Electro-absorption Modulators EE 232, Spring 2001 Pei-Cheng Ku April 24 & 26, 2001
Optical Modulation n Direct modulation on semiconductor lasers: n Output frequency drifts n n carrier induced (chirp) temperature variation due to carrier modulation Limited modulation depth (don’t want to turn off laser) External modulation n n Electro-optical modulation (low efficiency) Electroabsorption (EA) modulation (smaller modulation bandwidth)
Electroabsorption (EA) Modulator n n EA modulator is a semiconductor device with the same structure as the laser diode. In laser diodes, we inject large enough current to achieve stimulated emission. While in EA modulator, we apply electric field (reverse bias) to change the absorption spectrum. No carriers are injected into the active region. However, carriers are generated due to absorption of light.
Advantages of EA modulators n n n Zero biasing voltage Low driving voltage Low/negative chirp high bandwidth Integrated with DFB
Schematics of an EA modulator From textbook, figure 13. 10. n Waveguide type is more popular.
Outline of our discussion n Know how absorption spectrum changes with applied electric field (effective-mass approximation) See how extinction ratio of modulation can be enhanced in the quantum well (quantum confined Stark effect). Excitons will be introduced. We will follow the textbook (chapter 13). A recommended reference book is “Introduction to Semiconductor Optics” by N. Peyghambarian et al. (1993).
Physics behind EA Modulators n How absorption spectrum in semiconductors can be changed? n Physical model: effective-mass equation n n n Single-particle representation Two-particel representation Coulomb interaction between electrons and holes: Excitons Electric field effect: Franz-Keldysh effect Coulomb+Electric field: DC Stark effect Coulomb+Electric field+QW: QCSE
Absorption Spectrum Change under Applied Electric Field n Franz-Keldysh Effect n n DC Stark Effect n n Neglect Coulomb interaction between electrons and holes. Franz-Keldysh effect plus Coulomb interaction between electrons and holes (excitons). Quantum confined Start effect (QCSE) n n DC Stark Effect in quantum wells Excitons been confined in quantum well. Stark effect enhanced.
Effective mass approximation n n Electron moving in semiconductor crystal with periodic potential V (r) under the influence of a potential U (r). Effective mass approximation: if the potential is slowly varying compared to the period of the lattice, the above equation can be replaced by effective-mass equation as follows (section 4. 4. 1 and appendix B): that is the potential causes the envelope of the fast-varying Bloch function to change. Without potential U (r), the envelope f (r) reduces to plane wave
Wavefunction for electrons moving in periodic potential n For example, at the valence bandedge. The wavefunction is p-like (L = 3/2 in sp 3 orbitals).
Form of U (r) n n n Franz-Keldysh Effect DC Stark Effect (slow-varying approximation only holds for Wannier excitons) Quantum confined Start effect (QCSE)
Effective mass approximation in Two-Particle Representation n n Since U (r) is in general a function of both electron and hole, it’s more convenient to combine two single-particle equations of motion for electron and hole together. Effective-mass equation for electrons: Effective-mass equation for holes: Add them together
Effective mass approximation in Two-Particle Representation (cont. ) n In the path of generalizing single-particle effective mass equation to two-particle effective mass equation, we didn’t make new approximations. The new envelope wavefunction (which is a linear combination of the single-particle envelope functions for electrons and holes) still has to satisfy the slowly-varying approximation. This will be valid if the potential seen by electrons can be smoothly transferred to the potential profile seen by holes.
Validity of Slowly-Varying Approximation in two-particle representation n Franz-Keldysh Effect For 5 V applied voltage, there is 500 k. V/cm electric field across 0. 1 mm active region. The variation of electric field across a lattice site is 500 k. V/cm x 5. 6Å ~ 0. 028 V which gives an energy variation of 0. 028 e. V for an electron. For an electron moving in periodic atomic potential, it sees a Coulombic attraction from the nucleus. The potential energy variation across a lattice site is around V(a) ~ 2. 569 e. V >> 0. 028 e. V!
Excitons n The effective-mass equation for an electron-hole pair is: We have to sum over all possible interactions between electrons and holes. But if the carrier density is low enough, we can neglect the contribution from the rest of the electrons and holes and consider only bare Coulomb interaction, that is:
Effective-mass equation for Excitons n n With the bare Coulomb interaction between electron and hole, we have the effective-mass equation for an exciton as follows. This equation is formally identical to a hydrogen atom problem except the masses for the electron and proton are replaced by the effective masses for the electrons and holes, respectively. Any central-force problem (i. e. the potential is a function of relative coordinate between two particles) can be separated into center-ofmass motion and relative motion.
Center-of-mass motion and Relative motion n n Effective-mass equation in two-particle representation for any central-force problem can be rewritten as follows. Center-of-mass motion is a free-running solution:
Solutions for Excitons n n Same as hydrogen atoms (3 D or 2 D) Exciton Bohr Radius and Rydberg energy Energy levels: bound states and continuum n Ionization energy for ground state exciton in 2 D case is 4 times larger than it is in 3 D case. Therefore in QW, it’s easier to form bound exciton states.
Energy Diagram of Excitons Electron-hole energy E n=3 n=2 continuum n=1 In Ga. As, E 1=4. 2 me. V (3 D) or E 1=16. 8 me. V (2 D) K
Wavefunction for Excitons (3 D) n n n Coulomb potential is spherical symmetric with respect to the center of mass. Three quantum numbers: n, l, m. “Accidental degeneracy” for same quantum number n. Orbitals designated by 1 s, 2 p, 3 s, 3 p, 3 d, …
Radial Solution (3 D) probability 1 s 2 s 2 p r/a. B
Spherical Harmonics Ylm (3 D)
Wavefunction Representation for Excitons (3 D) Sitting at hole reference frame. P-like exciton envelope function (e. g. n=2) S-like exciton envelope function (e. g. n=1) s-like electron (L=1/2) bloch function p-like hole (L=3/2) bloch function
Wavefunction for Excitons (2 D) n Coulomb potential is cylindrical symmetric with respect to the center of mass. n Two quantum numbers: n, m. n “Accidental degeneracy” for same quantum number n.
Radial Solution (2 D) 2 D excitons are more tightly bounded.
Validity of Approximations used in the Calculation of Excitons n n Slowly-Varying Potential Profile Approximation: Ga. As/In. Ga. As/Al. Ga. As lattice density ~ 5. 6Å Bare Coulomb interaction approximation: Exciton radius a. B for Ga. As ~ 140Å Exciton density: 3 D: 8. 7 e 16/cm-3 2 D: 8. 1 e 10/cm-2 n n For EA modulators, no carriers are injected into active region. Carrier density ~ intrinsic doping level ~ 1 e 16/cm-3 (bulk) or 1 e 10/cm-2 (100Å QW). Bare coulomb interaction is a good approximation. But for laser applications, carrier density is high: 5 e 18/cm -3 (bulk) or 5 e 12/cm-2 (100Å QW). Coulomb interaction between electrons and holes is screened.
Optical Absorption without U (r) n From Fermi’s golden rule, the absorption coefficient can be derived as in equation (9. 1. 23) between two energy bands: where matrix element is calculated as in section 9. 3. 1 in the case of no external electric field and neglecting the Coulomb interaction between electrons and holes. The result is as follows (equation 9. 3. 11).
Optical Absorption with U (r)
Absorption Spectrum Calculation Summary n n n Absorption is proportional to the probability that we find electron and hole at the same “position” (i. e. r=0). For central force problem, we only need to evaluate the envelope wavefunction for the relative motion and take the square of its absolute value. Note the factor of 2 in the formula of the absorption spectrum comes from the fact that we consider two particles at one time. The delta function takes care of the (joint) density of states available. We have to sum over all possible states to get the total absorption.
Remarks on the first assertion n Same “position” actually means electron and hole are in the same lattice site. What happens in the unit cell is governed by the oscillation strength pcv which depends on the details of the Bloch functions inside the lattice site. This is because is an “envelope” function. The true wavefunction is the product of the envelope function with the Bloch function. is the probability of finding the electron and hole in the same lattice site.
Density of States function g (E) n Density of states means total number of degenerate states for a given energy level. For example in bulk materials, since there’s no restriction on the motion of the electrons, for a given energy E, the number of states that satisfy is the density of states at that energy.
g (E) as a function of dimensionality
Diagrammatic representation for g (E) 0 D g(E) 1 D 3 D 2 D E For a given energy range, the number of carriers necessary to fill out these density of states: 3 D>2 D>1 D>0 D.
Franz-Keldysh Effect n Franz-Keldysh effect is also a central-force problem.
Airy Function Ai (Z) n n Z>0: electron-hole energy+Eg < electric field potential Z<0: electron-hole energy+Eg > electric field potential, i. e. above bandgap oscillation wavefunction Smaller period Ai(Z) Z
Absorption Spectrum of Franz-Keldysh Effect n Absorption spectrum reduce to the familiar square root dependence of energy when F 0.
Exciton Absorption Spectrum n 3 D Excitons n 2 D Excitons
Exciton Absorption Spectra (schematics) Without damping n With damping 2 D and 3 D exciton absorption spectra with zero/finite linewidth. The spectra is very sensitive to the temperature.
Excitonic effect at different temperature T Bulk Ga. As Absorption peak blueshifts because the bandgap becomes larger at lower temperature. Quantum well (quasi 2 D)
Sommerfeld Enhancement n Even without excitonic peaks, bandedge absorption is enhanced due to Coulombic interaction between electrons and holes. Lots of bound states near the onset of continnum sum together to give Sommerfeld enhancement.
Quantum Confined Stark Effect* Quantum well potential n Because the potential of the quantum well breaks the spherical symmetry, it’s not a central force problem anymore! This symmetry breaking, unfortunately, forces us to solve this problem numerically. However, we’ll try to factorize this equation and get more physical insights. *D. A. B. Miller et al. , Phys. Rev. B. , 32, 1043 (1985)
Factorization of Schrödinger equation for QCSE
Solutions to QCSE n n n Consider solutions inside the quantum well. For given z, we can see that the problem reduces to 2 D-exciton problem. For given r, this is a quantum well problem. Therefore we can expect the solution to this equation can be expanded in terms of the product of 2 D-exciton envelope function and the quantum well wavefunction. We then expect that this expansion should only need a few terms to give a good approximation. Physically, because of the carrier confinement due to the quantum well, we should expect most of the excitons have their electron and hole either both in the quantum well or both outside the quantum well. As we have seen before, 2 D exciton has 4 times larger ionization energy and therefore is much more stable than excitons outside the quantum well.
Absorption Spectrum for QCSE n Exciton absorption peak red-shifts with increasing electric field strength.
Absorption peak shifts quadratically with applied electric field n n If the quantum well “only perturbs” the solution a little bit, the absorption peak position shift can be calculated from time-independent perturbation theory. The energy shift, for example for the ground state exciton in infinite quantum well, due to the electric field is: Potential profile has reflection symmetry along z-direction. Thus exciton wavefunction is either even or odd with respect to z=0. Absorption peak shifts quadratically with applied electric field.
Design considerations for EA modulators n n n Operation principle Contrast ratio Insertion loss Modulation efficiency Chirp considerations and optimization Packaging and Integration
EA Modulators use QCSE n By applying electric field parallel to the quantum well growth direction, the absorption can be changed dramatically at the desired wavelength.
Contrast Ratio n n The larger the contrast ratio, the better it is. Contrast ratio can be made as large as possible by increasing the length of the modulator. But propagation loss then becomes an issue.
Insertion Loss n n The longer the modulator is, the larger the insertion loss. Therefore, contrast ratio should not be increased by lengthen the device too much. Waveguide structure also needs careful design to increase the coupling efficiency from the singlemode-fiber output.
Modulation Efficiency n n Modulation efficiency quantifies how much voltage do we need to modulate the optical signal. Smaller detuning will increase the modulation efficiency. However, it also results in a larger insertion loss.
Chirp Fundamental linewidth n n Usually, the light to the input of the EA modulator comes directly from the output of a laser diode (e. g. DFB). This light signal is almost single frequency but will have a finite linewidth due to the frequency noise inside the laser diodes. This variation of frequency is called “chirp”. Chirp results from spontaneous emission and carrier induced phase fluctuation from the coupling between imaginary and real parts of the refractive index (Kramers-Kronig relation).
Chirp generated from Direct Modulation n Recall that the generalized rate equation for laser diodes we derived in the class: Neglect spontaneous emission chirp n n The amount of chirp generated from direct modulation depends on e and the signal waveform (note that S is the photon number inside the cavity and the output photon number is S (1 -R) where R is the reflectivity of the output facet). Large-signal modulation: blue-shift at the beginning of the pulse and red-shift at the tail.
Disadvantage of Chirp n n n In optical fiber, the dispersion is anomalous at 1. 55 mm, that is the refractive index of the fiber increases with the wavelength (cf. the regular optical glass in which the blue color sees higher refractive index than the red color does). The speed of light in a media is inverse proportional to the refractive index. Therefore in fiber, longer wavelength signal travels in a slower speed. Because of this property, if the signal pulse has higher frequency component (positive chirp) at the beginning of the pulse and lower frequency (negative chirp) at the tail, the pulse will experience broadening during the propagation through the fiber. If we can have opposite frequency chirp for the signal pulse, we can compensate the dispersion in the fiber.
Chirp generated in EA Modulator n n n We have loss instead of gain. With the increase of carrier density inside the active region due to the absorption of signal, we expect the linewidth enhancement factor to become negative if generated carriers can be removed from the cavity fast enough (i. e. no saturation of absorption or equivalently speaking no gain is generated by the small amount of inversion carriers). This will give us positive chirp when signal has negative slope (corresponding to the tail). Aside: In semiconductor optical amplifier, the chirp has opposite sign to laser diodes if gain is saturated.
Negative Chirp in EA Modulator n Chirp is a strong function of wavelength and bias
40 GHz Modulator Long device length for easy packaging short modulator section to reduce capacitance low driving voltage of 2. 8 V low insertion loss of 8 d. B
Integrated DFB-EA Transmitter n 10 Gb/s module, Ith = 20 m. A, Pmax = 4 m. W @80 m. A , extinction ratio = 15 d. B for -2. 5 V.
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