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Appendix 5 BEHAVIOUR MODEL FOR SEMICONDUCTOR LASER DIODES 1. WHY DO WE NEED THE BEHAVIOUR MODEL? ................................................... 1 2. SOME PHYSICAL CONCEPTS ................................................................................... 4 Recombination ................................................................................................................ 6 A) Spontaneous recombination.............................................................................. 6 B) Non-radiative recombination ............................................................................ 7 Photon absorption (stimulated carrier generation) .......................................................... 7 Stimulated emission. ....................................................................................................... 8 3. WORKING PRINCIPLES OF THE LASER ............................................................... 10 4. BALANCE EQUATIONS ............................................................................................ 11 Carrier Density Balance Equation ................................................................................ 12 Photon Number Balanced Equation: ............................................................................. 16 Photon Phase Rate Equation ......................................................................................... 18 5. STEADY STATE ANALYSIS..................................................................................... 23 1. WHY DO WE NEED THE BEHAVIOUR MODEL? The laser transforms electrical signals into optical signals. The behaviour model tells us 1) How the output optical power depends on DC input electrical current (the socalled STEADY-STATE ANALYSIS, Section 7) 1 a) The electrical signal is constant Iin; b) The optical signal is a sine wave with optical frequency v (“nu”), but its amplitude, and therefore its power Pout, is constant. The steady state analysis tells us how Pout depends on Iin. 2 2) How the output optical power depends on AC input electrical current. (SMALL SIGNAL ANALYSIS) a) The electrical signal changes slightly around some DC value with some comparatively small frequency f; b) The optical signal is a sine wave with optical frequency v, modulated by signal frequency f << v. Small signal analysis tells us how the output modulation P depends on the input modulation I and frequency f. Note The behaviour model depicts the laser as a black box with an input and an output. It does NOT tell us about the physics of laser operation, but we need physics to derive and understand the model. In order to complete the course, you only need to know how the behaviour model works. You do not need to understand all the physics behind it. However, we will briefly introduce some physical concepts that are necessary to build the simplest behaviour model. If you want to learn more about them please refer to the special literature on semiconductor devices. 3 2. SOME PHYSICAL CONCEPTS The carriers of electrical current in semiconductor material are free electrons and holes. Free (conduction) electrons are the electrons that are not bound to any atom. They are very mobile, move fast and therefore have high energy. Electron density N is the number of free electrons per unit volume. Bound (valence) electrons are electrons that are bound to an atom. They cannot move away and we can conclude that they have lower energy than the free electrons (in the same way that the potential energy of a body near Earth is less than the potential energy of a body flying freely in space). Note: Only 4 of 33 electrons are shown near each atom. Holes are associated with the vacancies of bound electrons. For example, if a nucleus has charge +33q (q = 1.6 x 10-19 is the charge of the electron) (i.e., Arsenic), then in a normal state it should have 33 electrons around it. If, for some reason, one of the electrons was taken away, so that only 32 are left, the total charge of the ion is +33q [nucleus] – 32q [remaining electrons] = +1q. An electron from a neighbouring atom can occupy the vacancy, leaving behind a hole. In this way, positively charged holes can travel from atom to atom and therefore they are current carriers. 4 Hole density P is the number of holes (i.e., atoms with such vacancies) per unit volume. In a regular crystal structure, holes are mainly created by making some electrons free (and letting them move away). Therefore, in a regular (undoped) crystal structure the electron density should be approximately equal to the hole density: NP This is NOT the only way to create holes (and free electrons). N-doping. It is possible to blend two materials in such a way that the resulting material has a significant number of free electrons and a very small number of holes. Such semiconductor materials are called n-doped (from the word “negative”, since the electron has a negative charge). P-doping. Similarly, it is possible to blend two materials in such a way that the resulting material has a significant number of holes and a very small number of free electrons. Such semiconductor materials are called p-doped (from the word “positive”, since the hole has a positive charge). Note: Only 4 of 33 electrons are shown near each atom 5 PN junction. If we bring n- and p-doped material into contact, the electrons driven by diffusion will go to the p-region and holes will go to the n-region. Recombination. When they meet each other at the p-n junction, the free electron can occupy the vacant places represented by holes. In this case, the electron ceases to exist as a free electron and so does the hole. The electron density N and the hole density P are each reduced by 1. As mentioned above, the energy of a bound electron is less than the energy of a free electron. According to the energy conservation law, when an electron goes from a higher energy state to a lower energy state it should liberate some energy in some form. We have two possibilities: A) Spontaneous recombination is the process whereby such an electron releases energy in the form of LIGHT. According to quantum mechanics principles, the frequency of this light is ν E state1 E state2 , h where h is the Plank constant h = 6.6x10-34 J.s. The explanation of these principles is beyond the scope of this course. I would like to point out that, although it may seem strange, WE DO NOT HAVE TO KNOW THE AMPLITUDE OF THIS LIGHT. We know from the energy conservation law that the TOTAL energy of this light is E E state1 E state2 . Therefore, if a certain number R of such electron-hole recombinations occurs per unit time, the energy of the light released per unit time is R( Estate1 Estate2 ) Rhv 6 The amount of energy released per unit time is by definition the power of the generated light. Since the power of the generated light is what we are looking for, all we need to know is R and v. Photon. A chunk of light generated by such recombination is called a photon. A photon has two characteristics: frequency and phase (no amplitude). The number of photons equals the number of recombinations, so we can say that R is the photon generation rate. R and v are the variables that totally determine the output power. In simple words, spontaneous recombination of an electron-hole pair is the process whereby the number of electrons and the number of holes are each reduced by 1 and 1 photon is generated. Balance: 1 EH pair 1 ph B) Non-radiative recombination is when an electron recombining with a hole releases energy in any other form. It is an undesirable phenomenon because we reduce the number of carriers without gaining any light. Balance: 1 EH pair some useless energy Photon absorption (stimulated carrier generation). Is spontaneous recombination a reversible process? Yes. It is possible that a photon gives its energy to a bound electron, making it a free (conduction) electron and leaving behind a hole. This process reduces the number of photons by 1 and increases the number of electrons and the number of holes each by 1. Balance: 1 ph 1EH pair. 7 Stimulated emission. Now we would like to consider a very interesting phenomenon that can be explained only by quantum mechanics. The explanation of this phenomenon is beyond the scope of this course. In fact, we are more interested in what happens rather than why it happens. The phenomenon is the following: Consider a semiconductor material that has free electrons and holes. As mentioned above, when an electron recombines with a hole, it releases some energy. So we can assign a certain energy difference E to each electron-hole pair (EH pair). This is the energy that WOULD BE released in the case of recombination. Now consider a photon of energy hv travelling through this material. Quantum mechanics says that when this photon passes by an EH pair with E = hv it can trigger recombination, so that the EH pair disappears and two photons travel further: the original one and the one that was released. The energy conservation law is satisfied: Before “collision”: TOTAL ENERGY = hv + E After “collision”: TOTAL ENERGY = hv + hv = hv + E. A photon cannot trigger recombination for an EH pair with E hv. So we can say that stimulated emission is the process whereby a photon makes an EH pair disappear and generates another photon. The generated photon is identical to the original one in frequency and phase. Balance: 1 ph + 1EH pair 2 ph. These 4 processes: 1) Spontaneous recombination: 1 EH pair 1 ph; 2) Photon absorption: 1 ph 1EH pair; 3) Stimulated emission: 1 ph + 1EH pair 2 ph; and 4) Non-radiative recombination: 1 EH pair some useless energy 8 are the main processes that define the main work principles of the laser. We will need them to build a simple behaviour model of the laser. 9 3. WORKING PRINCIPLES OF THE LASER The typical structure of a semiconductor is shown in this figure. The three main principles of how a laser works are listed and explained below. . 1. Power supply. Holes from the p-doped region and free electrons from the n-doped region are driven by the applied electrical field to the active region i. This region is normally undoped (or weakly doped), so the number of free electrons and the number of holes there are approximately the same (see 2. Some Physical Concepts, Hole Density, p. 3). In fact, we do want them to be the same, because it is an EH pair rather than a single electron or hole that generates light. Further, we will only consider the electron density N, since the hole density is the same. We will call it carrier density. Another property of this region is that it is easy for electrons and holes to get into it but difficult for them to get out (this phenomenon is explained in the textbook). To conclude, a high density of EH pairs that can release optical energy is achieved in the active region by applying the forward electric bias (“forward” means a positive terminal is applied to the p-doped region and a negative terminal is applied to the n-doped region). 10 2. Power transformation is achieved by recombination processes (both spontaneous and stimulated). 3. Feedback. At the beginning of the operation of the laser, the main process that generates light is spontaneous emission. The light generated in this way has a very broad frequency band and therefore is not a good signal carrier. By applying semitransparent mirrors at the front and rear facets of the laser, we make some portion of the outgoing light go back into the active region. Such a mirror structure filters out some frequency components, leaving some very narrow-band peaks. When this filtered light travels through the active region it stimulates the recombination of EH pairs and produces new photons of the same frequency and the same phase (see 2. Some Physical Concepts, Stimulated emission, p. 5). Thus only filtered, narrow band light experiences significant gain. 4. BALANCE EQUATIONS We would like to know the relation between the input current and the output optical power. As we said the optical power depends on the number of photons produced per unit time (see 2. Some Physical Concepts, Photon, p. 4). The photons are produced by two processes: Spontaneous Recombination and Stimulated 11 Recombination. In both cases, the rate of photon generation depends on the EH pair density, or carrier density N. Stimulated recombination is triggered by the photons, so its rate should depend on the number of photons S in the laser. The carrier density for its part depends on the input current, and, due to Photon Absorption (see 2. Some Physical Concepts, Photon Absorption, p. 5), on the photon number. If we assume the current to be a constant, we have two main variables defining the work of the device: N and S. Let us write down two equations that show how, respectively, carrier density N and the number of photons S change in time. All the terms in our equations should be in one of three groups: (1) variables to be determined; (2) variables that can be measured experimentally or determined from the design; and (3) well known constants. Carrier Density Balance Equation dN GENERATION RECOMBINATION dt GENERATION is the growth of the electron density due to the input current, which, by definition, is equal to the incoming charge per unit time: I dQ dt 12 The incoming charge is the number of incoming electrons multiplied by the electron charge q. The number of electrons is the electron density multiplied by the volume of the active region. Therefore Q = nq = VNgen q N gen Q qV and GENERATION dN gen dt 1 dQ I qV dt qV This is an ideal case, when all the electrons from the current go into the active region. In real cases, only some portion of them does. For that reason, we introduce a phenomenological variable called the current capture coefficient (eta). So, finally GENERATION ηI qV q is a known fundamental constant. I is the controlled input and and V can be measured experimentally. THE RECOMBINATION RATE consists of three terms: stimulated emission, spontaneous emission and non-radiative emission. (1) RECOMBINATION . The STIMULATED EMISSION RATE is responsible for the gain of light. Since gain can be measured experimentally, it is convenient to express our stimulated emission rate in terms of gain. Net gain is defined as the coefficient of the exponential growth of optical power, or equivalently, the number of photons travelling along the active region. Let S photons enter an active region of small length z. Through stimulated emission, they experience gain gst, and through photon absorption they experience decay ast. The net gain is g = gst- ast. The number of photons leaving this small area will be S out S exp( gΔz ) . 13 Therefore, (Sout – S) photons are generated and the same number of carriers is recombined in the time that it takes the light to travel through this area. This time is t Δz vg where vg is the light group velocity that can be determined experimentally. Thus, the carrier stimulated recombination rate is Rst v g ( S out S ) v g S[exp( gΔz ) 1] S out S Vt VΔz VΔz Note that we put the volume in the denominator because we need the carrier density rate, not the carrier’s number. It can be derived using Taylor expansion that if z is small enough then exp(gz) gz + 1. Substituting this in the above equation, we obtain Rs t Sgvg V S/V is the photon density in this small region and Rst is the stimulated recombination rate in this small region. To obtain an average stimulated recombination rate, S should be the total number of photons and V should be the total volume. Note: Since g is the net gain, Rst takes into account both stimulated emission and photon absorption. So strictly speaking, we should use the term net stimulated emission. Since we can only measure the net gain it is convenient to put these two phenomena into one term. Again, the above formula is the ideal case when ALL photons are confined to the active region. However, this is never the case in reality, so we have to introduce a correction term called optical confinement factor . It is defined as a ratio of 14 the photons in the active region to the total number of photons and can be estimated using the optical properties of the laser. Finally Rst Sgv g V where S is the total number of photons in the laser. Variables g, vg, can be determined experimentally, and V is known from the design. We can presume that the gain depends on the number of free electrons and holes, while the absorption depends on the number of the valence (bound) electrons (see 2. Some physical concepts. Photon absorption (stimulated carrier generation)). When the valence (bound) electron density exceeds the carrier density, gst< ast and g < 0. When the carrier density exceeds the valence (bound) electron density, gst> ast and g > 0. The carrier density at which g = 0 is called the transparency carrier density Ntr. It can be derived that, for bulk1 lasers, the net gain g = a(N-Ntr) where a is a coefficient that can be calculated based on the properties of the material. (2) RECOMBINATION. The SPONTANEOUS EMISSION RATE depends only on the electron and hole densities: 1 A bulk laser is a laser whose active region is one thick layer of semiconductor material. There is another type of laser where the active region consists of several thin layers (on the order of several nanometres) sandwiched between p and n regions. This type is called a quantum well (QW) laser. It has better properties and is more widely used in industry. For the QW laser g = a ln(N-Ntr) 15 Rsp =Bsp NP ≈ Bsp N 2 since P ≈ N in a weakly doped active region. Bsp is some coefficient. (3) RECOMBINATION. NON-RADIATIVE RECOMBINATION RATE depends only on the carrier density. It is defined phenomenologically as Rnr = AN + Bnr N2+CN3 For simplicity, the terms Bsp N 2 and Bnr N2 are combined into one: BN2 = (Bsp + Bnr) N2, so that the combined spontaneous and non-radiative recombination rate are expressed as Rsp nr AN BN 2 CN 3 N c where c (N ) 1 A BN CN 2 τ c is called “carrier life time”. Coefficients A, B, C can be found experimentally. Finally we can write Sgvg dN I N η Γ dt qV V τc We considered all the terms that contribute to the carrier density change rate. We took into account all 4 main processes and the injection current. We expressed the carrier density change rate through N, S and several coefficients that can be calculated theoretically, found experimentally, or determined from the design. Photon Number Balanced Equation: dS GENERATION CONSUMPTION dt GENERATION: comes from the stimulated and spontaneous emission (1) GENERATION. STIMULATED EMISSION G s t Γv g gS 16 This term is the same as that used for the carriers, but without V in the denominator, because now we consider the total number of photons, not the density. (2) GENERATION. SPONTANEOUS EMISSION could be expressed in the same way G s p VRs p . Unfortunately, most spontaneous emission is noise. Only a small part of it is coupled into the useful signal. So we have to introduce a coefficient K < 1 called Petermann’s factor, and G s p KG s p _ total . Sometimes it is convenient to relate the spontaneous and the stimulated emission: G sp _ total n sp G st where n s p is the so-called population inversion factor. Thus, the spontaneous emission term can be written as G s p KG s p _ total Kn s pG s t Kn s p Γv g gS CONSUMPTION is defined by the losses inside the cavity αint (this is different from photon absorption, described above, because it is already included the in stimulation emission term) and the losses caused by some of the light leaving the cavity αcav (this is nothing but our output light). αtotal αcav αint The formula for Gcons is similar to the one for Gst, except that it does not have factor Γ and instead of gain g we have loss αtotal. C v g (α cav α int ) S Sometimes it is convenient to write the above formula as C v g ( cav int ) S τp where S p 1 vg (αcav αint ) is called photon life time. Finally, we can write dS S Γv g gS Kn sp Γv g gS dt τp 17 Photon Phase Rate Equation This is the third important equation. It shows how the phase of the photon changes in time (which by definition equals light frequency). Phasse is closely connected to frequency: dφ ω dt Therefore, let us see what frequency is produced by laser. The field inside a laser is described by a traveling wave harmonic function: z E( x , y, z , t ) E( x , y) exp jω(t ) v where v is a velociy of the light inside the laser: v c nr c 3 108 m / s is the speed of light in vacuum, and nr 1 is the real part of the refractive index2 of the active region. n E( x , y, z , t ) E( x , y) exp jω(t z ) c 2 The real part of the refractive index the imaginary part nr is responsible for the change speed of light inside a medium and n i is responsible for the loss or gain. For simplicity we ignored the imaginary part, but it is easy to see what happens if we account for it: n jn i E( x , y, z , t ) E( x , y) exp jω(t r z ) c n ωn E( x , y) exp jω(t r z ) exp i c c z The last term stands for a simple exponential growth or decay (depends on the sign of n i ). 18 As it is shown in the pictures above after making a round trip withing the cavity the light can add with itself either in a constructive or destructive way. Consider two waves at a time moment t 0 in a point z z 0 : one of them has just started travelling (blue) and the other has already made a round trip (red) z E1 ( x , y, z , t ) E( x , y) exp jω(t 0 0 ) v z 2 L E 2 ( x , y, z , t ) E( x , y) exp jω t 0 0 v where L is the length of the cavity. In order to add constructively, the phase difference between them should be a multiple of 2π : 2mπ φ2 φ1 2ωL 2ωnL c c where m is any integer number. We can see that only those frequencies that satisfy ω ωm mπc nL can add constructively and therefore can exist for a long period inside laser. All the others add destructively and quickly disappear contributing very little in the laser’s work. 19 Now let us assume that the carrier density N changed. It changes the ability of the active region to produce light due to stimulated emission i.e. it changes gain. Gain is defined as an exponential coefficient of the growth of the optical power: P P0 exp( gz ) We know that optical power is proportional to the square of the field amplitude: P ~ E2 We know that n ωn E( x , y, z , t ) E( x , y) exp jω(t r z ) exp i c c and z n i stands for a simple exponential growth or decay of the field amlitude. Comparing the three above formulas we can see that g2 ωn i c Therefore, the change of the gain is automatically the change of the imaginary part of the index ni . According to the complex analisys of analytical functions the real and imaginary part are not independent and the change of n i immediately causes the change of nr . Their mutual dependence is described by so-called Kramers-Kronig relation which is beyong of the scope of this course. Instead, we introduce a phenomenological coeffitient called Linewidth Enhancement Factor that can be defined as α LEF Note that in general Δnr Δni (only due to the change of N and T ) n i can change due to several factors: carrier density N, temperature T, number of photons S. Mathematically it can be expressed as Δni , general ni n n ΔN i ΔT i ΔS N T S Linewidth Enhancement Factor DOES NOT take into account the last term. Sometimes it is defined as α LEF Δnr n i n i ΔN ΔT N T sometimes 20 α LEF Δn r n i ΔN N Since we do not consider the temperature change now the last definition is enough. Let us go back to our chain of changes: ΔN Δg Δni Δnr . From formula ω ωm we can see that the change of mπc nL nr causes the change of ω and ω is a time derivative of the phase. Thus our chain has two final links added ΔN Δg Δni Δnr Δω φ . 1st link 2nd link 3rd link 4th link 5th link That was the purpose of this section: to see how the carrier density changes the frequency of the output light. Now we need to write this chain in the form of the equations. Link # 5 dφ ω dt Link # 4 ω Δω mπc (n r Δnr ) L ω Δω(nr Δn r ) mπc L ωnr ωΔn r nr Δω Δn r Δω mπc L Assuming Δnr nr and Δω ω and neglecting the last term of the LHS we get ωn r ωΔn r n r Δω mπc L Since ωn r m πc L we can write 21 ωΔn r n r Δω 0 Δω ω Δn r nr Link # 3 Δnr αLEF Δni (remember: Δn i is only due the carrier density change (and, in more advanced analysis, temperature change), but NOT photon number change) Link # 4 Δni Δgc 2ω Finally Δω α LEF c α Δg LEF vg Δg 2 nr 2 If we are near the threshold where gain changes from zero to a certain value gnet then Δg gnet Γg α where α αint αcav Previously, we defined 1 vg (αcav αint ) τp Thus, finally, we can write dφ 1 1 α LEF Γv g g dt 2 τ p where α LEF Δnr Δni (only due to the change of N and T ) 22 5. STEADY STATE ANALYSIS As we mentioned in the first part, in the steady state analysis we assume the DC injection current that is converted in “DC” optical power. Of course, we should remember that the optical signal is ALWAYS high frequency (on the order of 100 THz) signal, so when we say “DC” optical power we just mean this signal is not modulated and the average power is constant. This means that the number of photons is also constant: dS 0. dt We mentioned before that carrier density is a function of injection current I and the number of photon S. Since neither of them changes, N should be the constant too. The balanced equations become: 0η Sgvg N I Γ qV V τc 0 Γvg gS Kns pΓvg gS S τp As mentioned above, the contribution of spontaneous emission in the useful signal is very small and can be neglected: Kn s p Γv g gS 0 . Thus Γv g g 1 S 0 τ p 23 This equation has two solutions: (1) S = 0; and (2) Γv g g 1 0. τ p IN THE FIRST CASE: 0η I N qV τc N ητ c I qV N is directly proportional to the injection current and S and consequently the output optical power are zero. Can N grow indefinitely with growing I ? Considering the second case will answer this question. IN THE SECOND CASE: Γv g g 1 0 τ p g 1 gth Γv g τ p No terms in RHS depend on the current I and, consequently, g does not depend on the current either. We remember that in a bulk laser g = a(N – Ntr) or N g N tr . a Since g is a constant w.r.t. the current, N should be a constant too N N th N tr g 1 N tr a aΓvg τ p 24 It means that N grows with growing I until N reaches the value N th N tr 1 called threshold carrier density. After this it is clamped aΓv g τ p at its threshold value no matter how high the input injection current is. Let us substitute Nth in the carrier density balance equation 0η Sgth v g N I Γ th qV V τc S V Γgth v g I N η th τc qV η qV I N th Γgv g q ητ c Define threshold current as I th qV 1 N th . Substituting Ith and gth in the ητ c Γvg τ p above equation, we obtain S ητ p q I I th Now we have the opposite picture: N is constant and S is linearly dependent on the input current. As mentioned above, the output power is the energy of photons coming out of the mirrors per unit time. 25 The number of photons leaving the cavity per unit time is expressed in the photon number balance equation through the cavity loss term cav C cav α cavv g S The output power is by definition the total energy of photons leaving the cavity per unit time and , therefore, equal to the energy of one photon h miltiplied by Ccav Pout hνC cav hναcavv g S 26