PHYSICS nd TERM Outline Notes (continued)

PHYSICS 2800 2 nd TERM Outline Notes (continued) Section 6. Optical Properties (see also textbook, chapter 15) This section will be concerned with how electromagnetic radiation (visible light, in particular) interacts with different solids (metals, semiconductors, insulators); absorption, reflection and transmission of light; applications like photoconductivity, lasers, optical fibres, photonic band-gap materials. 6.1 Basic properties of electromagnetic radiation (see also textbook, sections 15.2) An electromagnetic (em) wave describes the wave-like fluctuations of the electric field E and the magnetic field H in vacuum or in any medium. The E and H fields are related to one another through Maxwell s equations of electromagnetism: they travel at the same speed (the speed of light), and they are perpendicular to the direction of propagation (transverse waves) and also perpendicular to each other. E An electromagnetic wave showing electric field E and magnetic field H components, as well as wavelength λ. The spectrum of em radiation, including wavelength ranges for the various colours in the visible spectrum. The conversion between different units is also shown. 53

The visible part has wavelength λ from roughly 0.4 µm (or 400 nm) [violet/blue end] up to 0.7 µm (or 700 nm) [red end]. All em waves travel at the speed of light, so when in a vacuum this is 1 c = = 3 10 8 m/s approx ε µ 0 0 and c = νλ, where ν = frequency (in Hz) Recall that when em radiation is quantized, it comes in packets, or photons, of energy E = hν = hc/λ (Planck s hypothesis) When light passes from a vacuum region to a solid medium, some may be transmitted, some reflected back, and some absorbed in the medium. These fractions depend on the wavelength of the incident light and there may be some (usually small) conversion to different wavelengths. In terms of the corresponding intensities (defined by energy flow per unit time per unit area perpendicular to the direction of flow), we have Incident intensity I 0 = I T + I A + I R, by conservation of energy. Dividing by I 0, the above result becomes T + A + R = 1 where T = I T /I 0 = transmissivity, A = I A /I 0 = absorptivity, R = I R /I 0 = reflectivity. Some terminology applied to materials: transparent very little absorption (A 0) translucent light transmitted diffusively (objects cannot be seen clearly through the material) opaque very little transmission Example of light transmittance of 3 different Al 2 O 3 samples: a single-crystal sapphire (left) which is transparent; a dense (non-porous) polycrystalline sample (centre) which is translucent; a porous polycrystalline sample (right) which is opaque. These properties can be very dependent on the wavelength λ (and therefore on the corresponding energy scale). 54

In general, the process of absorption (or its absence in transparent materials) depends on the electronic transitions. For isolated atoms: Schematic illustration of photon absorption in an isolated atom by the excitation of an electron from one energy state to another. The photon energy must be exactly equal to the energy difference between the two states (E 4 E 2 in this case). The spacing between energy levels must be such that E = hν If this is satisfied, then there is absorption of a photon at that frequency all of the photon energy is absorbed the atom is then in an excited state (energy is above the ground state) the electron can drop to lower levels by emitting radiation (photon), allowing several possibilities, as discussed later. In solids, the atoms cannot usually be considered as isolated. They interact giving rise to bands and band gaps, so the situation is then different for metals, semiconductors and insulators. Optical properties of metals in metals there are energy bands without occurrence of any band gaps so all radiation can be absorbed (there are lots of energy levels close together, so there are no restriction on E or ν). Metals are opaque (unless they are extremely thin, e.g. 5 nm film of Fe). (a) Schematic representation of the photon absorption process in a metal, where an electron is excited into a higher-energy excited state. The change E in the electron energy is equal to the energy of the absorbed photon. (b) The subsequent reemission of a photon by direct transition of the electron from the The above is not applicable in the case of non-metals because there are band gaps for the electrons. There would be restrictions on E and hence on frequency ν for absorption to occur. To summarize, for a metal, only absorption and reflection (from a surface) are of interest, but for a non-metal, we are also interested in the transmission and refraction (as well as absorption and reflection). 55

6.2 Refraction and reflection (see also textbook section 15.3) Light within a solid travels at a lower velocity v than in vacuum c. We define refractive index n by n = c/v (so n > 1) gives bending of light (refraction) at any vacuum/solid interface also gives partial reflection at any vacuum/solid interface leads to well-known behaviour of lenses, prisms, etc. From the equations of electromagnetism, v = 1 εµ which is just the analog of the previous expression for c in vacuum. n c = = v εµ εµ 0 0 But if we denote µ = µ r µ 0 as before (for relative permeability µ r ) and ε = ε r ε 0 where ε r is the dielectric constant, then n = ε µ r r If the material is only weakly magnetic (i.e. diamagnetic or paramagnetic) then µ r 1 and so n = ε r Examples for some relatively transparent ceramics (including glasses) and polymers are: Note: average means an average taken over the wavelengths of visible light and (in the case of anisotropic materials) over all directions. In simple (isotropic) materials, n is the same for all directions of light propagation. For some anisotropic materials (including many polymers and magnetic materials) n may be slightly different in different directions. If there is a smooth planar boundary between two different materials, some of the light gets reflected back at the boundary even if both materials are completely transparent and some will be refracted (i.e., bent at the interface). 56

Refraction at a planar interface: When light passes obliquely from a less dense medium (e.g., air) to a denser transparent medium (e.g., glass), some of the light is transmitted and gets bent towards the normal direction in the denser material: n (low) n (high) θ This occurs according to Snell s law which states that n sin θ = n sin θ θ Hence n > n implies that sin θ < sin θ and so θ < θ. There is a reflected beam as well in the less dense medium (but it is not shown in the diagram). A more interesting effect occurs if the light is passing from a dense to less dense medium, because light will get bent away from the normal direction. This leads to the occurrence of a critical angle. n (low) n (high) θ C 1 2 Ray 1 corresponds to the critical angle θ C in the dense medium, which is obtained when θ in the less dense medium is 90 0, i.e., the beam grazes the interface. It is defined by n sin θ C = n sin 90 0 = n, so sin θ C = n /n < 1. For example, if n = 1.51 (for soda-lime silica glass) and n = 1 (air), then sin θ C = 0.66 implying θ C = 41 0. Whenever the incident angle in the dense medium is greater than θ C (as for ray 2 above), the only possibility is for the light to be reflected in the dense medium. This property is utilized in applications to optical fibres (see later). Reflection at a planar interface: Recall the earlier definition that R = I R /I 0 = reflectivity. 57

This ratio is calculated using em boundary conditions for the E and H fields in the light wave. The final result (in the case of normal incidence) can be written in terms of the refractive index for the materials: R n n 2 1 = n2 + n1 2 Here n 1 and n 2 are the refractive indices. In this case it makes no difference whether light goes from material 2 to 1, or vice-versa. For example, taking the same case as previously for an interface between soda-lime silica glass and air: 1.51 1 R = = 0.041. 1.51+ 1 Therefore only 4.1% of the light energy is reflected 2 6.3 Absorption and transmission (see also textbook section 15.4) Non-metals can be either opaque or partly transparent to visible light may have a characteristic colour. in the latter case they In semiconductors and insulators, there is a band gap E g and the occurrence of absorption depends on this gap. (a) Mechanism of photon absorption for nonmetallic materials in which an electron is excited from the top of the valence band (leaving behind a hole) into the conduction band by absorption of a photon with energy E greater than the band gap. (b) Emission of a photon by a direct electron transition back to the valence band. To summarize, absorption takes place only if hν > E g implying hc/λ > E g it occurs because of the valence band to conduction band electronic transitions. By substituting values of λ for visible light, we can work out the corresponding limitations on E g for absorption to occur: Taking λ = 0.4 µm (violet/blue) implies E g = 3.1 ev or less. Taking λ = 0.7 µm (red) implies E g = 1.8 ev or less. Alternatively, for a given E g, we can work out a condition for λ for absorption, e.g., in diamond, E g = 5.6 ev implying λ < 0.22 µm for absorption of radiation. This is outside the wavelength range for visible light, so no light is absorbed in a diamond crystal, i.e. it appears colourless. This would be true for any material with E g > 3.1 ev. By similar arguments, all materials with E g < 1.8 ev will absorb all visible light, so they appear opaque. 58

All materials with E g in the range from 1.8 to 3.1 ev will absorb visible light at some of the wavelengths, so they appear colored. Thus a material with (say) E g = 2.5 ev will absorb blue light and transmit red light, so it appears red-ish under transmission. When there are impurities present (e.g., as in doped semiconductors) there are additional absorption or emission processes that can also occur. Some possibilities are: For a material with an impurity level within the band gap: (a) Photon absorption via a valence band to conduction band transition. (b) Emission of two photons via an electron transition first into an impurity state and then a transition back to the ground state. (c) Another two-stage process via an impurity state involving emission of a phonon and then a photon. In an absorptive medium, the light intensity typically decays with penetration distance in a simple exponential fashion, i.e., like e β x where x measures distance for the radiation into the material and β is the absorption coefficient. A large value of β implies strong absorption. The absorption affects the overall transmission properties, e.g., consider a film of material of thickness l. The transmitted intensity I T will depend on l and β, and also on the reflections at the front and back faces: The approximate result is (full calculation is complicated because of summing over all possible ways that repeated internal reflections may occur): 2 I = I (1 ) 0 R e β l T 59

Colour perception this depends on the transmission I T (specifically how much intensity there is and at what wavelengths). The examples below show results for green glass, sapphire and ruby at different wavelengths including those of visible light: The variation with wavelength of the fractions of incident light transmitted, absorbed and reflected through a green glass. Transmission of light as a function of wavelength for sapphire (single crystal Al 2 O 3 ) and ruby (Al 2 O 3 containing some chromium oxide). The sapphire appears colourless, while the ruby has a red tint due to selective absorption over specific wavelength ranges. 6.4 Applications (see also textbook sections 21.11 21.13) In this part we will cover mainly the topics of luminescence and photoconductivity and lasers and optical fibres and photonic band-gap materials. A) Luminescence This is the property of some materials (typically semiconductors) of absorbing incident energy and then re-emitting it as visible light. 60

From previous discussion this requires for the emitted light 1.8 ev < hν < 3.1 ev The absorbed incident energy might be some other form of em radiation of similar or higher energy (e.g. uv radiation) or high-energy electrons or heat, etc. The emission process depends on the band-gap energy E g, as discussed, either through a direct or indirect process. Some more specific terms are:- Fluorescence: this is when the re-emission of energy (as visible light) occurs with a time delay but still takes place on a short time scale (<< 1 sec). Phosphorescence: this is when the time scale is of order 1 sec or more. Fluorescence and phosphorescence occur in only a few materials (e.g., some sulphides, oxides, phosphors, etc) typically they are compounds and the controlled addition of impurities is important. Applications are to various kinds of optical coating (e.g. cathode-ray TV screens). Electroluminescence: this is when an electrical process (such as charge flow in a p-n semiconductor diode) can be used to generate visible light. Under conditions of forward bias in a p-n junction, the electrons and holes move towards each other within the recombination region. They annihilate to release energy: electron + hole energy ( E) where E is of order of the band gap energy E g. Schematic diagram of a forwardbiased semiconductor p-n junction diode showing (a) the injection of an electron from the n-side into the p-side, and (b) the emission of a photon of light as this electron recombines with a hole. Therefore if E g is in the required range (1.8 to 3.1 ev), then visible light will be emitted. Such diodes that luminesce are the basis of the light-emitting diodes (LEDs) used in digital displays. Note that the band gaps are too small in the elemental semiconductors Si (E g = 1.1 ev) and Ge (E g = 0.7 ev), but many compound semiconductors are suitable, such as GaAs and InP and their alloys. For example, the colours red, orange and yellow are possible with LEDs based on the GaAs-InP system (depending on the mix), while blue and green LEDs have been developed using (Ga,In)N semiconducting alloys. In addition, several polymers have been developed that behave as large-e g semiconductors capable of doping as n-type and p-type, and these have been developed as LEDs. For some applications they have distinct advantages over the conventional semiconductor LEDs, e.g., they 61

can emit a mixture of colours rather than just a narrow range, and they can be more easily produced as thin films (larger area and bendable to different shapes). There are two main types: organic LEDs (or OLEDs), which use a low molecular weight organic polymer, and polymer LEDs (or PLEDs), which use a high molecular weight polymer. Schematic diagram showing the components and configuration of an organic light-emitting diode (OLED). B) Photoconductivity The electrical conductivity of a doped semiconductor material depends on the number of charge carriers (n for electrons in the conduction band and p for holes in the valence band) and the corresponding carrier mobilities: σ = n e µ e + p e µ h If incident light is absorbed by the semiconductor, then this energy can be used to excite electrons from the valence band to the conduction band (so increasing both n and p). This additional contribution to σ is known as photoconductivity. One application is to light meters, where CdS (E g = 2.4 ev) is often used. Another application is for solar cells as arrays of semiconductor p-n junction diodes basically operating as the reverse process in LEDs (the photoexcited electrons and holes are drawn away from the junction and become part of a current flow in an external circuit). C) Lasers All the radiative processes (i.e., those in which an electron drops from a higher energy state to a lower energy state) can be described as spontaneous emission in the sense that there is no specific external cause. The transitions to lower energy occur randomly and independently of one another, and so they produce emission of incoherent radiation (the different lights waves are randomly of different phases). By contrast, light from a laser is derived from stimulated emission (meaning that it is initiated by an external stimulus) and it is coherent. Hence the acronym: LASER = Light Amplification by Stimulated Emission of Radiation. Two types of lasers will be discussed here as examples: the ruby laser and the GaAs semiconductor laser. The ruby laser Ruby is Al 2 O 3 in single crystal form (i.e., sapphire) to which a small concentration (typically 0.05%) of Cr 3+ ions has been added. As explained earlier, the Cr impurities give ruby its 62

characteristic red colour (by allowing selective absorption), but in a ruby laser they provide electron states that are essential for the operation of the laser. Basically, the device consists of a rod of ruby surrounded by a conventional light source (a lamp). The ends of the ruby rod are flat and parallel and highly polished. Both ends are silvered as reflectors, so that one end is essentially totally reflecting and the other end allows some transmission (for the coherent laser light to get out). Schematic diagram of the ruby laser and xenon flash lamp. Initially all the Cr 3+ ions will be in their ground state with the electrons occupying the lowest available states. Next the external power source is switched on to illuminate the xenon flash lamp and then the incident light (photons with wavelength about 0.56 µm) excite the electrons to a higher level. Subsequently, some of these electrons will just drop back directly to the ground state, but the operation of the laser depends on some others making a transition to a lower excited state of Cr 3+ that is metastable. Schematic energy level diagram for the Cr 3+ ions in a ruby laser, showing electron excitation and decay paths. The transition from the original excited state (labelled E) to the metastable state M occurs just by normal spontaneous decay. The average lifetime of the electron in the metastable state M is about 3 ms, which is very long (by several orders of magnitude) compared with the typical lifetime for an electron in an excited state. Eventually each electron decays from M back to the ground state G by emission of a photon. It is this MG transition that eventually produces the laser light, but at start-up there will just be decay by spontaneous emission. The function of the laser (with its ruby 63

rod) is to build up an avalanche type of process with the excited electrons traveling back and forth along the rod so that stimulated emission component to the MG decay is obtained. Schematic representation of the process of stimulated emission and light amplification for a ruby laser. (a) The chromium ions before excitation. (b) The electrons in some chromium ions are excited into higher-energy states by the xenon light flash. (c) The emission from metastable electron states is initiated or stimulated by photons that are spontaneously emitted. (d) On reflection from the silvered ends of the ruby rod, the photons continue to stimulate emissions as they traverse the length of the rod. (e) The coherent and intense beam is finally emitted through one end (the partially silvered end). The initial spontaneous photon emission by a few electrons gives the stimulus to trigger an avalanche of emissions from the remaining electrons that are in the metastable state. Of the photons moving along the long axis of the rod, some are transmitted through the partially silvered end, while others incident on the totally silvered end are reflected back. (Photons moving in other directions are lost). The light travels back and forth repeatedly along the length, and during this time its intensity builds up as more and more emissions are stimulated. Eventually a highintensity coherent and collimated laser beam is transmitted in bursts of energy out at one end of the laser. The monochromatic light has a wavelength of 0.6943 µm (in the red) corresponding to the energy separation between M and G. Lots of other materials are capable of behaving as a laser, and they all operate according to the same basic principles, i.e., there needs to be one (or more) metastable level(s) into which electrons can be pumped and then through an avalanche process the stimulated emissions can occur as the electrons move back to the ground state. 64

Some common types of lasers, along with their characteristics and applications are shown in the following table:- The GaAs semiconductor laser Various semiconductors, including GaAs as a convenient example, may be used as lasers, especially in applications where low power and compactness are important. The ground state corresponds to an electron being in the valence band and the metastable excited state is the electron excited to the conduction band (leaving behind a hole in the valence band). The main challenges in designing a semiconductor laser consist of a) confining the excited electrons to the semiconductor during the stimulation process, and b) overcoming heating problems within the semiconductor. A typical arrangement is shown below. Schematic diagram of a GaAs semiconductor laser. The holes, excited electrons, and laser beam (photons) are all confined to the GaAs layer by the surrounding n- and p-type GaAlAs layers. 65

From previous discussion of semiconductors, it follows that the wavelength and the band gap are related by hc/λ = E g. Hence if λ is to lie in the visible range (between about 0.4 and 0.7 µm), E g must lie between about 1.8 and 3.1 ev. The laser process is shown schematically in the following set of diagrams. Schematic representation of the stimulated recombination of excited electrons in the conduction band with the holes in the valence band: (a) One electron recombines spontaneously with a hole and a photon is emitted. (b) This emitted photon stimulates the recombination of another excited electron and hole, so another photon is emitted. (c) The two photons, which have the same wavelength and are in phase, are reflected back at one end of the laser device. At the same time, new excited electrons and holes are generated by a current that passes through the semiconductor. (d) - (e) More and more stimulated excited electron-hole recombinations occur. (f) Some fraction of the laser beam is transmitted out through the partially reflecting mirror at on e end of the device. 66