Lecture 8: Optical Spectroscopy Tuesday, October 3, 2017 Introduction: The verification of quantum mechanical properties can be elegantly explored by exploiting the light-matter interaction. Interaction of matter with light can reveal details about electronic transitions described by quantum mechanics. One need is to probe ever-smaller volumes of materials, e.g. nano-materials. A typical experimental apparatus to probe the optically-induced transitions in a semiconductor nanostructure is shown below, a micro-spectroscopy method is employed to look at sub micron features at varying temperatures. Figure 1: A micro-spectroscopy set-up capable of examining sub-micron regions of materials at temperatures down to 4K. The solid immersion lens sits on the sample surface, and enhances the resolution of the far-field objective by increasing the refractive index. Shown are a micro-cryostat, microscope objective, and piezo-controlled focusing mechanism (pico-motor) and two solid immersion lenses. The optical signals are relayed to the spectrometer shown in the background, for spectral analysis. Light Scattering: Due to electron-photon coupling, energy and momentum can be exchanged with matter via interaction with electromagnetic radiation. Such processes are generally termed light scattering. These processes can be classified into two categories: Elastic (no energy exchanged) and Inelastic (energy lost or gained by electronic system). Examples of inelastic processes would be fluorescence/luminescence and Raman scattering; elastic processes might be reflection or Raleigh scattering. An important process to consider is the absorption of a photon by an electron in an atom or solid. In figure 2, an electron in a state described by the wave vector k and energy E interacts with a photo of energy hν, and momentum ~kp. Energy and momentum conservation require: E2 − E1 = hν −! Energy conserved ~k2 − ~k1 = ~kp −! Momentum conserved In crystals, typically electron momentum ~k ~kp and consequently the photon momentum is considered zero, yielding the additional condition: k2 − k1 ≈ 0 −! ∆k = 0 which is considered a selection rule for optically-induced transitions in solids. 1 Figure 2: A Feyman diagram describing the evolution of an optical transition wherein an electron of energy E1 and momentum k1 absorbs a photon of energy hν and momentum ~kp. Time axis is by convention implied from left to right, but shown here explicitly. Figure 3: A Feyman diagram describing the evolution of an optical transition wherein an electron of energy E1 and momentum k1 emits a photon of energy hν and momentum ~kp. Figure 4: A Feyman diagram describing the evolution of an electron transition wherein an electron of energy E1 and momentum k1 emits a phonon of energy ~Ω and momentum ~kph. Diagrams for other important processes are shown in figures 3-4, which describe the processes of emission of a photon (figure 3) and scattering of an electron off an ion in a lattice and creation of a phonon, a quantum of vibrational energy in a crystal (figure 4). Combinations of these diagrams would describe higher-order processes, for instance multi- photon absorption, Raman scattering or stimulated emission. We will now discuss in more detail these processes, namely, the processes of absorption of light, emission of light, and internal energy relaxation through non-radiative processes such as phonon scattering (figure 4). In particular, we will discuss experimental methods to probe the internal quantum states of matter by exploiting these interactions in optical spectroscopy measurements. 2 Selection Rules: Having computed the wave function, we wish to describe general properties of optically-induced transitions. We will assume the wave function has the general form: j i = j n; l; ml ; s i | {z } |{z} spatial spin where we have factored the spatial and spin portions of the wave function (spin-orbit interaction is usually treated as a perturbation), and j i satisfies the time independent Schr¨odinger equation: Hb j i = Ej i In general, a transition will involve a change of quantum state, for instance between two Eigenstates here represented as atomic levels: Figure 5: Energy level diagram showing an absorption of a photon of energy hν, resulting in an atomic transition between to quantum states. and the probability for the transition is given by Fermi's golden rule: 2π 2 Wfi = jhn; l; ml; sjHbint(0)jno; lo; mlo ; soij δ(∆Enno − ~!) ~ as we saw in our previous discussion of quantum mechanics and transitions between quantum states. Evaluating the expression for the matrix element, we can quote some selection rules: h f jHbintj ii = hsf jsiihn; l; mljHbint(0)jno; lo; mlo i hsf jsii = δif spin allowed for i = f hn; l; mljHbint(0)jno; lo; mlo i 6= 0 dipole allowed Where the spin dependent part of the wave function is independent of the spatial coordinates and we can therefore factor the expression out. We say the transition is spin allowed if the spin portion of the matrix element is non-zero, and, since we used Hbint = −p·E to calculate the matrix element (see discussion of Fermi's golden rule), if the spatial portion of the matrix element is non-zero we say the transition is dipole allowed. Energy Relaxation in Semiconductors and Molecules: The general notions discussed above can be equally applied to transitions between Bloch states in a semiconductor or a the localized states of a molecule. Here we discuss some general properties of semiconductors and then examine the energy relaxation processes which occur between the absorption and emission of a photon in both semiconductors and molecules. Band filling in semiconductors: Figure 6 shows the folded- or reduced-zone scheme for the energy dispersion relation of an electron in a periodic structure. Each band will have 2Nk states, associated with the Nk possible wave vectors in the first Brillouin zone and two possible spin states. Depending on the valence of each ion in the crystal, these states will be filled or empty. The highest energy k-state defines the Fermi energy. In figure 6, the band labelled !(k+G) is completely filled, and the next available state is separated by the band gap, if sufficiently small in energy, we would consider this material a semiconductor, as it would take on order of 1eV or more of energy to make a transition to the next highest band. In figure 7 we concentrate only on the region of k-space near k = 0 in the vicinity of the energy gap, which we call the band gap, Eg. The figure examines what happens upon optical injection, or absorption of a photon: when a photon 3 is absorbed, an electron must make what is called a vertical transition in k-space, in order to satisfy the selection rule ∆k ≈ 0, and the energy of the subsequent electron (hole) created must be equal to Ec(k) − Ev(k) ≥ Eg. Figure 6: Energy dispersion relation in the extended zone scheme. Note the filling of the highest state in each band, signified by up / down arrows. Figure 7: Scattering diagram showing the processes which lead to light emission from a semiconductor: light absorbed at E = hν ≥ Eg. Excess energy is lost by rapid phonon emission and emission at energy E = hν ≈ Eg. As seen in figure 7, the electron is promoted to the energy state Ec(k), but quickly loses excess energy through non-radiative processes, which can have various origins, but are usually dominated by phonon emission processes such as discussed in connection to figure 4, and represented as the curved dotted lines in figure 7. The "jumps" described are uniform in energy because typically in solids optical phonon emission will dominate on short timescales, and the optical phonon is nearly dispersionless near k = 0, with a finite energy ~!LO (longitudinal optical phonon). Similar statements apply for the holes, shown in the valence band. These non-radiative processes are referred to as \carrier cooling", as they relate to the excess energy above the minimum which is required to initiate the transition, that is Eg. This excess energy takes the form of kinetic energy and translates to carrier velocity. For instance, 1 ∗ 2 ∗ 5 assuming KE = 2 m v , where m is the “effective" mass, velocity of the electron can be on the order of 10 m=s for typical excess energies of say, 0.1-0.3 eV. Another measure of kinetic energy is to equate this excess energy to 3 thermal energy, equating the kinetic energy to the Boltzmann equal partition of energy: 2 kBT , a carrier temperature can be associated with the excess kinetic energy, which again, for 0.1 eV excess energy, equates to ∼ 1400◦K. Such carriers are thus called "hot-carriers". The rate of LO phonon emission can be extremely fast, and thus the carriers 4 lose this energy on a 100's of femtoseconds time scale. This process is termed \carrier cooling", as the characterstic temperature of the carriers will in a short time thermalize with the lattice temperature. Figure 8: An energy level diagram describing the absorption and emission of a photon in a molecule. Note the energy of the emitted photon is less than the absorbed photon due to internal energy relaxation. A similar mechanism in molecules is depicted in figure 8, the conduction and valence bands are now replaced with their molecular counterparts, the so-called LUMO (lowest unoccupied molecular orbital) and HOMO (highest occupied molecular orbital) energy levels. The molecular counterpart to the phonons in crystals are the normal modes of vibration of the molecule, with discrete energies. Since the molecule does not have translational symmetry, the energy levels are discrete and we say that k is not a \good quantum number". The wave functions of molecules are built up from atomic orbitals, a formalism which we will examine in more detail later.