Chapter 5 Photons in Semiconductors

Chapter 5 Photons in Semiconductors
Fundamentals of Photonics 2017/4/16

Fundamentals of Photonics
Semiconductors A semiconductor is a solid material that has electrical conductivity in between a conductor and an insulator. Semiconductors can be used as optical detectors, sources (light-emitting diodes and lasers), amplifiers, waveguides, modulators, sensors, and nonlinear optical elements. Fundamentals of Photonics 2017/4/16

A. Energy bands and charge carriers

Energy bands in semiconductors
The solution of the Schrödinger equation for the electron energy in the periodic potential created by the atoms in a crystal lattice, results in a splitting of the atomic energy levels and the formation of energy bands. ATOM SOLID E2’ E1’ Conduction band Valence band E21 Fundamentals of Photonics 2017/4/16

Each band contains a large number of finely separated discrete energy levels that can be approximated as a continuum. The valence and conduction bands are separated by a “forbidden” energy gap of width Eg bandgap energy Conduction band Valence band Bandgap E Unfilled bands Filled bands Fundamentals of Photonics 2017/4/16

Materials with a large energy gap (>3eV) are insulators, those for which the gap is small or nonexistent are conductors, semiconductors have gaps roughly in the range 0.1 to 3 eV Fundamentals of Photonics 2017/4/16

ev ev Si GaAs Conduction band Conduction band 5 5 Eg Eg 1.1ev 1.42ev Energy Energy Valence band Valence band -5 -5 -10 -10 -15 -15 (a) (b) Figure Energy bands: (a) in Si, and (b) in GaAs Fundamentals of Photonics 2017/4/16

Electrons and holes In the absence of thermal excitations, the valence band is completely filled and the conduction band is completely empty. Thus, the material cannot conduct electricity. As the temperature increases, some electrons will be thermally exited into the empty conduction band, result in the creation of a free electron in the conduction band and a free hole in the valence band. Fundamentals of Photonics 2017/4/16

Electrons and holes Conduction band Electron energy E Electron Hole Bandgap energy Eg Valence band Figure Electrons in the conduction band and holes in the valence band at T>0. Fundamentals of Photonics 2017/4/16

Energy-momentum relations
p is the magnitude of the momentum k is the magnitude of the wavevector associated with the electron’s wavefunction m0 is the electron mass Fundamentals of Photonics 2017/4/16

K [100] [111] E Si GaAs Figure Cross section of the E-K function for Si and GaAs along the crystal directions [111] and [100]. Eg=1.1ev Eg=1.42ev The energy of an electron in the conduction band depends not only on the magnitude of its momentum, but also on the direction in which it is traveling in the crystal Fundamentals of Photonics 2017/4/16

Effective mass Near the bottom of the conduction band, the E-k relation may be approximated by the parabola Ec,Ev: the energy at the bottom of the conduction band and at the top of the valence band mc,mv: effective mass of the electron in the conduction band and the hole in the valence band The effective mass depends on the crystal orientation and the particular band under consideration Fundamentals of Photonics 2017/4/16

Effective mass Typical ratios of the average effective masses to the mass of the free electron mass mc/m0 mv/m0 Si 0.33 0.5 GaAs 0.07 Fundamentals of Photonics 2017/4/16

Direct- and indirect-gap semiconductors
K [100] [111] E Si GaAs Figure Approximating the E-K diagram at the bottom of the conduction band and at the top of the valence band of Si and GaAs by parabols. Eg=1.1ev Eg=1.42ev The direct-gap semiconductors such as GaAs are efficient photon emitters, while the indirect-gap counterparts are not. Fundamentals of Photonics 2017/4/16

B. Semiconducting materials

Semiconducting materials
Si: widely used for making photon detectors but not useful for fabricating photon emitters due to its indirect bandgap. GaAs, InP GaN etc.: used for making photon detectors and sources. Ternary and quaternary semiconductors: AlxGa1-xAs, InxGa1-xAsyP1-y etc. tunable bandgap energy with variation of x and y. Fundamentals of Photonics 2017/4/16

Semiconducting materials
AlxGa1-xAs is lattice matched to GaAs, means it can be grown on the GaAs without introducing strains. Solid and dashed curves represent direct-gap and indirect-gap compositions respectively. We can see that a material may have direct bandgap for one mixing ratio x and an indirect bandgap for a different x. Fundamentals of Photonics 2017/4/16

Doped semiconductors Dopants: alter the concentration of mobile charge carriers by many orders of magnitude. n-type: predominance of mobile electrons p-type: predominance of holes Fundamentals of Photonics 2017/4/16

C. Electron and hole concentrations

Density of states The density of states describes the number of states at each energy level that are available to be occupied. Density of states near band edges Fundamentals of Photonics 2017/4/16

Rc(E) E Ec Ec Ec Eg Ef Ef Ef Rv(E) K Density of states (a) (b) (c) Figure (a) Cross section of the E-K diagram (e.g., in the direction of the K1 component with K2 and K3 fixed). (b) Allowed energy levels (at all K). (c) Density of states near the edges of the conduction and valence bands. Pc(E)dE is the number of quantum states of energy between E and E+dE, per unit volume, in the conduction band. P(E) has an analogous interpretation for the valence band. Fundamentals of Photonics 2017/4/16

Probability of occupancy
Under the condition of thermal equilibrium, the probability that a given state of energy E is occupied by an electron is determined by the Fermi function. Ef: Fermi level, the energy level for which the probability of occupancy is 1/2. f(E) is not itself a probability distribution, and it does not integrate to unity; rather it is a sequence of occupation probabilities of successive energy levels. Fundamentals of Photonics 2017/4/16

T>0K T=0K f(E) Ec Ec Ec Ef Eg Ef Ef Ev Ev Ev 1-f(E) 0.5 1 f(E) 0.5 1 f(E) Figure The Fermi function f(E) is the probability that an energy level E is filled with an electron; 1-f(E) is the probability that it is empty. In the valence band, 1-f(E) is the probability that energy level E is occupied by a hole. At T=0K, f(E)=1 for E<Ef, and f(E)=0 for E>Ef, i.e., there are no elctrons in the conduction band and no holes in the valence band. Fundamentals of Photonics 2017/4/16

The Fermi level is above the middle of the bandgap
Ec n(E) E ED Ef Donor level Ev p(E) 1 f(E) Carrier concentration Figure Energy-band diagram, Fermi function f(E), and concentrations of mobile electrons and holes n(E) and p(E) in an n-type semiconductor. The Fermi level is above the middle of the bandgap Fundamentals of Photonics 2017/4/16

The Fermi level is below the middle of the bandgap
Ec p(E) E Acceptor level Ef EA Ev n(E) 1 f(E) Carrier concentration Figure Energy-band diagram, Fermi function f(E), and concentrations of mobile electrons and holes n(E) and p(E) in an p-type semiconductor The Fermi level is below the middle of the bandgap Fundamentals of Photonics 2017/4/16

Exponential approximation of the Fermi function
When the Fermi level lies within the bandgap, but away from its edges by an energy of at least several times , the equations above gives: As Nc and Nv are Fundamentals of Photonics 2017/4/16

Law of mass action is independent of the location of the Fermi level Thus for intrinsic semiconductor We have Therefore the law of mass action can be written as: The law of mass action is useful for determining the concentrations of electrons and holes in doped semiconductors. Fundamentals of Photonics 2017/4/16

D. Generation, recombination, and injection

Generation and recombination in thermal equilibrium
点击查看flash动画 Thermal equilibrium requires that the process of generation of electron-hole pairs must be accompanied by a simultaneous reverse process of deexcitation,. The electron-hole recombination, ooocurs when an electron decays from the conduction band to fill a hole in the valence band. Fundamentals of Photonics 2017/4/16

Generation and recombination in thermal equilibrium
Radiative recombination: the energy released take the form of an emitted photon. Nonradiative recombination: transfer of the energy to lattice vibrations or to another free electron. Recombination may also occur indirectly via traps or defect centers, they can act as a recombination center if it is capable of trapping both the electron and the hole, thereby increasing their probability of recombining. 点击查看flash动画 ε(cm3/s) a parameter that depends on the characteristics of the material Fundamentals of Photonics 2017/4/16

Electron-hole injection
Define G0 as the rate of recombination at a given temperature: Now let additional electron-hole pairs be generated at a steady rate R A new steady state: As We get: With τ: the electron-hole recombination lifetime Fundamentals of Photonics 2017/4/16

Electron-hole injection
For an injection rate such that In a n-type material, where n0>>p0,the recombination lifetime τ≈1/εno is inversely proportional to the electron concentration. Similarly, for p-type material, τ≈1/εp0. This simple formulation is not applicable when traps play an important role in the process. Fundamentals of Photonics 2017/4/16

Internal quantum efficiency
The internal quantum efficiency ηi is defined as the ratio of the radiative electron-hole recombination rate to the total recombination. It determines the efficiency of light generation in a semiconductor material. For low to moderate injection rates, Fundamentals of Photonics 2017/4/16

Internal quantum efficiency
(cm3/s) τr τnr τ ηi Si 10-15 10ms 100ns 10-5 GaAs 10-30 50ns 0.5 The radiative lifetime for Si is orders of magnitude larger than its overall lifetime, mainly due to its indirect bandgap, result in a small internal quantum efficiency On the other hand, direct bandgap material as GaAs, the decay is largely via radiative transitions, consequently larger internal quantum efficiency, useful for fabricating light-emitting devices. Fundamentals of Photonics 2017/4/16

E. Junctions Fundamentals of Photonics 2017/4/16

Homojunctions: junctions between differently doped reguions of a semiconductor material Heterojunctions: junctions between different semiconductor materials The p-n junction Fundamentals of Photonics 2017/4/16

Carrier concentration
Ef Electron energy Carrier concentration P n P-type n-type position Figure Energy levels and carrier concentrations of a p-type and n-type semiconductor before contact. Fundamentals of Photonics 2017/4/16

点击查看flash动画 Figure A p-n junction in thermal equilibrium at T>0K. The depletion-layer, energy-band diagram, and concentrations (on a logarithmic scale) of mobile electrons n(x) and holes p(x) are shown as functions of position x. The built-in potrntial difference V0 corresponds to an energy eV0, where e is the magnitude of the electron charge. Fundamentals of Photonics 2017/4/16

The depletion layer contains only the fixed charges, the thickness of the depletion layer in each region is inversely proportional to the concentration of dopants in the region. 2. The fixed charges created a built-in field obstructs the diffusion of further mobile carriers. 3. A net built-in potential difference V0 is established. 4. In thermal equilibrium there is only a single Fermi function for the entire structure so that the Fermi levels in the p- and n- regions must align. 5. No net current flows across the junction. Fundamentals of Photonics 2017/4/16

点击查看flash动画 Energy-band and carrier concentrations in a forward-biased p-n junction. Fundamentals of Photonics 2017/4/16

The biased junction Forward biased A misalignment of the Fermi levels in the p- and n-regions Net current i=isexp(eV/kBT)-is Reverse biased Net current ≈-is as V is negative in exp(eV/kBT) and |V|>>kBT/e Acts as a diode with a current-voltage characteristic Fundamentals of Photonics 2017/4/16

The p-i-n junction diode
Made by inserting a layer of intrinsic (or lightly doped) semiconductor material between the p- and n-type region. The depletion layer penetrates deeply into the i-region. Large depletion layer: small junction capacitance thus fast response Favored over p-n diodes as photodiodes with better photodetection efficiency Fundamentals of Photonics 2017/4/16

The p-i-n junction diode
A p-i-n diode has the depletion layer penetrates deeply into the i-region Fundamentals of Photonics 2017/4/16

Heterojunctions Junctions between different semiconductor materials are called heterojunctions. P n Eg1 Eg2 Eg3 Electron energy They can provide substantial improvement in the performance of electronic and optoelectronic devices Fundamentals of Photonics 2017/4/16

Junctions between materials of different bandgap create localized jumps in the energy-band diagram Electron energy Ef Eg1 Eg2 Eg3 A potential energy discontinuity provides a barrier that can be useful in preventing selected charge carriers from entering regions where they are undesired. This property used in a p-n junction can reduce the proportion of current carried by minority carriers, and thus to increase injection efficiency Fundamentals of Photonics 2017/4/16

Discontinuities in the energy-band diagram created by two heterojunctions can be useful for confining charge carriers to a desired region of space Heterojunctions are useful for creating energy-band discontinuities that accelerate carriers at specific locations Semiconductors of different bandgap type can be used in the same device to select regions of the structure where light is emitted and where light is absorbed Heterojunctions of materials with different refractive indices create optical waveguides that confine and direct photons. Fundamentals of Photonics 2017/4/16

Quantum Wells and Super-lattices
When the Heterostructures layer thickness is comparable to, or smaller than, the de Broglie wavelength of thermalized electrons (≈50 nm in GaAs), the energy-momentum relation for a bulk semiconductor material no longer applies. A quantum well is a double heterojunction structure consisting of an ultrathin (≤50 nm) layer of semiconductor material whose bandgap is smaller than that of the surrounding material. The sandwich forms conduction- and valence band rectangular potential wells within which electrons in the conduction-band well and holes in the valence-band well are confined. Fundamentals of Photonics 2017/4/16

(a) Geometry of the quantum-well structure. (b) Energy-level diagram for electrons and holes in a quantum well. (c) Cross section of the E-k relation in the direction of k2 or k3. The energy subbands are labeled by their quantum number q1 = 1,2,... The E-k relation for bulk semiconductor is indicated by the dashed curves. Fundamentals of Photonics 2017/4/16

The energy levels Eq in one-dimensional infinite rectangular well are determined by Schrodinger equation: The energy-momentum relation: where k is the magnitude of a two-dimensional k = (k2, k3) vector in the y-z plane For the quantum well, k1 takes on well-separated discrete values. As a result, the density of states associated with a quantum-well structure differs from that associated with bulk material. Fundamentals of Photonics 2017/4/16

In a quantum-well structure the density of states is obtained from the magnitude of the two-dimensional wavevector (k2, k3) Fundamentals of Photonics 2017/4/16

quantum-well structure exhibits a substantial density of states at its lowest allowed conduction-band energy level and at its highest allowed valence-band energy level Fundamentals of Photonics 2017/4/16

Multiquantum Wells and Superlattices
Multiple-layered structures of different semiconductor materials that alternate with each other are called multiquantum-well (MQW) structures A multiquantum-well structure fabricated from alternating layers of AlGaAs and GaAs Fundamentals of Photonics 2017/4/16

Quantum Wires and Quantum Dots
The density of states in different confinement configurations: (a) bulk; (b) quantum well; (c) quantum wire; (d) quantum dot. The conduction and valence bands split into overlapping subbands that become successively narrower as the electron motion is restricted in more dimensions. Fundamentals of Photonics 2017/4/16

A semiconductor material that takes the form of a thin wire of rectangular cross section, surrounded by a material of wider bandgap, is called a quantum-wire structure Density of states Fundamentals of Photonics 2017/4/16

In a quantum-dot structure, the electrons are narrowly confined in all three directions within a box of volume d1d2d3. Quantum dots are often called artificial atoms. Fundamentals of Photonics 2017/4/16

INTERACTIONS OF PHOTONS WITH ELECTRONS AND HOLES
Chapter 15.2 INTERACTIONS OF PHOTONS WITH ELECTRONS AND HOLES Fundamentals of Photonics 2017/4/16

Mechanisms leading to absorption and emission of photons in a semiconductor: Band-to-Band (Inter-band) Transitions. Impurity-to-Band Transitions. Free-Carrier (Intraband) Transitions Phonon Transitions Excitonic Transitions. Fundamentals of Photonics 2017/4/16

Examples of absorption and emission of photons in a semiconductor. (a) Band-to-band transitions in GaAs can result in the absorption or emission of photons of wavelength (b) The absorption of a photon of wavelength results in a valence-band to acceptor-level transition in Hg-doped Ge (Ge:Hg). (c) A free-carrier transition within the conduction band. Fundamentals of Photonics 2017/4/16

Absorption coefficient for some semiconductor materials Fundamentals of Photonics 2017/4/16

For photon energies greater than the bandgap energy Eg, the absorption is dominated by band-to-band transitions Absorption edge: The spectral region where the material changes from being relatively transparent to strongly absorbing Direct-gap semiconductors have a more abrupt absorption edge than indirect-gap materials Fundamentals of Photonics 2017/4/16

Band-to-Band Absorption and Emission
Bandgap Wavelength: The quantity lg is also called the cutoff wavelength. Fundamentals of Photonics 2017/4/16

Absorption and Emission
(a) The absorption of a photon results in the generation of an electron-hole pair. This process is used in the photodetection of light. (b) The recombination of an electron-hole pair results in the spontaneous emission of a photon. Light-emitting diodes (LEDs) operate on this basis. (c) Electron-hole recombination can be stimulated by a photon. The result is the induced emission of an identical photon. This is the underlying process responsible for the operation of semiconductor injection lasers. Fundamentals of Photonics 2017/4/16

Conditions for Absorption and Emission
Conservation of Energy Conservation of Momentum Fundamentals of Photonics 2017/4/16

Conditions for Absorption and Emission
Energies and Momenta of the Electron and Hole with Which a Photon Interacts Fundamentals of Photonics 2017/4/16

Optical Joint Density of States The density of states with which a photon of energy interacts under conditions of energy and momentum conservation in a direct-gap semiconductor is determined by: And illustrated as follow: The density of states with which a photon of energy interacts increases with in accordance with a square-root law Fundamentals of Photonics 2017/4/16

Photon Emission Is Unlikely in an Indirect-Gap Semiconductor ！点击查看flash动画 Photon emission in an indirect-gap semiconductor The recombination of an electron near the bottom of the conduction band with a hole near the top of the valence band requires the exchange of energy and momentum. The energy may be carried off by a photon, but one or more phonons are required to conserve momentum. This type of multiparticle interaction is unlikely. Fundamentals of Photonics 2017/4/16

Photon Absorption is Not Unlikely in an Indirect-Gap Semiconductor！点击查看flash动画 Photon absorption in an indirect-gap semiconductor The photon generates an excited electron and a hole by a vertical transition; the carriers then undergo fast transitions to the bottom of the conduction band and top of the valence band, respectively, releasing their energy in the form of phonons. Since the process is sequential it is not unlikely. Fundamentals of Photonics 2017/4/16

Rates of Absorption and Emission
the probability densities of a photon of energy hn being emitted or absorbed by a semiconductor material in a direct band-to-band transition are mainly determined by three factors: Occupancy probabilities Transition probabilities Density of states Fundamentals of Photonics 2017/4/16

Occupancy Probabilities
Emission condition: A conduction-band state of energy E2 is filled (with an electron) and a valence-band state of energy E1 is empty (i.e., filled with a hole) Absorption condition: A conduction-band state of energy E2 is empty and a valence-band state of energy E1 is filled. The probabilities are determined from the appropriate Fermi functions and associated with the conduction and valence bands of a semiconductor in quasi-equilibrium Fundamentals of Photonics 2017/4/16

The probability that the emission condition is satisfied for a photon of energy is: The probability that the absorption condition is satisfied is : Fundamentals of Photonics 2017/4/16

Transition Probabilities
Satisfying the emission/absorption occupancy condition does not assure that the emission/absorption actually takes place. These processes are governed by the probabilistic laws of interaction between photons and atomic systems Transition Cross Section: Fundamentals of Photonics 2017/4/16

Probability density for the spontaneous emission: Probability density for the stimulated emission: If the occupancy condition for absorption is satisfied, the probability density for the absorption is also fit this formula Fundamentals of Photonics 2017/4/16

Overall Emission and Absorption Transition
Rates of Spontaneous Emission、Stimulated Emission and Absorption： the occupancy probabilities and is determined by the quasi-Fermi levels Efc and Efv Fundamentals of Photonics 2017/4/16

Spontaneous Emission Spectral Density in Thermal Equilibrium
A semiconductor in thermal equilibrium has only a single Fermi function so then where Fundamentals of Photonics 2017/4/16

Spectral density of the direct band-to-band spontaneous emission rate (photons per second per hertz per cm3) from a semiconductor in thermal equilibrium as a function of hv. The spectrum has a low-frequency cutoff at and extends over a width of approximately Fundamentals of Photonics 2017/4/16

Gain Coefficient in Quasi-Equilibrium
The net gain coefficient is Where the Fermi inversion factor is given by The gain coefficient may be cast in the form： with The sign and spectral form of the Fermi inversion factor are governed by the quasi-Fermi levels Efc and Efv, which, in turn, depend on the state of excitation of the carriers in the semiconductor. Fundamentals of Photonics 2017/4/16

Absorption Coefficient in Thermal Equilibrium
A semiconductor in thermal equilibrium has only a single Fermi level, so: Therefore the absorption coefficient: If Ef lies within the band gap but away from the band edges Fundamentals of Photonics 2017/4/16

The absorption coefficient resulting from direct band-to-band transitions as a function of the photon energy (eV) and wavelength for GaAs Fundamentals of Photonics 2017/4/16

Refractive Index The ability to control the refractive index of a semi-conductor is important in the design of many photonic devices Semiconductor materials are dispersive the refractive index is dependent on the wavelength The refractive Index is related to the absorption coe-fficient inasmuch as the real and imaginary parts of the susceptibility must satisfy the Kramers-Kronig relations The refractive index also depends on temperature and on doping level Fundamentals of Photonics 2017/4/16

Refractive index for high-purity, p-type, and n-type GaAs at 300 K, as a function of photon energy (wavelength). Fundamentals of Photonics 2017/4/16

Refractive Indices of Selected Semiconductor Materials at T = 300 K for Photon Energies Near the Bandgap Energy of the Material Fundamentals of Photonics 2017/4/16

Chapter 5 Photons in Semiconductors

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