Compressibility and Capacitance of Quantum Systems Dissertation zur Erlangung des akademischen Grades Dr. rer. nat. eingereicht an der Mathematisch-Naturwissenschaftlich-Technischen Fakult¨at der Universit¨atAugsburg von Kevin Patrick Steffen Augsburg, Juli 2017 Tag der m¨undlichen Pr¨ufung: 04. September 2017 Erstgutachter: Prof. Dr. Thilo Kopp Zweitgutachter: Prof. Dr. Gert-Ludwig Ingold Contents 1. Prologue 1 2. Introduction to Capacitance and Compressibility 5 2.1. Compressibility of a Thermodynamic System . .5 2.2. Coupled Electron Systems . .8 2.3. Compressibility and its Dependence on the Distribution of the Positive Back- ground Charge . 15 2.4. Experimental Findings of Negative Capacitance and Compressibility of Subsystems 20 2.5. Further Theoretical Approaches to Capacitance Analysis . 23 3. Exchange and Correlation Contributions at Finite Temperature 25 3.1. Hartree-Fock Approximation . 25 3.2. Groundstate of the Free Electron Gas . 31 3.3. Correlation Energy and Second Derivatives with Respect to Density . 32 3.4. Uniform Electron Gas and its Compressibility at Finite Temperature . 35 4. Density Dependent Spin-Orbit Coupling 43 4.1. Spin-Orbit Coupling . 43 4.2. Density Dependent Rashba Spin-Orbit Coupling . 44 4.2.1. Comparison of Parameters to Experiment . 47 4.2.2. Compressibility and Capacitance . 48 4.2.3. Capacitance of a System with Three-Dimensional Top Electrode . 53 4.3. Six-Band Model of a Layer with Spin-Orbit Coupling . 55 4.3.1. Introduction to the Model . 55 4.3.2. Capacitance and Compressibility for Different Electrostatic Layouts . 59 4.4. Summary of the Section . 65 5. Electron System with Rashba Spin-Orbit Coupling and Exchange Interaction 67 5.1. Introduction of the Hamiltonian and its Diagonalization . 67 5.2. Analytical Solutions to the Diagonalization . 69 5.3. Numerical Solution of the Hamiltonian . 74 6. The Two-Dimensional Extended Hubbard Model 81 6.1. The Extended Hubbard Model with Attractive Nearest-Neighbor Interaction . 81 6.2. The Slave-Boson Approach . 82 6.3. Hartree-Fock Evaluation and Comparison to Slave-Boson Results . 84 6.4. Compressibility and Capacitance of a System with Symmetric Electrodes . 86 6.4.1. Polar Heterostructure . 86 6.4.2. Symmetric and Asymmetric Standard Capacitor . 89 6.5. Compressibility and Capacitance of a System with Unequal Electrodes . 91 6.6. Capacitance as Function of the On-Site Interaction U ............... 93 7. Summary 97 III Contents A. Appendix 101 A.1. Compressibility for Constant Volume . 101 A.2. Calculation of the Capacitance from a Charge Distribution . 102 A.3. Anisotropic Mass in the Two-Dimensional Free Electron Gas . 103 A.4. Perturbation Theory at Finite Temperature . 107 A.5. System with Rashba Spin-Orbit Coupling . 113 A.6. Atomic SOC in Basis of t2g Orbitals . 115 A.7. Semi-Analytical Solution of a System with RSOC and Exchange Coupling . 117 A.8. Preliminary Results for the Semi-Analytical Solution of a System with RSOC and Exchange Coupling . 121 Bibliography 129 Acknowledgments 139 IV 1. Prologue The technical progress of electronic devices is accompanied by or even based upon the minia- turization of electrical circuits. As characteristic lengths reach the nanoscale regime, quantum mechanical effects play an increasing role and may impede technical advances. For example, in metal-oxide semiconductor field-effect transistors (MOSFETs) larger gate capacitances reduce the power dissipation as they render smaller gate voltages possible. The capacitance can be increased by either reducing the plate distance d or by increasing the dielectric constant of the insulator between the gates. But d possesses a lower boundary below which the gate currents induced by quantum mechanical tunnel processes exceed reasonable values. Even for arbitrarily large , a finite screening length inside the gate electrodes limits the effective gate distance and hence the capacitance [1]. On the other hand, quantum mechanical phenomena may also open up new possibilities for the realization of electronic devices with enhanced or unique properties. A class of materials that is promising in this respect are heterostructures with transition-metal oxides [2], for which already today interfaces and surfaces can be fabricated with properties sufficient for functionality [3]. The paradigm example of a representative of oxide heterostructures is LaAlO3/SrTiO3 (LAO/STO), as the two-dimensional electron system at the interface between the two bulk insulators LaAlO3 and SrTiO3 features electronic correlations. These give rise to a plethora of physical phenomena, as, e.g., superconductivity [4], magnetism [5] and even the coexistence of superconductivity and magnetism [6, 7]. A seizable spin-orbit coupling is also present at the LAO/STO interface [8]. As the electron density n at the interface can be varied by at least one order of magnitude by application of a backgate voltage, the LAO/STO system offers an excellent aptitude for the utilization in electronic devices. Measurements of the capacitance of a LAO/STO heterostructure by Lu Li et al. revealed an increase of more than 40% above the geometric capacitance at low electron densities [9]. The observed enhancement is much larger than in any other system so far. It can be traced back to the negative compressibility κ < 0 of the dilute electronic subsystem in one electrode [10, 11, 12]: In a homogeneous electron system at low densities the negative contribution of the exchange and correlation term to the compressibility dominates over the positive kinetic contribution. These findings and previous theoretical work [10, 11, 12] point to a new way to enhance the capacitance: As the modification of the material between the electrodes, i.e., the search for high dielectrics1, has reached its limits, the turn towards the electrode material promises substantial progress. This theoretical work examines how the properties of two-dimensional quantum mechanical electronic systems on the electrodes affect the capacitance and electronic stability of a two-plate capacitor. We investigate different models for the electronic systems and identify conditions under which the capacitance is enhanced beyond the geometric value. The layout we analyze consists of two static two-dimensional electron systems with Helmholtz free energy Fi, i = 1; 2, at a fixed distance d that are coupled to each other. The total electron 1The denotation is actually \high-κ", where κ refers to the real part of the dielectric constant . In this work, however, κ is exclusively the compressibility. 1 1. Prologue density is nt = n1 + n2, where ni is the electron density in subsystem i. Then the Helmholtz free energy of the total system is given by (for details see Section 2.2): Ft(n1; nt) = F1(n1) + F2(nt − n1) + Fes(n1; nt) − eVextn1A Here Vext = Vs, where Vs is the voltage from an external voltage source, e is the elementary charge and A is the area of one electrode. Fes is the energy of the direct Coulomb interactions between all charges. The differential capacitance is given by the differential change of charge on one electrode divided by the differential voltage that induced the change in charge. As n1 has to minimize Ft, one can extract Vs from the first derivative of the equation above. Thus the −1 inverse differential capacitance Cdiff of this layout is proportional to the second derivative of Ft with respect to the electron density on one layer: 2 2 2 2 2 1 @Vs 1 @ F1 @ F2 @ Fes 1 @ F1 @ F2 1 = = 2 2 2 + 2 + 2 = 2 2 2 + 2 + (1.1) Cdiff eA@n1 e A @n1 @n1 @n1 e A @n1 @n2 Cgeo This relation, read from left to right, demonstrates that the differential capacitance is a series connection of the geometric capacitance, as derived from the electrostatic term Fes, with two capacitances that originate from the quantum mechanical energies of the electronic systems. For effective distances d/ in the nanometer regime, these two \quantum capacitances" can be of the −1 same order of magnitude as the inverse geometrical capacitance, Cgeo = 4πd/A. If the sum of the \quantum capacitances" is negative, then the differential capacitance exceeds the geometric value, Cdiff > Cgeo. The second partial derivatives appearing in the capacitance are closely connected to the inverse compressibility κ−1. The sign of the compressibility determines the stability of a system; for κ < 0 the system is instable (see Section 2.1). For constant temperature and volume of the system, κ−1 is given by 2 1 2 dµ 2 d Ft=A = nt = nt 2 ; (1.2) κ dnt dnt where µ is the chemical potential. The total derivatives are taken with respect to the thermo- dynamic variable of the system, which is the total electron density nt. The electron densities on the plates, n1 and n2, are functions of nt and adjust in such a way that Ft is minimized. If the electronic systems of the electrodes were isolated, then the second partial derivatives appearing in Eq. (1.1) could be replaced by total derivatives and the quantum capacitances would be proportional to the respective compressibilities. Then, the condition for Cdiff > Cgeo corresponds to a negative compressibility in the isolated electron systems. This highlights the road map for capacitance enhancement beyond the geometrical value: Identify electronic systems that possess negative compressibility if they are considered as isolated and then incorporate them into the electrodes of a capacitor. Despite the instable electronic subsystems, the total combined system may possess a positive compressibility as (i) the electrostatic energy yields a positive contribution and (ii) the compressibility has to be evaluated according to Eq. (1.2). The latter yields, due to the total derivatives with respect to nt, a different result than the single partial derivatives with respect to ni, which occur in the negative compressibilities of the isolated subsystems. The relation between capacitance and compressibility is discussed in Section 2.3 in more detail.