ G 2 (E p ) mσ mσ χ pσ n + 1σ χ pσ ], (3.8) where the identity χ pσ = G(E p ) V χ pσ has been used[8]. Eq. (3.8) is evaluated using the identity π where G(nσ, mσ; E) = nσ G(E) mσ. The result is dp π 2π δ(e E p) nσ χ pσ mσ χ pσ = 1 [ ] G(nσ, mσ; E) G (nσ, mσ; E), (3.9) 2πi J 2 π Im m=±l Ẋ mσ [ n + 1σ G 2 (E) mσ [ G(nσ, mσ; E) G (nσ, mσ; E) ] nσ G 2 (E) mσ [ G(mσ, n + 1σ; E) G (mσ, n + 1σ; E) ]]. (3.10) Eq. (3.10) can be simplified if we use the fact that the 1-D Green s function

48 36 takes a plane-wave form at large distances[8] G(n + 1σ, mσ; E k ) = e ik G(nσ, mσ; E k ) for n. (3.11) The asymptotic condition n means the observation point is far away from the scattering center and is located on the right side. Inserting this into Eq. (3.10) produces J 2 π Im m=±l Ẋ mσ E [ G(nσ, mσ; E)G (n + 1σ, mσ; E) ]. (3.12) At the last step of the calculation, we use the identity nσ G 2 (E) mσ = E G(nσ, mσ; E). (3.13) This identity is derived by expanding G(E) in a basis of energy eigenstates of H, G(E) = E µ E µ µ E E µ+i0. This is the alternative way to derive the identity. The + integration over energies in Eq. (3.6) can be performed with the result (3.12) at zero temperature where the Fermi distribution function is a step function j σ (n) = J 2 π Ẋ mσ m=±l Im [ G(nσ, mσ; E)G (n + 1σ, mσ; E) ] E=EF for n. (3.14) We find the closed form for the instantaneous pumped current in terms of retarded full Green s functions. In contrast to most treatments of quantum pumping, our derivation does not introduce the scattering matrix. The identity (3.9) is crucial in our derivation. To evaluate the pumped current associated with the model Hamiltonian (3.1), we need to compute the full Green s function. The full Green s function can be

49 37 expressed in terms of the free Green s function by the algebraic identity[35] G(E) = G 0 (E) + G(E)V G 0 (E), where G 0 (E) = 1/(E H 0 + i0 + ). Eq. (3.14) is evaluated to be j σ (n) = J Ẋ mσ π m=±l [ 1 [ f( mσ)f ( mσ) + h( mσ)h ( mσ) ] Im[G 0 (0σ, 0σ; E)] 2 +Re[f( mσ)h ( mσ)]im[g 0 (mσ, mσ; E)] sign(m)im[f( mσ)h ( mσ)]re[g 0 (mσ, mσ; E)]], (3.15) E=EF where f(mσ) = 1 JX mσg 0 (0σ, 0σ; E) Z mσ, h(mσ) = JX mσg 0 (mσ, mσ; E) Z mσ, Z mσ = 1 J[X mσ + X mσ ]G 0 (0σ, 0σ; E) +J 2 X mσ X mσ G 2 0(0σ, 0σ; E) J 2 X mσ X mσ G 2 0(mσ, mσ; E). Eq. (3.15) can be evaluated since the explicit expression for the free Green s function [35] is just G 0 (nσ, mσ; E k ) = e ik n m /2iJ sin k. The pumped charge associated with spin σ after one pumping cycle is obtained by integrating Eq. (3.15) with respect to time q σ = dt j σ (n). (3.16) The pumped charge associated with spin σ (3.16) can be represented as a surface integral rather than a line integral [2] q σ = J 2 π S dx lσ dx lσ sign(m) m=±l Xmσ Im [ G(nσ, mσ; E)G (n + 1σ, mσ; E) ] E=EF,

50 38 (3.17) where S indicates the area which is enclosed by the pumping cycle in parameter space. There are two features of the pumped entity (charge or spin), evident in the last two equations, that we utilize to control the flow of spin and charge. First, in the line integral form in Eq. (3.16) it is clear that reversing the direction of the time cycle changes the sign of the integral, indicating a flow in the opposite direction over a pump cycle. Secondly, the surface integral form, Eq. (3.17), shows that the magnitude of the pumped quantity in a full cycle depends entirely on the enclosed surface S in parameter space. Therefore, surfaces that enclose areas with identical functional form and values of the integrand will yield identical magnitudes of the pumped charge or spin, while the direction of traversal of the bounding curve will determine the direction of flow. 3.3 Spin current without charge current In the previous section we established how we can control the direction and magnitude of the pumped charge associated with each spin state, and found expressions to determine them. In particular, varying the magnetic fields in the generalized parameters allows differential manipulations of up and down spin states, because the path in parameter space of the spin-up parameters {X l, X l } becomes distinct from the path in parameter space of the spin-down parameters {X l, X l }. We will now use these considerations to present two distinct types of pumping cycles which generate only a pure spin current, with zero transported charge after each cycle of pumping. The first cycle relies on the fact that the integrand of (3.17) is symmetric under exchange of X lσ and X lσ, as one expects given the form of the Hamiltonian (3.3). Consider two square cycles (we will use the term boxes ) of side length δ which are located symmetrically in parameter space about the line X l = X l (see Fig.3.2a). For the cycle taken by spin-down parameters {X l, X l }, we pick an arbitrary point (a, b) as the initial choice of parameters. For the cycle taken by spin-up parameters {X l, X l }, the initial point (b, a δ) is chosen. For those initial points, we find u l /J = (a + b)/2, u l /J = (a + b δ)/2 and E l Z /J = (a b)/2, EZ l /J = (a b δ)/2. We fix u l, El Z throughout the pump-

51 39 Figure 3.2. (a) Two identical square boxes are located symmetrically about the X l = X l line. For spin up, the parameters {X l, X l } begin at (b, a δ) and go counterclockwise. For spin down, the {X l, X l } begin at (a, b) and go clockwise. (b) The spin up and spin down cycles are located on the same rectangular box. For spin up, the cycle begins at (a, b) and goes counterclockwise. For spin down, the cycle begins at (a + δ, b) and goes clockwise. ing cycle. First, the Zeeman energy at site l divided by J, E l Z /J, is decreased by δ, from (a b)/2 to (a b 2δ)/2. The resulting motion in parameter space is parallel to the X l axis, with the spin-up parameters moving in the positive direction and the spin-down parameters moving in the negative direction. Next, the potential barrier at site l divided by J, u l /J, is increased by the amount of δ, from (a + b δ)/2 to (a + b + δ)/2. The spin-down and spin-up parameters both shift upward parallel to the X l axis. Next, the Zeeman energy E l Z /J is increased by δ, from (a b 2δ)/2 back to (a b)/2. Finally, the potential barrier u l /J is decreased by δ, from (a + b + δ)/2 back to (a + b δ)/2, to complete the cycle. The form of the definition (3.2) ensures that spin-up and spin-down parameters shift in opposite directions when the Zeeman energy is varied and shift in the same direction when the potential barrier, which results from an electrical potential, is varied. The combination of these two effects moves the spin-up parameters in a counterclockwise cycle, and the spin-down parameters in a clockwise cycle. In addition, our chosen steps generate two square cycles located symmetrically about the line X l = X l in parameter space. Because of the symmetry in (3.17), these cycles lead to zero total pumped charge q c = q + q = 0. On the other hand, the pumped spin is q s = q q 0 as long as q (= q ) 0. The result is a pure spin current. A second type of cycle generates a pure spin current without relying on the symmetry between X l and X l. Consider a rectangular box in parameter space

52 40 (see Fig.3.2b). By choosing two initial points appropriately, we can make the parameters execute cycles on the same rectangular box of width δ and height γ but in opposite directions. For the cycle of spin-up parameters, we pick an arbitrary point in parameter space (a, b). For the cycle of spin-down parameters, the initial point (a + δ, b) is chosen. These choices correspond to u l /J = a + δ/2, u l /J = b, E Z l /J = δ/2, and EZ l /J = 0. We vary u l and E Z l while fixing u l and El Z. First the Zeeman energy EZ l /J is decreased by the amount δ, from δ/2 to δ/2. Second, the potential barrier u l /J is increased by γ, from b to b + γ. Third, the Zeeman energy E l Z /J is increased by δ, from δ/2 back to δ/2. Finally, the potential barrier u l /J is decreased by γ, from b + γ back to b. As a result of these variations, the spin-up parameters traverse the rectangular box in the counterclockwise direction while the spin-down parameters traverse the rectangular box in the clockwise direction. Since the cycles enclose the same region, but move in opposite directions, a pure spin current arises. In this second type of cycle, note that El Z is fixed at zero throughout the pumping. This suggests a means of realizing the cycle experimentally. Rather than trying to produce a localized magnetic field B l, one could apply a global magnetic field. If all sites except for the site l have a negligible g-factor, the desired Hamiltonian (3.3) will arise [45]. We conclude based on the above analysis that, for any given pair of identical symmetrically located square boxes or for any given single rectangular box in parameter space, there always exists a pumping cycle which generates a pure spin current. This finding implies great flexibility in the control of pumped pure spin after one cycle. Since the quantity of pumped spin depends on the shape of the enclosed area and its location in parameter space, one can tune the quantity of pumped spin by changing these characteristics of the pumping cycle. The following plots made using the expressions derived in previous section demonstrate flexibility in controlling a pure spin current. Fig. 3.3 and Fig. 3.4 portray how the total pumped spin depends on the size of the box in parameter space and on its location. For the first type of cycle involving pairs of symmetrically positioned square boxes in parameter space, three different sizes for the box pairs are considered in Fig. 3.3a. All meet at the same point on the line X l = X l. The dependence of the pumped spin on the location of the cycle in parameter space

53 41 Figure 3.3. (a) Three pairs of square boxes are located symmetrically about the median X l = X l and share the same meeting point. We move those pairs along the X l = X l line. For the three pairs, the box side lengths are 2,4, and 6 respectively. The pairs shift a distance 2d along the X l = X l line. (b) Pumped spin q s vs d at fixed k F = 1.4 and for l = 1: The solid line is for the pair of large boxes of side length 6, the long-dashed line is for the pair of medium boxes of side length 4, and the short-dashed line is for the pair of small boxes of side length 2. The result shows monotonic increase of pumped spin with box size and monotonic decrease of pumped spin with projected distance d. is studied by moving that common meeting point a distance 2d along the line X l = X l. Each curve in Fig. 3.3b shows the variation of the pumped spin as a function of d for a specific box size, a fixed Fermi wave vector k F = 1.4 and impurities at ±l = ±1. The different curves correspond to the three different box sizes. The plots show a monotonic increase of pumped spin in a cycle as the box size increases (except where the pumped spin vanishes for all three box sizes). This is physically reasonable since the box size determines the difference between the minimum and maximum value of the potential barriers. For large boxes, the potential barriers change a lot during the cycle, resulting in more pumped current, and the opposite is true for small boxes. The pumped spin decreases as the distance d increases. As d increases, the minimum height attained by the potential barriers gets larger. As a result, the current must traverse an increased potential barrier, so that the transmission is decreased. For the second type of cycle discussed above for generating a pure spin current, we consider three square boxes that each straddle the median line X l = X l as

54 42 Figure 3.4. Plots of q s vs d. In each plot, the solid line is for a large box of side length 6, the long-dashed line is for a medium box of side length 4, and the short-dashed line is for a small box of side length 2. (a) We move each square box along the line X l = X l, where the position of the lower left corner is (d, d). (b) Plot at k F = 1.4 and for l = 1. For small d, negative pumped spin is found for the smaller two boxes (i.e. spin is pumped in the opposite direction). (c) Plot at k F = 3.1 and for l = 1. The maximum value 2 of pumped spin per cycle occurs on the solid line. The pumped spin q s is independent of box size for sufficiently large d. shown in Fig. 3.4a. The pumped spin in a cycle is plotted as a function of d where the lower left corner of the box is at the point (d, d). The two plots, Fig. 3.4b and Fig. 3.4c, correspond to different choices of Fermi wave vector, k F = 1.4 and k F = 3.1 respectively; the clear differences between the two sets of curves indicate that the Fermi wave vector is another essential factor in controlling the spin current flow. For both plots the impurity locations are ±l = ±1. In Fig. 3.4b, the pumped spin increases monotonically with box size at most locations d. For large d, the behavior of the curves in Fig. 3.4b shows monotonic decrease with d like the curves in Fig. 3.3b. However, for small d the pumped spin increases with d, so that each curve in Fig. 3.4b has maximum pumped spin between d = 3 and d = 5. This kind of maximum can be explained in terms of resonant transmission[46]. When the pumping cycle includes locations in parameter space for which the Fermi energy satisfies a resonance condition, an enhanced transmission coefficient leads to a large pumped current. For the parameters chosen in the figure, a line of resonant points in parameter space runs near (X l, X l ) = (6, 6). As each box shifts with increasing d to enclose these resonant points, the pumped current grows even though the

55 43 minimum heights of the potential barriers get larger. For d near zero in the case of the two smaller boxes in Fig. 3.4b, there is actually a region of negative values for the pumped spin, indicating a reversal of direction of the spin flow. The direction of pumped spin changes as the box moves or its size increases. This shows we can control the direction of the spin current without reversing the whole cycle, by simply adjusting the box size or location. For Fig. 3.4c, the maximum pumped spin for the largest box is 2, so that during each cycle one spin-up carrier goes to the right and one spin-down carrier goes to the left. This large maximum value is due to both the low minimum barrier heights and the resonance transmission. For k F = 3.1, a resonant line runs near (X l, X l ) = (1, 1) and boxes enclosing the resonant line also have the low minimum barrier heights, by adding up two effects, the pumped spin has the large maximum value such as 2. With increasing d from the origin, the pumped spin decreases to assume the same finite value for the three different box sizes, indicating that there is little variation in the integrand in Eq. (3.17) in the surface integral far from the origin whereas there are stronger variations close to the origin. 3.4 Selective spin pumping We consider another type of cycle which has a spin-filtering effect. This cycle selectively pumps one kind of spin, so that an equal spin and charge current flow (See Fig. 3.5). Consider a rectangular box in parameter space. We choose the same initial point (a, b) for both spin-up and spin-down parameters. This choice of initial point implies that u l /J = a, u l /J = b and E l Z = EZ l = 0. We fix El Z at zero. First, we increase u l and decrease E l Z simultaneously in such a fashion that (u l + E l Z )/J remains at the initial value a while (u l E l Z )/J is increased by the amount δ, from a to a + δ. At the end of this process, u l /J is a + δ/2 and E Z l /J is δ/2. The spin-up parameters shift parallel to X l, but the spindown parameters remain unchanged. Second, u l /J is increased by γ, from b to b + γ. The spin-down parameters and the spin-up parameters both shift parallel to the X l axis. Next, we decrease u l and increase E Z l simultaneously, keeping u l + E l Z fixed while (u l El Z )/J decreases by δ, from a + δ back to a. This produces a path parallel to X l for spin-up parameters, but does not shift the

56 44 Figure 3.5. (a) An example of pure spin current. There is no net flow of charge. (b) An example of selective spin current. Only one kind of spin contributes to current, there is no cancellation of charge. spin-down parameters at all. Finally, u l /J is decreased by γ, from b + γ back to b, shifting both spin-down parameters and spin-up parameters parallel to X l. For spin-up parameters, this cycle makes rectangular path enclosing a non-zero area (see Fig. 3.6a) that produces a current of spin-up carriers. On the other hand, the cycle for spin-down shifts along a straight line in parameter space that encloses no area (see Fig. 3.6b) and pumps no current. The result is perfect selective spin pumping of spin-up carriers. Naturally, a selective current of spin-down carriers can be generated with trivial modifications to this protocol. More generally, we can transfer charge and spin to achieve any rational value of q s /q c by combining and repeating spin selective cycles. Suppose that the value Figure 3.6. (a) Pumping cycle for spin-up parameters, beginning at the point (a,b) in the generalized parameter space and proceeding counterclockwise. It encloses a nonzero area, and generates a non-zero spin-up current. (b) Pumping cycle for spin-down parameters, also beginning at (a,b). This cycle encloses no area, so there is no spin-down current generated.

57 45 q s /q c = M/N is desired, where M and N are integers. It is always possible to find two integers n and m satisfying n/m = (N M)/(N + M). By performing m selective spin-up cycles and n selective spin-down cycles, we can generate an arbitrary rational value for q s /q c. (If m is positive, the spin-up cycles should be traversed in a counterclockwise direction, while for negative m they should be traversed in a clockwise direction. The same is true for n and the spin-down cycles.) 3.5 Arbitrary combinations of spin current and charge current In earlier sections, we gave definite cycles that could be used to pump spin with no charge, to selectively pump carriers of a given spin orientation, or to pump a rational ratio of spin to charge. Here, we argue that other cycles can produce arbitrary ratios of spin current to charge current, requiring only one cycle of pumping with no repetition. (However, we do not give a recipe for identifying the cycle; trial and error tuning may be needed.) Consider a single rectangular box and put two congruent rectangular cycles symmetrically at its ends as shown in Fig. 3.7a. The left cycle is traversed by the spin-up parameters, and the right cycle by the spin-down parameters. We first describe a protocol in which the spin-up and spin-down parameter cycles are both traversed in a counterclockwise direction. Choose initial points at the lower left corner of each cycle. For spin-up parameters, the point is (a, b). For spin-down parameters, the initial point is (a+β δ, b). These choices imply that the physical parameters are u l /J = a + (β δ)/2, u l /J = b, and E l Z /J = (β δ)/2, and El Z = 0. Suppose that the Zeeman energies E l Z and EZ l are fixed. As a first step, we increase u l /J by δ, from a + (β δ)/2 to a + (β + δ)/2. Next, we increase u l /J by γ, from b to b + γ. Then, we decrease u l /J by δ, from a + (β + δ)/2 back to a + (β δ)/2. Finally, we decrease u l /J by γ, from b + γ back to b. This is the complete protocol. For counterclockwise traversal, it is typically the case that the charge q produced by the spin-up cycle will be positive and so will the charge q produced by the spin-down cycle. As a result, the ratio q s /q c = (q q )/(q + q )

58 46 Figure 3.7. (a) We consider two congruent rectangular boxes symmetrically located within one large rectangular box. By varying δ, we shift the interior edges of the rectangular boxes. The left rectangular box shows the trajectory of spin-up pumping parameters, while the right rectangular box is for spin-down pumping parameters. For the first type of cycle (shown here), both the small rectangles are traversed in a counterclockwise direction. For the second type of cycle (not shown here), spin-up parameters traverse in a counterclockwise direction while spin-down parameters traverse in a clockwise direction. (b) Plot of q s /q c vs δ for first type of cycle at k F = 3.1 and for l = 1. We set (a, b) = (0, 0) and β = γ = 8. δ is varied from 1 to 8. Any q s /q c in the range 0 to 1 can be found. Note that the origin of plot is (1, 0), not (0, 0). will typically satisfy q s /q c < 1. Tuning the parameters will typically permit any desired value of the ratio, as shown in Fig. 3.7b. In order to attain a ratio q s /q c > 1, one can use the same two rectangular cycles, traversed in, say, a counterclockwise direction for spin-up and in a clockwise direction for spin-down. The lower left point in the spin-up rectangle and the lower right point in the spindown rectangle serve as initial points. It follows that the physical parameters take the initial values u l /J = a + β/2, u l /J = b, E l Z /J = β/2, and EZ l = 0. We fix u l, El Z during the cycle. First, E l Z /J is decreased by δ, from β/2 to β/2 δ. Next, u l /J is increased by γ, from b to b + γ. Third, E l Z /J is increased by δ, from β/2 δ back to β/2. Finally, u l /J is decreased by γ, from b + γ back to b. Since we are traversing the same two rectangles as the previous protocol, we see that the same value of q will be produced, but the spin-down charge will now be q, where q is defined as the spin-down charge produce by a counterclockwise traversal. The

59 47 result is that q s /q c = (q + q )/(q q ), which is simply the inverse of the value of the ratio obtained in the first protocol, so that now q s /q c > 1 typically. Given these two protocols, we should be able to achieve arbitrary combinations of spin current and charge current over the whole range 0 q s /q c + by varying δ (see Fig.3.7b) which determines each pumping cycle within the rectangular box. Although tuning is required to achieve a given arbitrary ratio, the exact inverse ratio for that combination can be attained predictably by reversing one of cycles. The extreme cases q s /q c = + and q s /q c = 1 correspond respectively to the cases of pure spin pumping and to spin selective pumping described above. If q s /q c is positive, a corresponding negative ratio, which has the same absolute value, can be obtained by exchanging cycles so that the left box is for the spin down parameters and the right box is for the spin up parameters. Thus, all possible ratios q s /q c are attainable. 3.6 Conclusion In this chapter, we have presented a deterministic way to produce a pure spin current, a spin selective current, or a rational ratio of spin to charge current. Our proposal relies on generalized pumping parameters, each of which depends on more than one physical parameter. In our calculations, the maximum value 2 for the pumped spin is observed for some pumping cycles, and we find that the direction of the spin current can be manipulated via the size or location of the pumping cycle in parameter space. We also presented an argument to show that it is typically possible in a single cycle to pump an arbitrary ratio of spin current to charge current (although some trial and error may be needed to find the right cycle). These results suggest that adiabatic quantum pumping could be a versatile tool for generating a desired current in a spintronics device.

60 Chapter 4 Pumping Singlet 4.1 Introduction Quantum information theory exploits the quantum properties of nature which are unavailable to the classical world [47]. One of most remarkable features of the quantum world is the entanglement of particles, and entanglement is at the heart of the power of quantum information. To realize quantum information applications, it is therefore crucial to have tools to generate entangled states. There were several proposals [48, 49, 50] to generate entangled electron pairs by tunneling through quantum dot in the Coulomb blockade regime. The generating entangled electronhole pairs by one body potential was proposed by Beenakker et al. [51]. In this chapter, we propose a tool to produce entangled pairs of electrons. We apply pumping mechanism to generate entangled pairs. We use on-site Coulomb interactions, which are two-body potentials, as pumping parameters rather than the usual one-body potentials. As a result, paired electrons experience the pumping parameter together and the pump generates a paired electron current. When two-body potentials are included in the Hamiltonian, the conventional scattering matrix formalism is not applicable. However, the formalism that we have developed is more flexible and can be extended to the present problem. In the following section, we show this extension for electron pair pumping and present some plots predicting the quantity of pumped pairs. The work of this chapter was published in our paper[16].

61 Pumping Singlet Pair We apply pumping mechanism to produce entangled pairs of electrons. We introduce the following Hamiltonian. H = H 0 + V, (4.1) H 0 = J a n+1σa nσ + a nσa n+1σ, (4.2) n,σ=, V = U l n l n l + U l n l n l. (4.3) This Hamiltonian has similar mathematical structure to the Hamiltonian we introduced in chapter 1. We apply the same methods we used in previous chapters to evaluate current for entangled pairs. V is impurity potential, which includes two Hubbard impurities located at lattice sites l and l. Hubbard impurity simulates on-site Coulomb interaction between two electrons when they are located at the same site. One property of Hubbard impurity U m n m n m is that it only detects singlet pairs. It doesn t sense any one electron and any triplet pair. U m n m n m one electron = 0, (4.4) U m n m n m triplet = 0. (4.5) We choose on-site Coulomb interactions {U l, U l } as pumping parameters, only singlet pairs experience pumping mechanism because one electron state and triplet pairs cannot detect Hubbard impurities. As a result, pumped current should be current of electrons which belong to the singlet state. We develop our formalism with analogous extension of one body impurity problem. We define incoming two particle state. k 1, k 2 = c k 1 c k 2 0. (4.6) We have zeroth order scattering state and first order scattering state for the

62 50 incoming state. χ k1,k 2 = (1 + G V ) k 1, k 2, (4.7) φ k1,k 2 = χ k1,k 2 i G χ k1,k 2. (4.8) We project the first order scattering state to real space singlet state, n 1, n 2 = a n 1 a n 2 0 a n 1 a n 2 0. n 1, n 2 φ k1,k 2 = n 1, n 2 χ k1,k 2 i n 1, n 2 G χ k1,k 2. (4.9) We calculate pumped current from first order scattering state[15, 52]. J pumped = π π dk 2 2π δ(e E k 1 E k2 ) def (E) [ 2JIm dk 1 n 2 π π 2π n 1, n 2 φ k1,k 2 n 1 + 1, n 2 φ k1,k 2 ], (4.10) where E is total energy of two electrons and F (E) is distribution function for two electrons. We follow the same procedure as that of one body problem. We calculate the integration in bracket first. π π 2JIm n 2 π 2JIm π n 2 π π π π dk 1 2π dk 1 2π dk 2 2π δ(e E k 1 E k2 ) n 1, n 2 φ k1,k 2 n 1 + 1, n 2 φ k1,k 2 dk 2 2π δ(e E k 1 E k2 ) [ n 1, n 2 χ k1,k 2 n 1 + 1, n 2 χ k1,k 2 +i n 1, n 2 χ k1,k 2 n 1 + 1, n 2 G χ k1,k 2 i n 1 + 1, n 2 χ k1,k 2 n 1, n 2 G χ k1,k 2 ] 2JIm π π dk 1 dk 2 n 2 π π 2π 2π δ(e E k 1 E k2 ) [ i n1, n 2 χ k1,k 2 n 1 + 1, n 2 G χ k1,k 2 i n 1 + 1, n 2 χ k1,k 2 n 1, n 2 G χ k1,k 2 ]. (4.11)

63 51 We use the identity χ = G V χ to evaluate Eq. (4.11). π π 2JIm dk 1 dk 2 n 2 π π 2π 2π δ(e E k 1 E k2 ) [ i n1, n 2 χ k1,k 2 n 1 + 1, n 2 G χ k1,k 2 i n 1 + 1, n 2 χ k1,k 2 n 1, n 2 G χ k1,k 2 ] = 2JIm n 2 m=±l U m π π π π dk 1 2π dk 2 2π δ(e E k 1 E k2 ) [ i n1 + 1, n 2 G 2 a m a m 0 n 1, n 2 χ k1,k 2 0 a m a m χ k1,k 2 i n 1, n 2 G 2 a m a m 0 n 1 + 1, n 2 χ k1,k 2 0 a m a m χ k1,k 2 ] (4.12) We derive the identity to evaluate Eq. (4.12). π π π = 1 2πi dp dq 2π 2π δ(e E p E q ) 0 a n2 a n1 χ p,q 0 a m a m χ p,q [ G(n1, n 2 ; m, m) G (n 1, n 2 ; m, m) ], (4.13) π where G(n 1, n 2 ; m, m) = 0 a n2 a n1 G a m a m 0 is two particle full Green s 1 function and G =. E H+i0 + We evaluate Eq. (4.12) by using the identity (4.13). π 2JIm π dk 1 dk 2 U m n 2 m=±l π π 2π 2π δ(e E k 1 E k2 ) [ i n1 + 1, n 2 G 2 a m a m 0 n 1, n 2 χ k1,k 2 0 a m a m χ k1,k 2 i n 1, n 2 G 2 a m a m 0 n 1 + 1, n 2 χ k1,k 2 0 a m a m χ k1,k 2 ] = J π Im [ U m n1 + 1, n 2 G 2 a m a m 0 n 2 m [ n1, n 2 G a m a m 0 n 1, n 2 G a m a m 0 ] n 1, n 2 G 2 a m a m 0 [ n 1 + 1, n 2 G a m a m 0 n 1 + 1, n 2 G a m a m 0 ] ] = J π Im [ [ U m E n1, n 2 G a m a m 0 n 1 + 1, n 2 G a m a m 0 ] n 2 m

64 52 + n 1 + 1, n 2 G 2 a m a m 0 n 1, n 2 G a m a m 0 + n 1, n 2 G 2 a m a m 0 n 1 + 1, n 2 G a m a m 0 ] (4.14) = 4J [ U m E Im G(n 1, n 2 ; m, m)g (n 1 + 1, n 2 ; m, m) ]. π m n 2 (4.15) In this calculation, we use 0 a n2 a n1 G 2 a m a m 0 = EG(n 1, n 2 ; m, m) and G(n 1, n 2 ; m, m) = G(n 2, n 1 ; m, m). For Eq. (4.14), we prove the following statement. [ n 1 + 1, n 2 G 2 a m a m 0 n 1, n 2 G a m a m 0 n 2 + n 1, n 2 G 2 a m a m 0 n 1 + 1, n 2 G a m a m 0 ] = a real number for n 1. (4.16) We evaluate the instantaneous pumped current at zero temperature by putting Eq. (4.15) into Eq. (4.10). J pumped = 4J π = 4J π m m [ U m def (E) E Im U m Im n2 G(n 1, n 2 ; m, m)g (n 1 + 1, n 2 ; m, m) ] n 2 G(n 1, n 2 ; m, m)g (n 1 + 1, n 2 ; m, m) E=EF, (4.17) where we approximate F (E) θ(e F E). We derive some relations to calculate Eq (4.17). G(n 1, n 2 ; m, m) = f( m)g 0 (n 1, n 2 ; m, m) + h( m)g 0 (n 1, n 2 ; m, m), (4.18) where

65 53 f(m) = 1 U mg 0 (0, 0; 0, 0), X (4.19) h(m) = U mg 0 (m, m; m, m), X (4.20) X = 1 [U m + U m ]G 0 (0, 0; 0, 0) + U m U m G 2 0(0, 0; 0, 0) U m U m G 2 0(m, m; m, m). (4.21) We calculate Eq. (4.17) with the relation (4.18). J pumped = 4J U m Im G(n 1, n 2 ; m, m)g (n 1 + 1, n 2 ; m, m) π E=EF (4.22) m n2 = 4J U m Im [ f( m) f ( m)g 0 (n 1, n 2 ; m, m)g π 0(n 1 + 1, n 2 ; m, m) n2 m + h( m) f ( m)g 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) + f( m) h ( m)g 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) + h( m) h ( m)g 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) ] E=EF (4.23) = 4J [ U m f( m) f ( m) Im G 0 (n 1, n 2 ; m, m)g π 0(n 1 + 1, n 2 ; m, m) n2 m + h( m) h ( m) Im G 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) n 2 +Re [ f( m) h ( m) ] Im [ G0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) n 2 +G 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) ] +Im [ f( m) h ( m) ] Re [ G0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) n 2 G 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) ] ] E=EF. (4.24) We derive another identities to evaluate Eq. (4.24). Im n 2 G 0 (n 1, n 2 ; m, m) G 0(n 1 + 1, n 2 ; m, m) = 1 2J Im[ G 0 (0, 0; 0, 0) ] for n 1 > m (4.25)

66 54 Im n 2 [ G0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) + G 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) ] = 1 J Im[ G 0 (m, m; m, m) ] for n 1 > m (4.26) Re [ G0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) n 2 G 0 (n 1, n 2 ; m, m)g 0(n 1 + 1, n 2 ; m, m) ] = sign(m) 1 J Re[ G 0 (m, m; m, m) ] for n in 1 > m (4.27) For proofs for these identities, see Appendix C. We divide integral variable into two regions, in-region and out-region, by doing that, we take advantage of the explicit form of one particle free Green s function. In the expression Re[G 0 (m, m; m, m)] in, subscript in indicates that Re[G 0 (m, m; m, m)] in is the in-region integration of Re[G 0 (m, m; m, m)]. For details, see Appendix C. We put identities (4.25), (4.26) and (4.27) into the equation (4.24). J pumped = 4 [ 1[ U m f( m) f ( m) + π 2 h( m) h ( m) ] Im [ G 0 (0, 0; 0, 0) ] m +Re [ f( m) h ( m) ] Im [ G 0 (m, m; m, m) ] sign(m) Im [ f( m) h ( m) ] Re [ G 0 (m, m; m, m) ] in ] E=EF.(4.28) We have similar form to the final expression of one particle problem. This is expected because two problems have similar mathematical structures. Our approach to the present two-body problem can be considered as an analogous extension of one-body problem. The instantaneous pumped current depends on three evaluative numbers G 0 (0, 0; 0, 0), G 0 (m, m; m, m) and Re[G 0 (m, m; m, m)] in at a

67 55 given Fermi energy E F. From the instantaneous pumped current, we calculate pumped charge. Q pumped = 4 π m dt U [ 1[ m f( m) f ( m) + 2 h( m) h ( m) ] Im [ G 0 (0, 0; 0, 0) ] +Re [ f( m) h ( m) ] Im [ G 0 (m, m; m, m) ] sign(m) Im [ f( m) h ( m) ] Re [ G 0 (m, m; m, m) ] in ] E=EF. (4.29) Finally, we present plots for pumped singlet Figure 4.1. Q pumped for U min and U max : We consider square cycle where left-down corner has (U min, U min ) and right-up corner has (U max, U max ). This plot is for E F = 3.9. We set J = 1.

68 Figure 4.2. Q pumped for U min and U max : This plot is for F F = Figure 4.3. Q pumped for U min and U max : This plot is for F F = 1.

69 Figure 4.4. Q pumped vs E F : We consider square cycle where left-down corner has (0, 0) and right-up corner has (8, 8). This plot is for l = 1 which determines distance between two impurities. We set J = Figure 4.5. Q pumped vs E F : This plot is for l = 2.

70 Chapter 5 Pumping on extended lattice 5.1 Introduction In this chapter, we consider quantum pumping in two distinct configurations of interest both of which can be modeled by a discrete chain of tight-binding sites. The first configuration comprises of an Aharonov-Bohm (AB) loop geometry, subject of many studies associated with AB phase effects and Fano effects[53, 54, 55, 56]. Here, interference arises between the real space trajectories through the two loop arms. The model consists of a parallel double quantum dot (PDQD) system, where we take each arm of an AB loop to comprise of a single quantum dot. We derive analytical expressions for the pumped current in the PDQD system. We investigate the behavior of the model as a function of applied magnetic field. Early experimental work on adiabatic quantum pumping [12] observed a pumped current that is symmetric under magnetic field reversal, while theory predicted no definite symmetry[13]. It was suggested that the observed currents may be due to rectification effects rather than pumping [14]. It is thus desirable to have a model in which pumped current and rectification current can be definitively distinguished. Rectification currents should be symmetric under field reversal, so our model is designed so that the pumped charge is antisymmetric q pump ( B) = q pump (B) under magnetic field reversal. Thus, experiments conducted on a realization of this model would have some natural explanations of an experimentally observed current: If the current is anti-symmetric under magnetic field reversal, it is likely due to quantum pumping, otherwise there should be unexpected mechanism in-

71 59 volved in generating the resulting current. The second model that we consider is a tight-binding chain with next-nearestneighbor (nnn) hopping. When the chain has nnn hopping terms, the minimum of the conduction band energy need not lie at k = 0; the conduction band can look like that of an indirect semiconductor. As a result, the conduction band dispersion relation can have four Fermi wave vectors at a given Fermi energy E F, leading to interference effects between different k-space contributions to the electronic wave function. This model is physically relevant for pumping in semiconductor systems; band structure calculations of Si nanowires grown along, say, the [11 2] direction have predicted an indirect gap [57]. This chapter is based on the pre-print[17]. 5.2 Pumping Through a Loop Geometry We consider a 1-D tight-binding chain with a loop in the central region as shown in Fig The Hamiltonian is H(t) = H 0 + V (t), H 0 = J n (a n+1a n + a na n+1 ), V (t) = (J J e iϕ/4 )a 0a 1 + (J J e iϕ/4 )a 1a 0 +(J J e iϕ/4 )a 0a 1 + (J J e iϕ/4 )a 1a 0 J e iϕ/4 b 0a 1 J e iϕ/4 a 1b 0 J e iϕ/4 b 0a 1 J e iϕ/4 a 1b 0 +u a a 0a 0 + u b b 0b 0. (5.1) Here, a 0 is the creation operator on one site in the loop, and b 0 is the creation operator on the other site in the loop. We consider a nearly homogeneous chain where J is the hopping amplitude along the chain and J is the hopping amplitude on and off the loop sites. (The Hamiltonian H 0 includes a link with tunneling amplitude J to one of the loop sites and not the other; this link is cancelled out by the J dependent terms in V.) The on-site energy of the two loop sites are u a

72 60 Figure 5.1. Two quantum dots are connected to left and right leads in parallel. They form a closed loop in the central region which is threaded by a magnetic field B. One dot has on-site energy u a and the other has on-site energy u b. and u b. The magnetic flux Φ penetrating the loop gives rise to a phase difference ϕ = 2πΦ/Φ 0, Φ 0 = hc/e between the two real space paths in the loop so that they interfere. We identify J J e iϕ/4 as V 0, 1 and J J e iϕ/4 as V 0,1. We redefine b 0 as a 0, so that J e iϕ/4 = V 0, 1 and J e iϕ/4 = V 0,1. Also we identify u a as u 0 = V 0,0 and u b as u 0 = V 0, 0. We apply formalsim developed in Refs. [15, 8] to this model and by choosing pumping parameters as X 0 = u a /J and X 0 = u b /J, we have instantaneous pumped current j pump (n) = ej 2 π Im x Ẋ x G (E F )(n + 1, x)g(e F )(n, x), (5.2) 1 where G(E F ) = E F and the expression is valid for n + which means H+0 + current is observed on the right side far away from scattering center. G(E F )(n, x) is evaluated from Dyson s equation[35, 58] and this result should be consider as exact expression for a model which has a loop geometry. In terms of scattering matrix formalism, it is equivalent to getting pumped current with exact scattering matrix. Main feature of our model is that it keeps mirror symmetry for arbitary set of two pumping parameters in the absence of magnetic field because pumping parameters are lined up along with symmetric axis. In the presence of magnetic field, the mirror image of given Hamiltonian is achieved by reversing magnetic

73 61 Figure 5.2. (a) Pumped charge q vs ϕ: We set J /J = 0.5 and traverse a squareshaped pumping cycle with corners (X 0, X 0 ) = (0, 0) and (100, 100) as shown in (b). The solid line is for k F = 1.7, and the dashed line is for k F = 2.1. This plot shows the antisymmetry and periodicity of pumped charge as a function of the magnetic field. The solid line curve near the origin shows that the pumped current is highly sensitive to small magnetic fields. Figure 5.3. Pumped charge q vs k F : We set J /J = 0.5 and traverse a square-shaped pumping cycle with corners (X 0, X 0 ) = (0, 0) and (100, 100). The solid line is for ϕ = π 2, which corresponds to Φ = 1 4 Φ 0, and the dashed line is for ϕ = 3π 2, which corresponds to Φ = 3 4 Φ 0. Since the current is periodic for ϕ and is antisymmetric when the sign of ϕ is reversed, the two curves shown have opposite values of pumped charge at any given k F. field for any given arbitrary set of two pumping parameters. From this feature and conservation of charge for a closed cycle, we conclude that pumped charge has anti-symmetric relation for magnetic field dtj pump (n, ϕ) = dtj pump (n, ϕ). (5.3) This relation is clearly shown in Figs. 5.2(a) and 5.3 and agrees with the results

74 62 in Ref. [13]. One interesting result from this relation is that no pumped current is generated after a cycle even though two parameters are varing with time. The anti-symmetric relation is exactly opposite to symmetric relation for rectification current, our model can be used to answer the question which mechanism generates the resulting current. 5.3 Pumping on a chain with next-nearest-neighbor hopping We now turn our attention to the pumped current on a chain with next-nearestneighbor (nnn) hopping (see Fig. 5.4). We consider the following Hamiltonian H(t) = H 0 + V (t), H 0 = J n (a n+1a n + a na n+1 ) J n (a n+2a n + a na n+2 ), V (t) = u l n l + u l n l, (5.4) where J is the nnn hopping amplitude and n l = a l a l is the number operator on site l. The dispersion relation for this model (taking the chain constant to be 1) is E k = 2J cos k 2J cos 2k. So, in the case of a positive J and negative J the dispersion relation can have a double-well shape as shown in Fig This indirect gap shape is reasonable physically; it is reminiscent, for instance, of the band structure of certain Si nanowires [57]. Figure 5.4. We consider chain with next-nearest-neighbor hopping. J is the hopping amplitude for nearest neighbor hopping, and J is the hopping amplitude for nextnearest-neighbor hopping. On this chain, we assume only two sites (gray dots) located at l and l have on-site energies u l, u l.

75 63 Figure 5.5. This is the dispersion relation for J = 1 and J = 1. It has doublewell shape curve. Dashed line is for E k = E F = 1. Dashed line has four cross points with doublewell curve, those give four Fermi wave vectors {k 1F, k 2F, k 1F, k 2F }. For simplicity, we choose a simple time-dependent perturbation term V to drive the pumping. We need to extend the formalism in Ref. [15] to get the pumped current. First, it is necessary to define the current operator on this extended chain. Since there is a nnn hopping process, we define the current operator using the continuity equation t [ρ(n) + ρ(n + 1)] + J(n + 1) J(n 1) = 0, where ρ(n) = ψ a na n ψ and J(n) = ψ J n ψ. operator J n = J i (a n+1 a n a na n+1 ) J We are led to the definition of the i (a n+1 a n 1 a n 1 a n+1) J i (a n+2 a n a na n+2 ). (5.5) We find the instantaneous pumped current J pump (n) = ej π Im x=±l ej π Im x=±l ej π Im x=±l u x G(E F )(n, x)g (E F )(n + 1, x) u x G(E F )(n 1, x)g (E F )(n + 1, x) u x G(E F )(n, x)g (E F )(n + 2, x). This result is valid for n +. We have assumed that the parameters J

76 64 and J yield a double-well shaped dispersion relation (Fig. 5.5) with four solutions {k 1F, k 2F, k 1F, k 2F } to the equation E F = E k. Our convention is that k 1F > k 2F > 0. Some care is needed in applying this equation. Because of the shape of the dispersion relation, the velocity v k = 1 E k for +k k 2 is negative while the velocity for k 2 is positive. Thus incoming waves from the left reservoir correspond to wave vectors {k 1, k 2 } rather than {k 1, k 2 }. To keep this straight, we define the full Green s function G(E) = 1/(E H + iη α ), where η α=1 = 0 + and η α=2 = 0. As a result, left incoming waves with k 1 or k 2 have transmitted waves with the correct physical wave vectors {k 1, k 2 } and reflected waves with the correct physical wave vectors { k 1, k 2 }. Using Dyson s equation to compute the full Green s function, we derive an explicit expression for the instantaneous pumped current J pump = 2eJ where π x=±l u x [ g 2 1 d( x) + e 2ik 1x h( x) 2 sin k 1 (cos k 1 cos k 2 ) +g 2 2 d( x) + e 2ik 2x h( x) 2 sin k 2 (cos k 1 cos k 2 ) ] E=EF, (5.6) d(x) = 1 u xg 0 (E)(0, 0) Z x, (5.7a) h(x) = u xg 0 (E)(x, x) Z x, (5.7b) Z x = [1 u x G 0 (E)(0, 0)][1 u x G 0 (E)(0, 0)] u x u x G 2 0(E)(x, x), g 1 = 1 2iJ sin k 1 + 4iJ, sin 2k 1 g 2 = 1 2iJ sin k 2 + 4iJ, sin 2k 2 (5.7c) (5.7d) (5.7e) G 0 (E)(x, y) = g 1 e ik 1 x y + g 2 e ik 2 x y. (5.7f) The free Green s function G 0 (E)(x, y) is found to be the sum of two terms g 1 e ik 1 x y and g 2 e ik 2 x y. As a result, interference effects arise between the two wave vectors {k 1, k 2 }. In particular, the G 2 0(x, x) term in Z x contains a factor

77 65 of the form e 2i(k 1 k 2 ) x which depends on the sum of the two wave vectors, k 1 k 2 and the distance 2 x between the locations of the two time-dependent perturbation potentials in V. This factor e 2i(k 1 k 2 ) x admits an interpretation in terms of interference between the k 1 wave function and the k 2 wave function. Physically speaking, the dependence of the pumped current on the distance 2 x = 2l between the two points of action of the potentials in V depends not only on k 1 and k 2 individually but also on their relative wave vector k 1 k 2. This complicated pattern clearly distinct from that of pumped current on regular chain which does not have nnn hopping (see Fig. 5.6). In Fig. 5.7, we investigate how the pumped current depends upon the Fermi wave vector k 1F (where k 1F determines the Fermi energy E F and therefore also k 2F ). The high peak near k 1F = 1.9 results from resonance. To see this, note that the transmission will be greatest when the denominator Z l in Eq. (5.7c) is as small as possible. For large u l, u l, this means minimizing terms in Z l that are multiplied by the product u l u l. This leads to the condition G 2 0(E)(l, l) G 2 0(E)(0, 0) = 0. But this condition leads to three resonance conditions for wave vectors: e 4ik 1F l = 1, e 4ik 2F l = 1, and e 2i(k 1F k 2F )l = 1; when the Fermi wave vector k 1F is near 1.9, all three conditions are satisfied and a large pumped charge q 1 arises. For the low peak between 1.7 and 1.8 in Fig. 5.7, we find the corresponding k 1F satisfies the first Figure 5.6. Pumped charge q vs l (there is exactly one value of q for each choice of l in both (a) and (b)). In both (a) and (b), the Fermi level is E F = 1.5, and the pumping cycle is square shaped with lower-left corner (0, 0) and upper-right corner (100, 100). The distance between the two time-dependent potentials is 2l. (a) Standard chain with J = 1 and no nnn hopping, J = 0. A regular periodicity is evident. (b) Chain with nnn hopping, J = 1 and J = 1. The dependence of q on l is quite irregular and complicated.

78 66 Figure 5.7. Pumped charge q vs k 1F : We set J = 1, J = 1 and l = 10, square-shaped pumping cycle has (4, 4) and (100, 100) for left-lower corner and right-upper corner. High peak near k 1F = 1.9 is related to resonance transmission and low peak between k 1F = 1.7 and k 1F = 1.8 indicates destructive interference between two wave vectors. two resonance conditions but not the third. Physically, this can be interpreted as destructive interference between the two wave vectors, resulting in a small pumped charge. 5.4 Conclusion In this chapter, we find an anti-symmetric dependence on the magnetic field for our first model which has a AB loop and special feature that pumping parameters keep mirror symmetry in the absence of magnetic field. Our first model provides a clear means of distinguishing pumped current from rectification current. We have also derived an expression for the pumped current in a tight-binding chain with next-nearest-neighbor coupling. The behavior of the pumped current in this system results from interference effects between different wave vectors in k-space, in contrast to the interference effects between real space contributions in the AB loop.