0 Mathematical Models of Heat Flow in Edge-Emitting Semiconductor Lasers

Edge-emitting lasers started the era of semiconductor lasers and have existed up to nowadays, appearing as devices fabricated out of various materials, formed sometimes in very tricky ways to enhance light generation. However, in all cases radiative processes are accompanied by undesired heat-generating processes, like non-radiative recombination, Auger recombination, Joule effect or surface recombination. Even for highly efficient laser sources, great amount of energy supplied by pumping current is converted into heat. High temperature leads to deterioration of the main laser parameters, like threshold current, output power, spectral characteristics or lifetime. In some cases, it may result in irreversible destruction of the device via catastrophic optical damage (COD) of the mirrors. Therefore, deep insight into thermal effects is required while designing the improved devices. From the thermal point of view, the laser chip (of dimensions of 1-2 mm or less) is a rectangular stack of layers of different thickness and thermal properties. This stack is fixed to a slightly larger heat spreader, which, in turn, is fixed to the huge heat-sink (of dimensions of several cm), transferring heat to air by convection or cooled by liquid or Peltier cooler. Schematic view of the assembly is shown in Fig. 1. Complexity and large size differences between the elements often induce such simplifications like reduction of the dimensionality of equations, thermal scheme geometry modifications or using non-uniform mesh in numerical calculations. Mathematical models of heat flow in edge-emitting lasers are based on the heat conduction equation. In most cases, solving this equation provides a satisfactory picture of thermal behaviour of the device. More precise approaches use in addition the carrier diffusion equation. The most sophisticated thermal models take into consideration variable photon density found by solving photon rate equations. The heat generated inside the chip is mainly removed by conduction and, in a minor degree, by convection. Radiation can be neglected. Typical boundary conditions for heat conduction equation are the following: isothermal condition at the bottom of the device, thermally insulated side walls, convectively cooled upper surface. It must be said that obtaining reliable temperature profiles is often impossible due to individual features of particular devices, which are difficult to evaluate within the quantitative analysis. Mounting imperfections 1


Introduction
Edge-emitting lasers started the era of semiconductor lasers and have existed up to nowadays, appearing as devices fabricated out of various materials, formed sometimes in very tricky ways to enhance light generation.However, in all cases radiative processes are accompanied by undesired heat-generating processes, like non-radiative recombination, Auger recombination, Joule effect or surface recombination.Even for highly efficient laser sources, great amount of energy supplied by pumping current is converted into heat.High temperature leads to deterioration of the main laser parameters, like threshold current, output power, spectral characteristics or lifetime.In some cases, it may result in irreversible destruction of the device via catastrophic optical damage (COD) of the mirrors.Therefore, deep insight into thermal effects is required while designing the improved devices.From the thermal point of view, the laser chip (of dimensions of 1-2 mm or less) is a rectangular stack of layers of different thickness and thermal properties.This stack is fixed to a slightly larger heat spreader, which, in turn, is fixed to the huge heat-sink (of dimensions of several cm), transferring heat to air by convection or cooled by liquid or Peltier cooler.Schematic view of the assembly is shown in Fig. 1.Complexity and large size differences between the elements often induce such simplifications like reduction of the dimensionality of equations, thermal scheme geometry modifications or using non-uniform mesh in numerical calculations.Mathematical models of heat flow in edge-emitting lasers are based on the heat conduction equation.In most cases, solving this equation provides a satisfactory picture of thermal behaviour of the device.More precise approaches use in addition the carrier diffusion equation.The most sophisticated thermal models take into consideration variable photon density found by solving photon rate equations.The heat generated inside the chip is mainly removed by conduction and, in a minor degree, by convection.Radiation can be neglected.Typical boundary conditions for heat conduction equation are the following: isothermal condition at the bottom of the device, thermally insulated side walls, convectively cooled upper surface.It must be said that obtaining reliable temperature profiles is often impossible due to individual features of particular devices, which are difficult to evaluate within the quantitative analysis.Mounting imperfections Fig. 1.Schematic view of the laser chip or laser array mounted on the heat spreader and heat sink (not in scale).like voids in the solder or overhang (the chip does not adhere to the heat-spreader entirely) may significantly obstruct the heat transfer.Surface recombination, the main mirror heating mechanism in bipolar devices, strongly depends on facet passivation.Since quantum cascade lasers (QCL's) exploit superlattices (SL's) as active layers, they have brought new challenges in the field of thermal modelling.Numerous experiments show that the thermal conductivity of a superlattice is significantly reduced.The phenomenon can be explained in terms of phonon transport across a stratified medium.As a consequence, mathematical models of heat flow in quantum cascade lasers resemble those created for standard edge-emitting lasers, but the stratified active region is replaced by an equivalent layer described by anisotropic thermal conductivity.In earlier works, the cross-plane and in-plane values of this parameter were obtained by arbitrary reduction of bulk values or treated as fitting parameters.Recently, some theoretical methods of assessing the thermal conductivity of superlattices have been developed.The present chapter is organised as follows.In sections 2, 3 and 4, one can find the description of static thermal models from the simplest to the most complicated ones.Section 5 provides a discussion of the non-standard boundary condition assumed at the upper surface.Dynamical issues of thermal modelling are addressed in section 6, while section 7 is devoted to quantum cascade lasers.In greater part, the chapter is a review based on the author's research supported by many other works.However, Fig. 7, 8, 12 and 13 present the unpublished results dealing with facet temperature reduction techniques and dynamical thermal behaviour of laser arrays.Note that section 8 is not only a short revision of the text, but contains some additional information or considerations, which may be useful for thermal modelling of edge-emitting lasers.The most important mathematical symbols are presented in Table 1.Symbols of minor importance are described in the text just below the equations, in which they appear.

Models based on the heat conduction equation only
Basic thermal behaviour of an edge-emitting laser can be described by the stationary heat conduction equation: accepting the following assumptions (see Fig. 2): -the laser is a rectangular stack of layers of different thickness and thermal conductivities;1 -there is no heat escape from the top and side walls, while the temperature of the bottom of the structure is constant; -the active layer is the only heat source in the structure and it is represented by infinitely thin stripe placed between the waveguide layers.
The heat power density is determined according to the crude approximation: which physically means that the difference between the total power supplied to the device and the output power is uniformly distributed over the surface of the selected region. 2 The problem was solved analytically by Joyce & Dixon (1975).Further works using this model introduced convective cooling at the top of the laser, considered extension and diversity of heat sources or changed the thermal scheme in order to take into account the non-ideal heat sink (Bärwolff et al. (1995); Puchert et al. (1997); Szyma ński et al. (2007;2004)).Such approach allows to calculate temperature inside the resonator, while the temperature in the vicinity of mirrors is reliable only in the near-threshold regime.The work by Szyma ński et al. (2007) can be regarded as a recent version of this model and will be briefly described below.
Assuming no heat escape from the side walls: and using the separation of variables approach (Bärwolff et al. (1995); Joyce & Dixon (1975)), one obtains the solution for T in two-fold form.In the layers above the active layer (n -even) temperature is described by while under the active layer (n -odd) it takes the form: 5) In ( 4) and ( 5)   4) and ( 5) can be used.
The results obtained according to the model described above are presented in qualitative assessment.A similar problem was described in Manning (1981), where even greater discrepancies between theory and experiment were obtained.For the properly mounted device C1 excellent convergence is found.
Improving the accuracy of calculations was possible due to taking into account the finite thermal conductivity of the heat sink material by thermal scheme modifications (see Fig. 3).
Assuming constant temperature at the chip-heat spreader interface leads to significant errors, especially for p-side-down mounting (see Fig. 4).
The analytical approach presented above has been described in detail since it has been developed by the author of this chapter.However, it should not be treated as a favoured one.In recent years, numerical methods seem to prevail.Pioneering works using Finite Element Method (FEM) in the context of thermal investigations of edge-emitting lasers have been described by Sarzała & Nakwaski (1990;1994).Broader discussion of analytical vs. numerical methods is presented in 8.3.Thermal effects in the vicinity of the laser mirror are important because of possible COD during high-power operation.Unfortunately, theoretical investigations of these processes, using the heat conduction only, is rather difficult.There are two main mirror heating mechanisms (see Rinner et al. (2003)): surface recombination and optical absorption.Without including additional equations, like those described in sections 3 and 4, assessing the heat source functions may be problematic.An interesting theoretical approach dealing with mirror heating and based on the heat conduction only, can be found in Nakwaski (1985;1990).However, both works consider the time-dependent picture, so they will be mentioned in section 6.

Models including the diffusion equation
Generation of heat in a semiconductor laser occurs due to: (A) non-radiative recombination, (B) Auger recombination, (C) Joule effect, (D) spontaneous radiative transfer, (E) optical absorption and (F) surface recombination.The effects (A)-(C) and (E,F) are discussed in standard textbooks (see Diehl (2000) or Piprek (2003)).Additional interesting information about mirror heating mechanisms (E,F) can be found in Rinner et al. (2003).The effect (D) will be briefly described below.
Apart from stimulated radiation, the laser active layer is a source of spontaneous radiation.
The photons emitted in this way propagate isotropically in all directions.They penetrate the wide-gap layers and are absorbed in narrow-gap layers (cap or substrate) creating the additional heat sources (see Nakwaski (1979)).Temperature calculations by Nakwaski (1983a) showed that the considered effect is comparable to Joule heating in the near-threshold regime.
On the other hand, it is known that below the threshold spontaneous emission grows with pumping current and saturates above the threshold.Thus, the radiative transfer may be recognised as a minor effect and will be neglected in calculations presented in this chapter.Note that processes (A)-( C) and (F) involve carriers, so g(x, y, z) should be a carrier dependent function.To avoid crude estimations, like equation ( 2), a method of getting to know the carrier distribution in regions essential for thermal analysis is required.

Carrier distribution in the laser active layer
An edge-emitting laser is a p-i-n diode operating under forward bias and in the plane of junction the electric field is negligible.Therefore, the movement of the carriers is governed by diffusion.Bimolecular recombination and Auger process engage two and three carriers, respectively.Such quantities like pumping or photon density are spatially inhomogeneous.Far from the pumped region, the carrier concentration falls down to zero level.At the mirrors, surface recombination occurs.Taking all these facts into account, one concludes that carrier concentration in the active layer can be described by a nonlinear diffusion equation with variable coefficients and mixed boundary conditions.Solving such an equation is really difficult, but the problem can often be simplified to 1-dimensional cases.For example, if problems of beam quality (divergence or filamentation) are discussed, considering the lateral direction only is a good enough approach.In the case of a thermal problem, since surface recombination is believed to be a very efficient facet heating mechanism responsible for COD, considering the axial direction is required and the most useful form of the diffusion equation can be written as where linear gain G(N)=α gain (N − N tr ) and non-linear carrier lifetime τ(N)= (A nr + BN + C A N 2 ) −1 have been assumed.The surface recombination at the laser facets is 9 Mathematical Models of Heat Flow in Edge-Emitting Semiconductor Lasers www.intechopen.comexpressed through the boundary conditions: The problem of axial carrier concentration in the active layer of an edge-emitting laser was investigated by Szyma ński (2010).Three cases were considered: (i) the nonlinear diffusion equation with variable coefficients (equation ( 6) and boundary conditions ( 7)) ; (ii) the linear diffusion equation with constant coefficients derived form equation ( 6) by assuming the averaged carrier lifetime τ av and averaged photon density S av ; (iii) the algebraic equation derived form equation ( 6) by neglecting the diffusion (D = 0).The results are shown in Fig. 5.It is clear that the approach (iii) yields a crude estimation of the carrier concentration in the active layer.However, for thermal modelling, where phenomena in the vicinity of facets are crucial due to possible COD processes, the diffusion equation must be solved.In many works (see for example Chen & Tien (1993), Schatz & Bethea (1994), Mukherjee & McInerney (2007)), the approach (ii) is used.It seems to be a good approximation for a typical edge-emitting laser, which is an almost axially homogeneous device in the sense that the depression of the photon density does not vary too much or temperature differences along the resonator are not so significant to dramatically change the 10 Heat Transfer -Engineering Applications www.intechopen.comnon-linear recombination terms B and C A .The approach (i) is useful in all the cases where the above-mentioned axial homogeneity is perturbed.In particular, the approach is suitable for edge-emitting lasers with modified regions close to facets.These modifications are meant to achieve mirror temperature reduction through placing current blocking layers (Rinner et al. (2003)), producing non-injected facets (so called NIFs) (Piersci ńska et al. (2007)) or generating larger band gaps (Watanabe et al. (1995)).

Carrier-dependent heat source function
The knowledge of axial carrier concentration opens up the possibility to write the heat source function more precisely compared to equation ( 2), namely g(x, y, z)=g a (x, y, z)+g J (x, y, z),( 8) where the first term describes the heat generation in the active layer and the second -Joule heating.According to Romo et al. (2003): The terms in the right hand side of equation ( 9) are related to non-radiative recombination, Auger processes, absorption of laser radiation and surface recombination at the facets, respectively.Assessing the value of S av was widely discussed by Szyma ński (2010).The Π's are positioning functions: expresses the assumption that the defects in the vicinity of the facets are uniformly distributed within a distance d sur = 0.5μm from the facet surface (Nakwaski (1990); Romo et al. (2003)), while Π a (x, y, z)=1forx, y, z within the active layer and Π a (x, y, z)=0elsewhere.
The effect of Joule heating is strictly related to the electrical resistance of a particular layer.High values of this parameter are found in waveguide layers, substrate and p-doped cladding due to the lack of doping, large thickness and low mobility of holes, respectively Szyma ński et al. (2004).Thus, it is reasonable to calculate the total Joule heat and assume that it is uniformly generated in layers mentioned above of total volume V hr : where R s is the device series resistance.

Selected results
Axial (mirror to mirror) distribution of relative temperature3 in the active layer of the edge-emitting laser is shown in Fig. 6.It has been calculated numerically solving the Fig. 6.Axial (mirror to mirror) distribution of relative temperature in the active layer.

Models including the diffusion equation and photon rate equations
The most advanced thermal model is described by Romo et al. (2003).It takes into account electro-opto-thermal interactions and is based on 3-dimensional heat conduction equation ∇(λ(T)∇T)=−g(x, y, z, T), (12) 1-dimensional diffusion equation ( 6), and photon rate equations The mirrors impose the following boundary conditions: Fig. 9. Self-consistent algorithm (Romo et al. (2003)).
Note the quadratic terms in equations ( 13) and ( 14), which describe the spontaneous radiation.
To avoid problems with estimating the spatial distribution and extent of heat sources related to radiative transfer, Romo et al. (2003) have 'squeezed' the effect to the active layer.Such assumption resulted in inserting the term (1 − 2β)B(T)N 2 hν into equation ( 9).The set of four differential equations mentioned above was solved numerically in the self-consistent loop, as schematically presented in Fig. 9. Several interesting conclusions formulated by Romo et al. (2003) are worth presenting here: -the calculations confirmed that the temperature along the resonator axis is almost constant, while it rises rapidly in the vicinity of the facets (cf.Fig. 6); -taking into account the non-linear temperature dependence of thermal conductivity significantly improves the accuracy of predicted temperature; -using the 1-dimensional (axial direction) diffusion or photon rate equations is a good enough approach; -heat conduction equation should be solved in 3 dimensions, reducing it to 2 dimensions is acceptable, while using the 1-dimensional form leads to significant overestimations of temperature in the vicinity of facets.

Discussion of the upper boundary condition
Typical thermal models for edge-emitting lasers assume convectively cooled or thermally insulated (which is the case of zero convection coefficient) upper surface.In Szyma ński (2007), using the isothermal condition T(x, y t )=T up .16 Heat Transfer -Engineering Applications www.intechopen.comby separation-of-variables approach.Due to ( 16), the expression (4) describing temperature in the layers above the active layer must be modified in the following way: Full analytical expressions can be found in Szyma ński (2007).
The investigations have been inspired by temperature maps obtained by thermoreflectance method (Bugajski et al. (2006); Wawer et al. ( 2005)) for p-down mounted devices.These maps suggest the presence of the region of constant temperature in the vicinity of the n-contact.
Besides, the isothermal lines are rather elliptic, surrounding the hot active layer, than directed upward as calculated for convectively cooled surface.The results are presented in Fig. 10 and 11.It is clear that assuming the isothermal condition and convection at the top surface one gets nearly the same device thermal resistances, but with the first assumption closer convergence with thermoreflectance measurements is found.
Fig. 12.Time evolution of central emitter active layer temperature.

Dynamical picture of thermal behaviour
Time-dependent models of edge-emitting lasers are considered rather seldom for two main reasons.First, edge-emitting lasers are predominantly designed for continuous-wave operation, so there is often no real need to investigate transient phenomena.Second, the complicated geometry of these devices, different kinds of boundary conditions and uncertain values of material parameters make that even static cases are difficult to solve.
The authors who consider dynamical models usually concentrate on initial heating (temperature rise during the first current pulse) of the laser inside the resonator (Nakwaski (1983b)) or at the mirrors (Nakwaski (1985;1990)).The papers mentioned above developed analytical solutions of time-dependent heat conduction equation using sophisticated mathematical methods, like for example Green function formalism or Kirchhoff transformation.Numerical approach to this class of problems appeared much later.As an example see Puchert et al. (2000), where laser array was investigated.It is noteworthy that the heat source function was obtained by a rate equation model.Remarkable agreement with experimental temperature values showed the importance of the concept of distributed heat sources.The author has theoretically investigated the dynamical thermal behaviour of p-down mounted 25-emitter laser array.Temperature profiles during first 10 pulses have been calculated (Fig. 12 and 13).The transverse structure of the device, material parameters and the distribution of heat sources are presented in Table 3.The time-dependent heat conduction equation has been solved numerically. 6The following boundary and initial conditions have been assumed: -constant temperature T = 300K at the heat spreader-heat sink interface;7 -convective cooling of the top surface (α = 35 * 10 3 WK −1 m −2 ); -all side walls (including mirrors) thermally insulated; -T(x, y, z, t = 0)=300K.
Note that the considered laser array has been driven by rectangular pulses of period 3.33 ms and 50% duty cycle, while the carrier lifetime is of the order of several nanoseconds.Thus, the electron response for the applied voltage can be regarded as immediate8 and equation ( 8) can be transformed to the time-dependent function in the following way: where Θ(t)=1 or 0 exactly reproduces the driving current changes.

Heat flow in a quantum cascade laser
Quantum-cascade lasers are semiconductor devices exploiting superlattices as active layers.
In numerous experiments, it has been shown that the thermal conductivity λ of a superlattice Fig. 14.Calculated cross-plane thermal conductivity the active region of THz QCL (Szyma (2011)).Square symbols show the values measured by Vitiello et al. (2008).
Particularly, the cross-plane value λ ⊥ may be even order-of-magnitude smaller than than the value for constituent bulk materials.The phenomenon is a serious problem for QCLs, since they are electrically pumped by driving voltages over 10 V and current densities over 10 kA/cm 2 .Such a high injection power densities lead to intensive heat generation inside the devices.To make things worse, the main heat sources are located in the active layer, where the density of interfaces is the highest and-in consequence-the heat removal is obstructed.Thermal management in this case seems to be the key problem in design of the improved devices.
Theoretical description of heat flow across SL's is a really hard task.The crucial point is finding the relation between phonon mean free path Λ andSLperiodD Yang & Chen (2003).In case Λ > D, both wave-and particle-like phonon behaviour is observed.The thermal conductivity is calculated through the modified phonon dispersion relation obtained from the equation of motion of atoms in the crystal lattice (see for example Tamura et al. (1999)).In case Λ < D, phonons behave like particles.The thermal conductivity is usually calculated using the Boltzmann transport equation with boundary conditions involving diffuse scattering.Unfortunately, using the described methods in the thermal model of QCL's is questionable.They are very complicated on the one hand and often do not provide satisfactionary results on the other.The comprehensive comparison of theoretical predictions with experiments for 20 Heat Transfer -Engineering Applications www.intechopen.comnanoscale heat transport be found in Table II in Cahill al. (2003).This topic was also widely discussed by Gesikowska & Nakwaski (2008).the investigations in this field usually deal with bilayer SL's, while one period of QCL active layer consists of dozen or so layers of order-of-magnitude thickness differences.Consequently, present-day mathematical models of heat flow in QCLs resemble those created for standard edge emitting lasers: they are based on heat conduction equation, isothermal condition at the bottom of the structure and convective cooling of the top and side walls are assumed.QCL's as unipolar devices are not affected by surface recombination.Their mirrors may be hotter than the inner part of resonator only due to bonding imperfections (see 8.4).
Colour maps showing temperature in the QCL cross-section and illustrating fractions of heat flowing through particular surfaces can be found in Lee et al. (2009) and Lops et al. (2006).In those approaches, the SL's were replaced by equivalent layers described by anisotropic values of thermal conductivity λ ⊥ and λ arbitrarily reduced (Lee et al. (2009)) or treated as fitting parameters (Lops et al. (2006)).
Proposing a relatively simple method of assessing the thermal conductivity of QCL active region has been a subject of several works.A very interesting idea was mentioned by Zhu et al. (2006) and developed by Szyma ński (2011).The method will be briefly described below.
The thermal conductivity of a multilayered can be approximated according to the rule mixtures Samvedi & Tomar (2009); Zhou et al. (2007): where f n and λ n are the volume fraction and bulk thermal conductivity of the n-th material.However, in case of high density of interfaces, the approach ( 20) is inaccurate because of the following reason.The interface between materials of different thermal and mechanical properties obstructs the heat flow, introducing so called 'Kapitza resistance' or thermal boundary resistance (TBR) Swartz & Pohl (1989).The phenomenon can be described by two phonon scattering models, namely the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM).Input data are limited to such basic material parameters like Debye temperature, density or acoustic wave speed.Thus, the thermal conductivity of the QCL active region can be calculated as a sum of weighted average of constituent bulk materials reduced by averaged TBR multiplied by the number of interfaces: where TBR has been averaged with respect to the direction of the heat flow r (av) The detailed prescription on how to calculate r (av) Bd can be found in Szyma ński (2011).The model based on equations ( 21) and ( 22) was positively tested on bilayer Si 0.84 Ge 0.16 /Si 0.74 Ge 0.26 SL's investigated experimentally by Huxtable et al. (2002).Then, GaAs/Al 0.15 Ga 0.85 As THz QCL was considered.Results of calculations exhibit good convergence with measurements presented by Vitiello et al. (2008) as shown in Fig. 14.On the contrary, values of λ ⊥ calculated according to equation (20), neglecting the influence of interfaces, show significant discrepancy with the measured ones (Fig. 15).

Summary
Main conclusions or hints dealing with thermal models of edge-emitting lasers will be aggregated in the form of the following paragraphs.

Differential equations
A classification of thermal models is presented in Table 4. Basic thermal behaviour of an edge-emitting laser can be described according to Approach 1.It is assumed that the heat power is generated uniformly in selected regions: mainly in active layer and, in minor degree, in highly resistive layers.Considering the laser cross-section parallel to mirrors' surfaces and reducing the dimensionality of the heat conduction equation to 2 is fully justified.For calculating the temperature in the entire device (including the vicinity of mirrors) Approach 2 should be used.The main heat sources may be determined as functions of carrier concentration calculated from the diffusion equation.plane of junction (2 dimensions) or reduced to the axial direction (1 dimension).Approach 3 is the most advanced one.It is based on 4 differential equations, which should be solved in self-consisted loop (see Fig. 9).Approach 3 is suitable for standard devices as well as for lasers with modified close-to-facet regions.

Boundary conditions
The following list presents typical boundary conditions (see for example Joyce & Dixon (1975), Puchert et al. (1997), Szyma ński et al. (2007)): -isothermal condition at the bottom of the device, -thermally insulated side walls, -convectively cooled or thermally insulated (which is the case of zero convection coefficient) upper surface.
In Szyma ński (2007), it was shown that assuming isothermal condition at the upper surface is also correct and reveals better convergence with experiment.Specifying the bottom of the device may be troublesome.Considering the heat flow in the chip only, i.e. assuming the ideal heat sink, leads to significant errors (Szyma ński et al. (2007)).
On the other hand taking into account the whole assembly (chip, heat spreader and heat sink) is difficult.In the case of analytical approach, it significantly complicates the geometry of the thermal scheme.In order to avoid that tricky modifications of thermal scheme (like in Szyma ński et al. ( 2007)) have to be introduced.In case of numerical approach, using non-uniform mesh is absolutely necessary (see for example Puchert et al. (2000)).
In Ziegler et al. (2006), an actively cooled device was investigated.In that case a very strong convection (α = 40 * 10 4 W/(mK)) at the bottom surface was assumed in calculations.

Calculation methods
Numerous works dealing with thermal modelling of edge-emitting lasers use analytical approaches.Some of them exploit highly sophisticated mathematical methods.For example, Kirchhoff transformation (see Nakwaski (1980)) underlied pioneering theoretical studies on the COD process Nakwaski (1985) and Nakwaski (1990), where solutions of the three-dimensional time-dependent heat conduction equation were found using the Green function formalism.Conformal mapping has been used by Laikhtman et al. (2004) and Laikhtman et al. (2005) for thermal optimisation of high power diode laser bars.Relatively simple separation-of-variables approach was used by Joyce & Dixon (1975) and developed in many further works (see for example Bärwolff et al. (1995) or works by the author of this chapter).Analytical models often play a very helpful role in fundamental understanding of the device operation.Some people appreciate their beauty.However, one should keep in mind that edge-emitting devices are frequently more complicated.This statement deals with the internal chip structure as well as packaging details.Analytical solutions, which can be found in widely-known textbooks (see for example Carslaw & Jaeger (1959)), are usually developed for regular figures like rectangular or cylindrical rods made of homogeneous materials.Small deviation from the considered geometry often leads to substantial changes in the solution.In addition, as far as solving single heat conduction equation in some cases may be relatively easy, including other equations enormously complicates the problem.Recent development of simulation software based on Finite Element Method creates the temptation to relay on numerical methods.In this chapter, the commercial software has been used for computing dynamical temperature profiles (Fig. 12 and 13)9 and carrier concentration profiles (Fig. 7 and 8). 10 Commercial software was also used in many works, see for example Mukherjee & McInerney (2007); Puchert et al. (2000); Romo et al. (2003).In Ziegler et al. (2006;2008), a self-made software based on FEM provided results highly convergent with sophisticated thermal measurements of high-power diode lasers.Thus, nowadays numerical methods seem to be more appropriate for thermal analysis of modern edge-emitting devices.However, one may expect that analytical models will not dissolve and remain as helpful tools for crude estimations, verifications of numerical results or fundamental understanding of particular phenomena.

Limitations
While using any kind of model, one should be prepared for unavoidable inaccuracies of the temperature calculations caused by factors characteristic for individual devices, which elude qualitative assessment.The paragraphs below briefly describe each factor.Real solder layers may contain a number of voids, such as inclusions of air, clean-up agents or fluxes.Fig. 12 in Bärwolff et al. (1995) shows that small voids in the solder only slightly obstruct the heat removal from the laser chip to the heat sink unless their concentration is very high.In turn, the influence of one large void is much bigger: the device thermal resistance grows nearly linearly with respect to void size.The laser chip may not adhere to the heat sink entirely due to two reasons: the metallization may not extend exactly to the laser facets or the chip can be inaccurately bonded (it can extend over the heat sink edge).In Lynch (1980), it was shown that such an overhang may contribute to order of magnitude increase of the device thermal resistance.
In Pipe & Ram it was shown that convective cooling of top and side walls plays a significant role.Unfortunately, determining of convective coefficient is difficult.The values found in the literature differ by 3 order-of-magnitudes (see Szyma ński ( 2007)).Surface recombination, one of the two main mirror heating mechanisms, strongly depends on facet passivation.The significant influence of this phenomenon on mirror temperature was shown in Diehl (2000).It is noteworthy that the authors considered values v sur of one order-of-magnitude discrepancy.11Modern devices often consist of multi-compound semiconductors of unknown thermal properties.In such cases, one has to rely on approximate expressions determining particular parameter upon parameters of constituent materials (see for example Nakwaski (1988)).

Quantum cascade lasers
Present-day mathematical models of heat flow in QCL resemble those created for standard edge emitting lasers: they are based on heat conduction equation, isothermal condition at the bottom of the structure and convective cooling of the top and side walls are assumed.The SL's, which are the QCLs' active regions, are replaced by equivalent layers described by anisotropic values of thermal conductivity λ ⊥ and λ arbitrarily reduced (Lee et al. (2009)), treated as fitting parameters (Lops et al. (2006)) or their parameters are assessed by models considering microscale heat transport (Szyma ński (2011)).

Fig. 2 .
Fig. 2. Schematic view of a laser chip cross-section (A).Function describing the heat source (B).
/b is the separation constant and thus it appears in both directions (x and y).Integer number k numerates the heat modes.Coefficients w be found inSzyma ński (2007).They are determined by the bottom boundary condition, continuity conditions for the temperature and heat flux at the layer interfaces and the top boundary condition.

Fig. 3 .
Fig. 3. Thermal scheme modification.Assuming larger b allows to keep the rectangular cross-section of the whole assembly and hence equations (4) and (5) can be used.

Fig. 4 .
Fig.4.Maximum temperature inside the laser for p-side down mounting.It is clear that the assumption of ideal heat sink leads to a 50% error in calculations(Szyma ński et al. (2007)).

Fig. 5 .
Fig. 5. Axial (mirror to mirror) carrier concentration in the active layer calculated according to algebraic equation (dotted line), linear diffusion equation with constant coefficients (dashed line) and nonlinear diffusion equation with variable coefficients (solid line) (Szyma ński (2010)).

Fig. 7 .
Fig. 7. Axial distribution of carriers in the active layer for the laser with non-injected facets.The inset shows the step-like pumping profile.
Fig. 11.Contour plot of temperature calculated under the assumption of convective cooling at the top surface (a), isothermal condition at the top surface (b) and measured by thermoreflectance method (c) (Szyma ński (2007)).

Fig. 13 .
Fig. 13.Transverse temperature profiles at the front facet of the central emitter.Dashed vertical lines indicate the edges of heat spreader and substrate.
It is recommended to use three-dimensional heat conduction equation.The diffusion equation can be solved in the Approach A classification of thermal models.Abbreviations: HC-heat conduction, D-diffusion, PR -photon rate.

Table 1 .
List of symbols.
⊥ ,λ thermal conductivity of QCL's active layer in the direction perpendicular and parallel to epitaxial layers, respectively ν frequency ρ n density, subscript n (if added) denotes the medium number τ, τ av carrier lifetime, averaged carrier lifetime c, e, h, k B physical constants: light velocity, elementary charge, Planck and Boltzmann constants, respectively.5 Mathematical Models of Heat Flow in Edge-Emitting Semiconductor Lasers www.intechopen.com

Table 2 .
The calculated values are slightly underestimated due to bonding imperfections, which elude

Table 3 .
Transverse structure of the investigated laser array.