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:

|ψi = | 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 |ψi satisﬁes the time independent Schr¨odinger equation:

Hb |ψi = E|ψi 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 = |hn, l, ml, s|Hbint(0)|no, lo, mlo , soi| δ(∆Enno − ~ω) ~ 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:

hsf |sii = δif spin allowed for i = f

hn, l, ml|Hbint(0)|no, 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 ω(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 ~ωLO (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. However, statements about excess energy lost to heat during carrier cooling would also apply to the molecular case. There has been, in the past and more recently, much interest in schemes to re-capture this lost energy to do useful work and thereby raise the efficiency of electro-optic devices. For instance, the Carnot efficiency of a solar cell operating on earth at T=300◦K is almost 85%, however the so-called Shockley-Quessier limit for an ideal solar cell is only around 30%. The difference is entirely due the carrier cooling processes and our inability to control them. Manipulation of matter at the nano-scale is one strategy to achieve such control over materials and device properties. Optical Spectroscopy: We will now discuss in more detail specific measurements and measurement techniques to probe the electronic transitions which ultimately dictate material properties and device performance. A canonical example would be a fluorescence or photoluminescence experiment. Figure 9 shows a typical apparatus for measuring fluorescence or luminescence, e.g. what might exist in a fluorescence microscope. An example luminescence spectra is also shown, the details of the various signatures will be discussed in what follows.

Figure 9: (left): Experimental apparatus for measuring luminescence or fluorescence. An excitation source is reflected off a dichroic mirror, and focused onto the sample by an achromatic lens. As a result of absorption of light energy and internal relaxation, emission at a longer wavelength is emitted from the sample and transmited by the dichroic mirror to be analyzed by spectroscopic instruments. (right): Example µ-photoluminescence spectra from GaxIn1−xP.

5 Absorption and Emission: Absorption measurements are the most direct measurement of the transitions which take place in atoms and solids. However, direct absorption may not be easy to measure, so we often measure secondary events which are proportional to the absorption, in lieu of actually trying to measure the absorption directly. For instance, absorption measurements which rely on emission as a secondary event will be discussed here. In ﬁgure 10 the parabolic band approximation for a type-I semiconductor (e.g. GaAs) is shown. Plotted alongside the dispersion relation is the density of states (DOS). Since the curvature of the bands may be diﬀerent in the conduction and valence bands, we calculate a separate density of states in each band, and treat the electron/hole in each band as separate particles according to convention. The DOS, or ρc(v)(E), is dependent on the volume of a shell in k-space, and will be zero at the gaps, thus the shape of ρc(v)(E) should be similar to what is sketched. It shows that the number of states approaches zero at the gap, and should dictate the number of transitions which can be excited at a given energy – a quantity directly proportional to the absorption.

Figure 10: The density of states (DOS), or ρc/v(E), is shown alongside the energy dispersion relation in the parabolic band approximation for a semiconductor. We are interested in the probability an electron can make the transition from the valence band to the conduction band, described by the joint density of states (JDOS), which takes into account the properties of the valence and conduction bands. Based on the parabolic band approximation, a hypothetical JDOS, ρ(E), is plotted in the leftmost portion of ﬁgure 11:

Figure 11: (Left): JDOS ρ(E) in parabolic approximation, (Right): Absorption spectra in semiconductor, including excitonic absorption resonance at energy Eg − Eb, where Eb is the exciton binding energy.

Figure 11 shows on the left the density of states in the parabolic band approximation. Ideally, the absorption spectra would be directly proportional to this. The absorption as a function of energy can be measured in a variety of ways, as will be discussed in what follows, but often other considerations beyond the shape of ρ(E) factor in. For example, the excitonic resonance shown in the rightmost portion of the ﬁgure, which causes the absorption spectra to deviate

6 from the shape of ρ(E). We will discuss excitons and other excitations in semiconductors in following paragraphs, but now will show an experimental method which is used to directly measure absorption in semiconductors.

Photoluminescence Excitation: The method commonly used in semiconductor spectroscopy to measure the absorption spectra is known as photoluminescence excitation spectroscopy, or PLE, described in figure 12. The principle is based on the energy relaxation which takes place between absorption and emission, which results in emission at a lower energy than absorption, as discussed in connection with figures 7 and 8. In the left part of figure 12 an emission spectra is shown, centered around an energy Eg − Eb, if a spectrometer can be set to an energy below this, a laser excitation source (whose energy range is represented by the green arrow in the figure) can be scanned continuously towards higher energy. This can be achieved using a tunable dye laser or a solid-state laser such as Ti:Sapphire based lasers. By monitoring the luminescence intensity at the PLE detection energy, all carriers injected at higher energy will eventually emit near the lowest energy state, which in the figure is ∼ Eg − Eb. Plotting this resulting intensity as a function of the excitation energy produces a graph similar to the rightmost part of figure 12. We see that this curve faithfully reproduces ρ(E), the JDOS, with the exception of the added excitonic resonance (to be discussed in following text). In fact, the added resonance adds states so this is not an artifact, it is the theory that needs to be corrected here.

Figure 12: (Left): A photoluminescence (PL) spectra resulting from excitonic recombination, a narrow energy range near the low energy tail of the spectra is selected as the PLE detection energy. The green arrow shows the range of excitation energies which will be used to obtain the PLE spectra. (Right): Photoluminescence intensity at the PLE detection energy plotted as a function of excitation energy, which faithfully reproduces the exciton absorption resonance and the density of states ρ(E).

TRPL: In the above measurements, we considered the time-integrated intensity of photons emitted from a semiconductor or molecule. It is often of interest to know the time evolution of this emission, which is related to the emission rate calculated using Fermi’s golden rule, but can have other important factors not related to the quantum transition rate, but certainly important in terms of material properties and device performance. To measure this evolution, we must time-resolve the emission. The most basic time-dependent measurement is time-resolved PL or TRPL, a proto-typical example is shown in figure 13. The measurement can be made using a number of methods, including time-correlated single photon counting (TCSPC) with a time-resolution on the order of 500ps, or using a streak-camera, with a resolution in the sub-picosecond range. Without going into detail about how the measurements are made, a few general comments about the information which is obtainable are discussed now. The measured fluorescence or luminescence decay time can often be characterized by a single exponential decay, e.g. −t/τ I(t) = Ioe , but the decay rate τ may have other contributions beyond the radiative decay rate based on the quantum transition rate, which are usually sub-divided into radiative and non-radiative components, related as so: 1 1 1 = + τ τR τNR where the possibility that a transition which does not involve emitting a photon may contribute to the observed decay τ. For instance, phonon emission where the optical energy of the photon is converted to heat. For semiconductors, and even molecular crystals, where carriers may become mobile, we must consider effects related to transport. For

7 Figure 13: (Left): Photo-luminescence, or PL, is initiated by an excitation pulse (blue), and time-resolved to show the recombination as function of time (red). (Right): A typical decay trace will be dominated by the recombination lifetime, but in principle is influenced by carrier thermalization, which takes place on a timescale typically very short compared to recombination. instance, as seen in figure 14, carriers (electrons and holes) can freely diffuse, and therefore the recombination rate will depend on the probability that two carriers of opposite type (electron/hole) will occupy the same site in the crystal. This is expressed in the formulas shown in the figure, namely:

nenh Rc = 2 R ni where ne(h) is the free electron(hole) density, which comes about by doping, a process of adding additional carriers above the ‘intrinsic’ carrier density ni, which is related to the valence of the ions which make up the crystal. The formula takes into account any imbalance in carrier concentrations which would affect the recombination rate. The measured decay times in this case are sensitive to details of the sample, such as concentration of impurities and the presence of surfaces and interfaces, which act as recombination centers. In well-understood geometries, these affects can be modeled using diffusion theory, and material and device parameters, such as carrier mobility can be extracted from the time-resolved data.

Figure 14: In a system where the excited state is mobile, such as a semiconductor or molecular crystal, transport effects influence the measured recombination rate or fluorescence / luminescence decay.

Nonlinear time-resolved methods: Using non-linear gating techniques, such as sum-frequency generation, it is possible to resolve light emission on a femtosecond time scale. For optically thin samples, differential transmission, also known as transient absorption, is a well-known method which measures directly the changes in absorption on a femtosecond time scale. As shown in figure 15, an intense “pump” pulse fills some fraction of the available states, resulting in increased transmission (reduced absorption) which will be dependent on the choice of time delay τ between the “pump” and “probe” pulse. As seen in the figure, the ultrafast relaxation can be measured with a precision which

8 in principle can exceed the width of the pulses themselves, which can be 10’s of fs (10−15s). Interpretation of the data obtained is not straight-forward however, and multiple experiments at select energies, intensities and perhaps temperature are often needed to clearly assign the relaxation mechanisms, which are typically dominated by optical phonon emission.

Figure 15: (Left): Differential transmission utilizes an intense pump pulse and typically a less intense probe pulse, delayed by a time τ. Transmission is then a function of time delay τ due to state-filling effects. (Right): Typically the differential transmission, ∆T/T (τ) is plotted vs. delay τ. Illustrating the ultrafast hot carrier cooling and slower recombination processes for τ τc, the carrier cooling rate.

Excitons: As mentioned in the previous discussions relating to absorption is semiconductors, additional absorption resonances occur beyond the Bloch states which are solutions to the Schr¨odingerequation in a solid. One of the most prominent is the formation of electron-hole pairs, bound by coulomb attraction. An electron in a conducting state may be bound to the positively charged hole, and the centro-symmetric system forms a so-called “quasi-particle” known as an exciton. We consider the pair a system, with a center of mass which can diﬀuse and even interact with other excitons. The coulomb attraction between the electron and hole results in a binding energy:

m∗q4 1 13.6eV m∗ 1 Eb = 2 2 2 = 2 2 2h c n (/o) mo n with a quantum number n, which speciﬁes the s-wave character exciton states. In other words, the energies of the exciton are quantized in Bohr orbitals, similar to the hydrogen atom. The ground state of the exciton for GaAs corresponds to ∼ 10meV, which is less than kBT at room temperature (23meV), so excitons in GaAs are not observable at room temperature (they are ionized). It’s interesting to note that the exciton has an integral spin, due to the combination of electron and hole spin, which results in the exciton behaving as a Boson and following Bose statistics, very diﬀerent than the Fermi statistics of the electrons which make it up. It has been a long-held goal of condensed matter physics to observe Bose-Einstien condensation in a collection of excitons.

Figure 16: Schematic of quasi-particle referred to as an Exciton – an electron-hole pair bound together in an orbital system by coulomb attraction. The two particles would revolve around the center of mass, which should be very close to the hole, with a radius abx, which might be 10nm in GaAs, for instance.

The above mentioned exciton is termed a “Wannier” exciton, and is seen in semiconductors. The binding energy will in turn depend on the dielectric function and eﬀective mass, and is typically larger in wide-gap semiconductors such

9 as GaN. In molecular crystals the binding energy is much larger, typically 400meV, and there the correlation between the electron and the ion it is associated with is never lost, thus the exciton is more properly considered an excited state of an individual constituent molecule in the crystal. Such excitons are usually termed “Frenkel” excitons, which have different properties than their solid-state counterparts. However, the exciton can still move by “hopping” from one molecule to another, the mechanism is either through a dipole-dipole energy transfer (Förstermechanism), or by the tunneling of an electron to a neighboring molecule (Dexter mechanism). Transport is still important in this case, as the exciton can still meet an impurity or interface, or another exciton (so-called exciton-exciton annihilation), all these processes will affect the observed emission and it’s time dependence, and are influenced by the geometry of the sample studied. Impurities in semiconductors: The inclusion of an impurity in a perfect crystal breaks the translational symmetry and creates a localized defect state. These defects can give rise to a variety of interesting effects, the most important in semiconductors is probably the effect of doping, where the impurity can give up a charge which is transferred into one of the Bloch states of the crystal. Depending on energy of the impurity, it may lie above or within the band gap. If the defect state lies in the gap, but within a few kBT of either band edge, a charge associated with the defect may move into a Bloch state. This would be called a “donor” and is sketched in figure 17. Similarly, if the defect state is close to the valence band, an electron may be promoted to the defect level and as a consequence leave behind a hole in the valence band, this is called an “acceptor”. The carrier concentration due to the presence of these impurities can be calculated from statistical physics, for instance:

∼ 1/2 −Ed/2kB T 2 3/2 n = (noNd) e where no = 2(mekB T/2π~ ) is the carrier concentration due to the density Nd of donors. These levels can also trap excitons, and as a result, below-gap emission can occur, as seen in ﬁgure 17. In fact, if many donors and acceptors are present in the crystal, the probability that they will be located on adjacent lattice sites can become appreciable. In that case carriers can be bound to levels associated with both the donor and the acceptor, and recombination can occur from these levels, deemed donor-acceptor pair emission (DAP).

Figure 17: Energy level diagram showing donor and acceptor levels and recombination for donor acceptor pairs.

Figure 18: Donor-acceptor emission is modulated by discrete ﬂuctuations in the nearest neighbor distance, which modulate the coulomb interaction energy.

The energy of this emission is observed in discrete lines, corresponding to integral numbers of lattice spacings, for

10 which the coulomb attraction varies discretely according to:

q2 ∆E = E − E − E − g d a εr thus a series of lines corresponding to varying donor-acceptor separation are observed, as sketched in ﬁgure 18.

Effective mass in semiconductors: Due to the anisotropy in E(k) with direction and the varying character of the atomic orbitals which build up the Bloch states, there is a degeneracy in the valence band that results in multiple degenerate states at k = 0. The degeneracy is lifted away from k = 0, and due to spin-orbit interaction, resulting in a multiplet of energy bands such as shown in figure 19. One notes the varying curvature of the bands in the figure, and this is described in terms of the “effective mass”, derived from the curvature of the energy dispersion relation:

2k2 1 1 d2E E = ~ −→ = 2m m∗ ~2 dk2 where we see that the band with higher curvature will have a smaller effective mass. For this reason, the two degenerate bands at the valence band maxima are labelled “heavy” and “light” hole bands (designated hh and lh in the figure). The slit off band is due to a perturbation on the Bloch states due to the spin-orbit interaction. Signatures of these bands are seen in temperature dependent photoluminescence experiments, for example, when the thermal occupation of the light hole band becomes appreciable a second peak in the luminescence can appear.

Figure 19: Parabolic band approximation for typical direct gap semiconductor, showing degeneracy in valence band at k = 0, so-called heavy and light hole bands and split-oﬀ band.

REFERENCES

• Jaques Pankove, “Optical Processes in Semiconductors,” Dover, NY, 1975. • Maneul Cardona, Peter Yu, “Fundamentals of Semiconductors,” Springer Berlin New York, 2001. • Simon Sze, Ng Kwok, “Phyiscs of Semiconductor Devices,” Wiley, NY, 2006. • Gabriel Bastard, “Wave Mechanics Applied to Semiconductor Heterostructures,” Wiley, NY, 1988.

• Jasprit Singh, “Physics of Semiconductors and their Heterostructures,” McGraw Hill, 1993. • Svˆatoslav Anatol’evi˘cMoskalenko, D. W. Snoke, “Bose Einstien Condensation of Excitons and Biexcitons: and Coherent Nonlinear Optics with Excitons,” Cambridge University Press, Feb 28, 2000.