Condensed matter physics I: 2015-2016 Introduction, Drude model R. Ganesh 1 Introduction Condensed matter physics (CMP) is the study of systems with many interact- ing particles. The adjective `condensed' should be understood as the opposite of `dilute' { the constituent particles are so close to each other that their interac- tions cannot be ignored. Some typical examples of `condensed' systems include electrons in a metal, spins in an insulating magnet, neutrons in a neutron star, Na atoms in an ultracold atomic gas, He atoms in liquid He, etc. Historically, the field grew out solid state physics. It used to be called 'solid state physics' until 1967 when Phil Anderson and Volker Heine coined the term `condensed matter physics'. CMP serves as a counterpoint to the `reductionism'. In the reductionist ap- proach, complex systems are studied by breaking them down into smaller, funda- mental units. This is exemplified by particle accelerator physics in which atoms are broken down into nucleons, which are further broken down into quarks, etc. In the condensed matter point of view, the whole is greater than the sum of its parts. For example, we may understand how a single He atom behaves according to the laws of quantum mechanics. However, that does not tell us how a beaker with liquid He will behave. Completely unforeseen behaviour (superfluidity in this case) can emerge from the interactions of many constituent particles. The new physics that emerges from collective behaviour at different length scales is the domain of condensed matter physics. There are two broad (not necessarily mutually exclusive) divisions within CMP. `Soft' CMP deals with classical systems { the constituent entities obey classical equations of motion, e.g., grains of sand which obey Newton's laws. In `hard' CMP, the quantum nature of constituents is important and can not be ignored, e.g., La2CuO4, a cuprate, is an antiferromagnet composed of spin-1=2 spins. Among contemporary research areas in physics, CMP stands out for its particularly rich interplay of theory and experiment. Theoretical ideas have suggested new experimental pathways and vice versa. By and large, new dis- coveries in this field have come from experiments. Some prominent examples include superfluidity (Kapitza and Allen, 1930's), high-temperature supercon- ductivity (Bednorz and M¨uller,1986), quasicrystals (Shechtman, 1982), etc. 1 Theoretical studies have also lead to new discoveries. A prominent topical ex- ample is topological insulators (Kane and Mele, 2005). CMP is a truly complex field. It encompasses a wide spectrum of phenomena that cannot meaningfully be reduced to a set of basic equations. This can be compared with: • Electromagnetism $ Maxwell's equations • Fluid mechanics $ Navier-Stokes equations • Classical mechanics $ Newton's equations. Unlike these areas, there is no underlying `basic' law of CMP. Of course, any many particle system is ultimately described by a many body Schr¨odingerequa- tion. However, in practice, it is impossible to deduce the system's macroscopic behaviour from this. In part due to this complex nature, CMP remains an active area of research despite one century of experimental and theoretical work. CMP is an excellent window into the scientific process. Models are con- structed for complex systems, based on the knowledge available at a given point in time. They are continually continually tested for theoretical consistency and experimental validity and updated accordingly. The classic example is the Drude model for metals that was proposed in 1900; it continues to be used, albeit with several modifications and caveats. In this course, we will focus on solid state systems to develop several basic ideas. Within solids, we will mostly concern ourselves with crystalline materials. The translational symmetry of crystals allows us to make great progress. 2 Drude model What makes metals metallic? Why does copper conduct electricity, while com- mon salt does not? The simplest model to explain conduction in metals was put forward by Paul Drude, a German, in 1900. Some of the crucial discoveries that paved the way for the Drude model include • Ohm's law, 1827 • Joule heating of metals, 1841 • Equipartition theorem, 1845 • Discovery of the electron, J. J. Thomson, 1897 J. J. Thomson showed that cathode rays emitted by different atoms all contain the same negatively charged particle. This subatomic particle was called this the electron. Drude identified these as the charge carriers responsible for con- duction in metals. His model pictures a metal as a gas of electrons. Positive charge centres, which are required for charge neutrality, are taken to be very heavy and immobile { thereby playing no role in conduction. This remarkably 2 simple-minded model continues to be invoked to this day, despite the presence of sophisticated numerical techniques and complex quantum analyses. This is an example of the tremendous utility of simple physically-motivated models in CMP. Drude's original model assumed that electrons wander around and scatter of the positive-charge centres. We now know this to be incorrect. We will dis- cuss some scattering mechanisms later on. In a modern re-interpretation of the Drude model, valence electrons in metals delocalise and scatter off defects. For future reference, we note that the typical length scale in metals is 1-3A˚. The typical density of electrons in (1-30) ×1016m−3 (or 1022cm−3). Assumptions of the Drude model: • Electrons move freely inside the metal, except for collisions with `ions'. Between collisions, electrons move in a straight line with fixed velocity { Independent electron approximation: no interaction between elec- trons { Free electron approximation: no interaction between electrons and ions • Collisions occur with some scattering centres. Collisions are instantaneous. • Probability of a given electron undergoing a collision in an infintesimal interval dt is given by dt/τ. This is independent of any details such as the electron's position or velocity. τ is a system parameter called `relaxation time' or `mean free time'. • Collisions immediately lead to thermal equilibrium. After a collision, the electron comes out with a randomly oriented velocity. Its speed is related to T , the `local' temperature (assuming that a temperature can be as- signed to a region within the metal { more on this later). This is major assumption { the outgoing velocity is taken to be independent of details such as velocity before the collision, presence of neighbouring ions, etc. Exercise 1: Show that for a given electron, the average until the next collision is given by τ. What is the average time between two collisions? Exercise 2: Show that the average time elapsed since the immediately pre- ceding collision is also given by τ. Note that τ encodes a lot of information, including number of defect struc- tures, temperature, etc. See the book for a table showing how τ decreases with increasing temperature { we will try and interpret this at a later point in this course. DC electrical conductivity: The Drude model immediately implies Ohm's law V = IR. Resistance, an extrinsic quantity, can be described using resistivity, an intrinsic quantity R = ρL/A. The conductivity σ = 1/ρ can be deduced from the Drude picture as follows. 3 Consider a metal slab with an imposed voltage V . An electron in the bulk of this slab travels freely and independently between collisions. In addition, it now experiences an electric field E due to the imposed voltage. The velocity of a given electron, at time t since the immediately preceding collsion, is now given by ve = vcollision − teE=m. vcollision is the velocity with which it emerged after the immediately preceding collision { a randomly oriented vector. Therefore, the average velocity of an electron is given by vavg = −teE=m. The mass and charge of an electron are denoted by m and −e respectively. If we now take an ensemble of electrons in one region of the metal (assuming that the local temperature is the same everywhere), the average velocity is given by vavg = −τeE=m. We have used hti = τ, i.e., the average time since the last collision for a given electron is τ. Apart from τ, we need one more piece of information about the material { the electron density n. Consider an imaginary square of cross-sectional area A within the metal, oriented normal to the electric field. How many electrons pass through this in a time interval dt? We can geometrically argue that all the electrons in a cuboid of cross sectional area A and length dt × vavg will pass through this square. Thus, the current through the square is given by the amount of charge passing 2 in time dt divided by dt: −neA × dt × vavg=dt = nAτe E=m. This is a vector parallel to the direction of electric field. The current density is this quantity divided by A. This can be rewritten as J = σ0E; (1) where ne2τ σ = : (2) 0 m One can easily show that this leads to Ohm's law with R = l/σ0A for a bulk conductor. Drude estimated the mean free time τ from the experimentally known con- ductivities of metals (∼ µ ohm-cm). He used equipartition theorem to obtain vavg, and estimated that the `mean free path' (vavgτ ) is of the same order as the lattice spacing. This is consistent with Drude's picture that electrons scat- ter of ions. However, with materials available today, we find mean free paths ranging from ∼ 1 − 108 times the lattice constant. Clearly, we need a more sophisticated understanding of the scattering mechanism. Frictional damping in the Drude model: Let us denote the average momentum of an electron at time t as p(t). How does this momentum change in an infinitesimal interval dt? During this interval, a fraction of the electrons (number of electrons ×dt/τ) will undergo a collision.