combination (with positive coefficients in the Ising case) of known objects
called the conformal blocks. Equalling these two decompositions, one ends up
with constraints on the possible exponents.

There is a priori no reason to be able to determine the critical exponents as
the unique values satisfying a (finite) number of constraints thus obtained.
Indeed, the set of possible values may not shrink when considering more and
more conditions, but it happens that in the case of the Ising model, the
region of the plane for possible critical exponents
$(\bm{\Delta}_{\sigma},\bm{\Delta}_{\varepsilon})$ for the spin and energy
local operators can be reduced drastically, to a point where estimates –
namely
$(\bm{\Delta}_{\sigma},\bm{\Delta}_{\varepsilon})=(0.5181489(10),1.412625(10))$
– using this bootstrap technique become way better than Monte-Carlo
simulations. We refer to [94, 79, 41] for some of the original papers and [98]
for a review of the most recent progress in this very exciting area of modern
theoretical physics.

Let us conclude that even if one may use conformal bootstrap to exactly
identify the critical exponents, this would leave the question of proving that
the critical 3D Ising model indeed converges to a CFT widely open. In some
sense, getting sufficient information on the possible scaling limits and
proving that these scaling limits indeed exist are two almost entirely
disjoint questions even though, of course, one may hope that information on
the former question would help answer the latter.

##  9 A tail to this story

The Ising model has always played the role of a locomotive in the developments
of statistical physics. Its central place and incredible properties turn it
into an amazing playground for both mathematicians and physicists. As a
consequence, during most of its history novel techniques were developed to
solve problems on it, which later led to whole independent fields of
mathematical physics (integrable systems, graphical representations, rigorous
renormalisation methods, etc).

Let us mention several long-standing problems remaining widely open for this
model. At the top of the list, universality of the 2D behaviour (see Section
8.3), critical properties of the 3D model (see Section 8.4), and the
roughening phase transition (see Section 5.3) are among the most important
unsolved puzzles. Solving them will probably require the development of new
techniques that will again, through cross-fertilisation, benefit the whole
field of statistical mechanics.

We thank all our coauthors for the wonderful years of joint research, both
past and future. We also wish to thank D. Cimasoni, T. Gunaratnam, D. Krachun,
I. Manolescu, R. Panis, V. Tassion, and Y. Velenik who, without knowing its
ultimate aim, took the time to take a look at this review and to give the
author feedback. We thank T. Hutchcroft for bringing [71] to our attention.

This work has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreements No. 757296). The author acknowledges funding from the NCCR
SwissMap, the Swiss FNS, and the Simons collaboration on localization of
waves.

## References

  * [1] M. Aizenman. Translation invariance and instability of phase coexistence in the two dimensional Ising system. Commun. Math. Phys. 73 (1980), no. 1, 83–94.
  * [2] M. Aizenman. Geometric analysis of $\varphi^{4}$ fields and Ising models. I, II, Commun. Math. Phys. 86 (1982), no. 1, 1–48.
  * [3] M. Aizenman, D.J. Barsky, and R. Fernández. The phase transition in a general class of Ising-type models is sharp, J. Statist. Phys. 47 (1987), no. 3–4, 343–374.
  * [4] M. Aizenman and H. Duminil-Copin. Marginal triviality of the scaling limits of critical 4D Ising and $\phi_{4}^{4}$ models. Ann. Math. 194 (2021), no. 1, 163–235.
  * [5] M. Aizenman, H. Duminil-Copin, and V. Sidoravicius. Random Currents and Continuity of Ising Model’s Spontaneous Magnetization, Commun. Math. Phys. 334 (2015), 719–742.
  * [6] M. Aizenman, H. Duminil-Copin, V. Tassion, and S. Warzel. Emergent planarity in two-dimensional Ising models with finite-range interactions. Invent. Math. 216 (2019), no. 3, 661–743.
  * [7] M. Aizenman and R. Fernández. On the critical behavior of the magnetization in high-dimensional Ising models, J. Statist. Phys. 44 (1986), no. 3–4, 393–454.
  * [8] J.R. Baker and A. George. Application of the Padé approximant method to the investigation of some magnetic properties of the Ising model. Phys. Rev. 124 (1961), no. 3, 768.
  * [9] R.J. Baxter. Exactly solved models in statistical mechanics. Academic Press Inc. London, 1989\. Reprint of the 1982 original.
  * [10] R.J. Baxter and I.G. Enting. 399th solution of the Ising model. J. of Phys. A: Math. and Gen. 11 (1978), no. 12, 2463.
  * [11] R. Bauerschmidt, D.C. Brydges, and G. Slade. Scaling limits and critical behaviour of the 4-dimensional $n$-component $|\phi^{4}|$ spin model. J. Stat. Phys. 157 (2014), 692–742.
  * [12] A.A. Belavin, A.M. Polyakov, and A.B. Zamolodchikov. Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Physics B 241 (1984), no. 2, 333–380.
  * [13] S. Benoist, H. Duminil-Copin, and C. Hongler. Conformal invariance of crossing probabilities for the Ising model with free boundary conditions. Ann. de l’IHP 52 (2016), no. 4, 1784–1798.
  * [14] S. Benoist and C. Hongler. The scaling limit of critical Ising interfaces is CLE(3). Ann. Probab. 47 (2019), no. 4, 2049–2086.
  * [15] T. Bodineau, D. Ioffe, and Y. Velenik. Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 (2000), no. 3, 1033–1098.
  * [16] M. Biskup. Reflection positivity and phase transitions in lattice spin models. In Methods of contemporary mathematical statistical physics, volume 1970 of Lecture Notes in Math., pages 1–86. Springer, Berlin, 2009\.
  * [17] T. Bodineau. The Wulff construction in three and more dimensions. Commun. Math. Phys. 207 (1999), no. 1, 197–229.
  * [18] T. Bodineau. Translation invariant Gibbs states for the Ising model. Prob. Th. Rel. Fields 135 (2006), no. 2, 153–168.
  * [19] W. Bragg and E. Williams. The effect of thermal agitation on atomic arrangement in alloys. (–II) Proc. of the Royal Soc. of London. 145 (1934), no. 855, 699–730 – 151 (1935), no. 874, 540–566.
  * [20] S.G. Brush. History of the Lenz-Ising model. Rev. Mod. Phys. 39 (1967), no. 4, 883.
  * [21] D. Brydges, J. Fröhlich, and T. Spencer. The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys. 83 (1982), no. 1, 123–150.
  * [22] F. Camia, C. Garban, and C.M. Newman. Planar Ising magnetization field I. Uniqueness of the critical scaling limit. Ann. Probab. 43 (2015), no. 2, 528–571.
  * [23] M. Campanino, D. Ioffe, and Y. Velenik. Ornstein-Zernike theory for finite range Ising models above $T_{c}$. Prob. Th. Rel. Fields 125 (2003), no. 3, 305–349.
  * [24] R. Cerf and A. Pisztora. On the Wulff crystal in the Ising model. Ann. Probab. 28 (2000), 947–1017.
  * [25] D. Chelkak. Ising model and s-embeddings of planar graphs. arXiv:2006.14559 (2020).
  * [26] D. Chelkak, D. Cimasoni, and A. Kassel. Revisiting the combinatorics of the 2D Ising model. Ann. IHP D 4 (2017), no. 3, 309–385.
  * [27] D. Chelkak, C. Hongler, and K. Izyurov. Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181 (2015), no. 3, 1087–1138.
  * [28] D. Chelkak, and S. Smirnov. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189 (2012), no. 3, 515–580.
  * [29] D. Cimasoni. The Critical Ising Model via Kac-Ward Matrices. Commun. Math. Phys. 316 (2012), 99–126.
  * [30] P. Curie. Propriétés magnétiques des corps à diverses températures. 4, Gauthier-Villars et fils.
  * [31] R.L. Dobrushin. Gibbs state describing coexistence of phases for a three-dimensional Ising model. Th. Probab. & Its Appl. 17 (1973), no. 4, 582–600.
  * [32] R.L. Dobrushin, R. Kotecký, and S. Shlosman. Wulff construction: a global shape from local interaction. Providence: American Math. Society. 104 (1992), x+204.
  * [33] N.P. Dolbilin, ,Y.M. Zinov’ev, A.S. Mishchenko, M.A. Shtan’ko, and M.I. Shtogrin. The two-dimensional Ising model and the Kac-Ward determinant. Izvestiya: Math. 63 (1999), no. 4, 707.
  * [34] C. Domb and M.F. Sykes. On the susceptibility of a ferromagnetic above the Curie point. Proc. Royal Soc. London. Series A. Mathematical and Physical Sciences 240 (1957), no. 1221, 214–228.
  * [35] H. Duminil-Copin. Random currents expansion of the Ising model. arXiv:1607.06933 (2016).
  * [36] H. Duminil-Copin, S. Goswami, and A. Raoufi. Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature. Commun. Math. Phys. 374 (2020), no. 2, 891–921.
  * [37] H. Duminil-Copin, C. Hongler, and P. Nolin. Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 (2011), no. 9, 1165–1198.
  * [38] H. Duminil-Copin and I. Manolescu. Planar random-cluster model: scaling relations. arXiv:2011.15090 (2020).
  * [39] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343 (2016), no. 2, 725–745.
  * [40] R.G. Edwards, and A.D. Sokal. Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and monte carlo algorithm. Phys. Rev. D 38 (1988), no. 6, 2009.
  * [41] S. El-Showk, M.F. Paulos, D. Poland, V.S. Rychkov, D. Simmons-Duffin, and A. Vichi. Solving the 3D Ising model with the conformal bootstrap. Phys. Rev. D 86 (2012), no. 2, 025022.
  * [42] J.W. Essam and M.E. Fisher. Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point. J. Chem. Phys. 38 (1963), no. 4, 802–812.
  * [43] J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor. Construction and Borel Summability of Infrared $\Phi_{4}^{4}$ by a Phase Space Expansion. Commun. Math. Phys. 109 (1987), 437–480.
  * [44] J.S. Feldman and K. Osterwalder. The Wightman axioms and the mass gap for weakly coupled $\phi_{3}^{4}$ quantum field theories. Ann. Phys. 97 (1976), no. 1, 80–135.
  * [45] M.E. Fisher. Correlation functions and the critical region of simple fluids. J. Math. Phys. 5 (1964), no. 7, 944–962.
  * [46] M.E. Fisher. On the dimer solution of planar Ising models. J. Math. Phys. 7 (1963) 1776.
  * [47] C.M. Fortuin and P.W. Kasteleyn. On the random-cluster model. I. Introduction and relation to other models. Physica 57 (1972), 536–564.
  * [48] S. Friedli and Y. Velenik. Statistical mechanics of lattice systems: a concrete mathematical introduction. Cambridge University Press (2017).
  * [49] J. Fröhlich. On the triviality of $\lambda\phi_{d}^{4}$ theories and the approach to the critical point in $d(-)>4$ dimensions. Nuclear Physics B 200 (1982), no. 2, 281–296.
  * [50] J. Fröhlich, B. Simon, and T. Spencer. Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50 (1976), no. 1, 79–95.
  * [51] K. Gawedzki and A. Kupiainen. Massless Lattice $\Phi_{4}^{4}$ Theory: Rigorous Control of a Renormalizable Asymptotically Free Model, Commun. Math. Phys. 99 (1985), 197–252.
  * [52] H.O. Georgii. Gibbs measures and phase transitions, volume 9 of de Gruyter Studies in Mathematics. Walter de Gruyter and Co., Berlin, second edition, 2011.
  * [53] A. Giuliani, R.L. Greenblatt, and V. Mastropietro. The scaling limit of the energy correlations in non-integrable Ising models. J. Math. Phys. 53 (2012), no. 9, 095214.
  * [54] J. Glimm and A. Jaffe. Positivity of the $\phi_{3}^{4}$ hamiltonian. Fortschritte der Physik. 21 (1973), no. 7, 327–376.
  * [55] J. Glimm and A. Jaffe. Quantum physics: a functional integral point of view. Springer Science & Business Media (2012).
  * [56] L. Greenberg and D. Ioffe. On an invariance principle for phase separation lines. Ann. l’IHP (B) Probab. Stat. 41 (2005), no. 5, 871–885.
  * [57] R.B. Griffiths. Correlation in Ising ferromagnets I, II. J. Math. Phys. 8 (1967), 478–489.
  * [58] R.B. Griffiths. Dependence of critical indices on a parameter. Phys. Rev. Letters 24 (1970), no. 26, 1479.
  * [59] R.B. Griffiths, C.A. Hurst, and S. Sherman. Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11 (1970), 790–795.
  * [60] R.B. Griffiths and B. Simon, The $(\Phi_{2})^{4}$ Field Theory as a Classical Ising Model, Commun. Math. Phys. 33 (1973), 145–164.
  * [61] F. Guerra, L. Rosen and B. Simon, The $P(\phi)^{2}$ Euclidean Quantum Field Theory as Classical Statistical Mechanics, Ann. of Math. 101 (1975), 111–189.
  * [62] T. Hara and H. Tasaki. A Rigorous Control of Logarithmic Corrections in Four-Dimensional $(\phi_{4})^{4}$ Spin Systems. II. Critical Behaviour of Susceptibility and Correlation Length, J. Stat. Phys. 47 (1987), no. 1/2, 99–121.
  * [63] W. Heisenberg. Zur Theorie des Ferromagnetismus. Zeitsch. für Physik, 49 (1928), no. 9, 619–636.
  * [64] Y. Higuchi. On limiting Gibbs states of the two-dimensional Ising models. Publ. Res. Instit. Math. Sciences. 14 (1978), no. 1, 53–69.
  * [65] C. Hongler. Conformal invariance of Ising model correlations. PhD thesis, université de Genève, 2010.
  * [66] C.A. Hurst and H.S. Green. New solution of the Ising problem for a rectangular lattice, J. Chem. Phys. 33 (1960) 1059.
  * [67] D. Ioffe and R.H. Schonmann. Dobrushin-Kotecký-Shlosman theorem up to the critical temperature. Commun. Math. Phys. 199 (1998), no. 1, 117–167.
  * [68] R.P. Isaacs. A finite difference function theory. Univ. Nac. Tucumán. Revista A., 2 (1941), 177–201.
  * [69] R.P. Isaacs. Monodiffric functions. Construction and applications of conformal maps. Proc. of a symp. (1941), no. 18, 257–266.
  * [70] E. Ising. Beitrag zur Theorie des Ferromagnetismus. Zeitsch. Phys., 31 (1925), 253–258.
  * [71] P. Jossang. “En Bref” de La Recherche, (1996, juillet-août), no. 289, 7 pages.
  * [72] M. Kac and J.C. Ward. A combinatorial solution of the two-dimensional Ising model, Phys. Rev. 88 (1952), 1332.
  * [73] L.P. Kadanoff. Scaling laws for Ising models near $T_{c}$. Physics Physique Fizika 2 (1966), no. 6, 263.
  * [74] L.P. Kadanoff. Critical behaviour. Universality and scaling. In Critical phenomena Ed. M.S. Green. Proceedings of the International School ‘Enrico Fermi’ 51. Italian Physical Society, 100–107. New York: Academic Press.
  * [75] L.P. Kadanoff and H. Ceva. Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B 3 (1971), no. 11, 3918.
  * [76] P.W. Kasteleyn. The statistics of dimers on a lattice. Physica 27 (1961) 1209 – Dimer statistics and phase transitions. J. Math. Phys. 4 (1963) 287.
  * [77] B. Kaufman and L. Onsager. Crystal statistics. III. Short-range order in a binary Ising lattice. Phys. Rev. 76 (1949), no. 8, 1244.
  * [78] G. Kirchhoff. Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geflührt wird. Ann. der Physik. 148 (1847), no. 12, 497–508.
  * [79] F. Kos, D. Poland, and D. Simmons-Duffin. Bootstrapping mixed correlators in the 3D Ising model. J. High Energy Phys. (2014), 1411, 109.
  * [80] H.A. Kramers and G.H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (1941), no. 3, 252.
  * [81] W. Lenz. Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern. Phys. Zeitschr. 21 (1920), 613–615.
  * [82] T.D. Lee and C.N. Yang. Statistical theory of equations of state and phase transitions. I. Theory of condensation. II. Lattice gas and Ising model. Phys. Rev. 87 (1952), no. 3, 404, 410.
  * [83] E.H. Lieb. A refinement of Simon’s correlation inequality. Commun. Math. Phys.. 77 (1980), no. 2, 127–135.
  * [84] B.M. McCoy and T.T. Wu. The two-dimensional Ising model. Harvard Univ. Press (2013).
  * [85] S. Miracle-Solé and A. Messager. Correlation functions and boundary conditions in the Ising ferromagnet. J. Stat. Phys. 17 (1977), no. 4, 245–262.
  * [86] E.W. Montroll. Statistical mechanics of nearest neighbour systems. J. of Chem. Phys. 9 (1941), no. 9, 706–721.
  * [87] M. Niss. History of the Lenz-Ising model 1920–1950: from ferromagnetic to cooperative phenomena. 1950–1965: from irrelevance to relevance. 1965–1971: the role of a simple model in understanding critical phenomena. Archive for history of exact sciences. 59 (2005), no. 3, 267-318. 63 (2009), no. 3, 267-318 243-287. 65 (2011), no. 6, 625-658.
  * [88] L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev.. 65 (1944), no. 3-4, 117.
  * [89] K. Osterwalder and R. Schrader. Axioms for Euclidean Green’s functions I. II. Comm. Math. Phys. 31 (1973) 83–112. Comm. Math. Phys. 42 (1975) 281–305.
  * [90] J. Palmer. Planar Ising Correlations. Springer Science & Business Media. 49 (2007).
  * [91] R. Peierls. On Ising’s model of ferromagnetism. Math. Proc. of the Cambridge Phil. Soc. 32 (1936), no. 3, 477–481.
  * [92] A.M. Polyakov. Conformal symmetry of critical fluctuations. JETP Lett. 12 (1970), 381–383.
  * [93] A. Raoufi. Translation-invariant Gibbs states of the Ising model: general setting. Ann. of Probab. 48 (2020), no. 2, 760–777.
  * [94] R. Rattazzi, V.S. Rychkov, E. Tonni, and A. Vichi. Bounding scalar operator dimensions in 4D CFT. J. High Energy Phys. (2008) 0812, 031.
  * [95] S. El-Showk, M.F. Paulos, D. Poland, V.S. Rychkov, D. Simmons-Duffin, and A. Vichi. Solving the 3d Ising model with the conformal bootstrap. Phys. Rev. D, 86 (2012), no. 2.
  * [96] T.D. Schultz, D.C. Mattis, and E.H. Lieb. Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36 (1964), no. 3, 856.
  * [97] P.D. Seymour and D.J.A. Welsh. Percolation probabilities on the square lattice. Ann. Discrete Math. 3 (1978), 227–245.
  * [98] D. Simmons-Duffin. The conformal bootstrap. New Frontiers in Fields and Strings: TASI 2015 Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics. (2017), 1–74.
  * [99] B. Simon and R.B. Griffiths. The $\phi^{4}_{2}$ field theory as a classical Ising model, Commun. Math. Phys. 33 (1973), no. 2, 145–164.
  * [100] S. Smirnov. Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2). 172 (2010), no. 2, 1435–1467.
  * [101] G.M. Viswanathan, M.A.G. Portillo, E.P. Raposo, M.G.E. da Luz. What does it take to solve the 3D Ising model? Minimal necessary conditions for a valid solution. arXiv.:2205.12357 (2022).
  * [102] B.L. van der Waerden. Die lange Reichweite der regelmassigen Atomanordnung in Mischkristallen. Z. Physik 118 (1941), 473–488.
  * [103] P. Weiss. L’hypothèse du champ moléculaire et la propriété ferromagnétique. J. Phys. Theor. Appl. 6 (1907), no. 1, 661–690.
  * [104] B. Widom. Equation of state in the neighborhood of the critical point. J. Chem. Phys. 43 (1965), no. 11, 3898–3905.
  * [105] A.S. Wightman. _Quantum Field Theory in Terms of Vacuum Expectation Values_. Phys. Rev. 101 (1956), 860.
  * [106] K.G. Wilson. Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture, Phys. Rev. B 4 (1971).
  * [107] C.N. Yang. The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85 (1952), no. 5, 808.


# Coherent Phonons-Driven Hot Carrier Effect in a Superlattice Solar Cell

I. Makhfudz  IM2NP, UMR CNRS 7334, Aix-Marseille Université, 13013 Marseille,
France  N. Cavassilas  IM2NP, UMR CNRS 7334, Aix-Marseille Université, 13013
Marseille, France  Y. Hajati  Department of Physics, Faculty of Science,
Shahid Chamran University of Ahvaz, 6135743135 Ahvaz, Iran  H. Esmaielpour
Walter Schottky Institut, Technische Universität München, Am Coulombwall 4,
D-85748 Garching, Germany  F. Michelini  IM2NP, UMR CNRS 7334, Aix-Marseille
Université, 13013 Marseille, France

###### Abstract

Carrier thermalization in a superlattice solar cell made of polar
semiconductors is studied theoretically by considering a minimal model where
electron-phonon scattering is the principal channel of carrier energy loss.
Importantly, the effect of an intrinsic quantum mechanical property; the
phonon coherence, on carrier thermalization is investigated, within
semiclassical picture in terms of phonon wave packet. It turns out that
coherent longitudinal optical (LO) phonons weaken the effective electron-
phonon coupling, thus supposedly lowering the carrier energy loss rate in
solar cell. The resulting thermalization power is indeed significantly reduced
by the coherent phonons, resulting in enhanced hot carrier effect,
particularly for thin enough well layer where carrier confinement is also
strong. A recent experiment on superlattice solar cell prototype is shown to
manifest the coherent phonons-driven phenomenon. Our results demonstrate the
practical implications of the fundamental quantum coherence property of
phonons in semiconductors for improving superlattice solar cell performance,
via hot carrier effect.

##  I Introduction

Superlattice has emerged as a remarkably useful semiconductor heterostructure
since its conception by L. Esaki and R. Tsu EsakiIBMjournal  and exposition by
L. Esaki and L. L. Chang of novel transport phenomenon in such artificially
made structure, such as negative differential resistance EsakiChangPRL . The
band structure of superlattice has been studied using various methods
RMPsuperlattice , consisting of mainly mini-subbands mimicking discrete energy
levels of a quantum well but each of which has a finite band width describing
the tunneling of wave function overlap between states localized at neighboring
quantum wells constituting the superlattice.

Arising as a historically imminent component in the study of semiconductor
physics and nanoelectronics TsuBook , superlattice also has immediate
practical applications, including for photovoltaic solar cell devices. In the
latter, the interest is to maximize the generated output from such solar cell
device. One of the ways in which this can be achieved is by keeping the
photogenerated charge carriers to remain energetic before their extraction. In
other words, it is necessary to keep the charge carriers “hot”, hence the so-
called hot carrier effect Conibeer1 -JShah , emerging as a leading strategy to
increase solar cell efficiency. Recent work by us PRapplied  found that a
quantum well heterostructure significantly reduces the thermalization power,
equivalent to energy loss rate, of the photogenerated charge carriers, leading
to hot carrier effect in such quantum well or ultra-thin solar cells
PRappliedCavassilas .

Motivated by existing works BarnhamDuggan -EsmaielpourAPL  to provide a
microscopic theory of carrier thermalization in a superlattice solar cell, we
extend our study on a quantum well solar cell to a superlattice-based solar
cell, focusing on the role of electron-phonon-induced scattering. Our present
work in particular addresses the quantum coherence of phonons and its effect
on carrier thermalization in superlattice solar cell, within the framework of
semi-classical model of kinetic theory which incorporates the effects of
quantum confinement, tunneling, and coherence, in a model with phonon band
folding folding Kleinreview . The investigation of carrier thermalization is
critical to the study of hot carrier effect Conibeer1 -JShah , by which the
efficiency of the solar cell can be improved RossNozik  beyond the
paradigmatic Shockley-Queisser theoretical limit ShockleyQueisser . Our theory
will show, first, that phonon coherence reduces the effective electron-phonon
coupling, and second, that when phonons are coherent and the carriers are
highly confined, carrier thermalization is strongly suppressed, driving a
significant hot carrier effect, thus offering an alternative method to enhance
solar cell performance.

The paper starts with the Hamiltonian that describes the carriers (electrons),
the phonons, and the interactions between them. The concept of coherent
phonons is then discussed and its role on modifying the electron-phonon
coupling is elucidated. The resulting electron-phonon scattering rate in the
superlattice with coherent phonon effect is derived. The following section
analyzes the thermalization power that measures the rate of electron energy
loss via electron-phonon scattering. The theoretical prediction on
thermalization power is compared with experimental data, indicating
manifestation of coherent phonons-driven hot carrier effect. The paper ends
with discussion and conclusion.

##  II Model Hamiltonian

###  II.1 Electron Hamiltonian

With a profile of superlattice as shown in Fig. 1, we consider a tight-binding
Hamiltonian written in the basis of states localized in the well
$|\alpha=1,2,3,\cdots\rangle$ with tunneling only between adjacent wells
separated by a barrier. In this case, the Hamiltonian in the (first-quantized)
matrix form can be written as

$H_{n}=\sum_{\alpha\beta}E_{n}|\alpha\rangle\langle\beta|\delta_{\alpha\beta}-V_{n}|\alpha\rangle\langle\beta|(\delta_{\alpha,\beta-1}+\delta_{\alpha,\beta+1})$
(1)

where the $\alpha,\beta$ represent the well indices while the subscript
$n=1,\cdots,N_{s}$ represents the $n^{\mathrm{th}}$ subband of $N_{s}$
confined states of carrier inside the well. The $E_{n}$ are the energy level
of the $n^{\mathrm{th}}$ subband of the individual well which for a finite
potential well is given by

$E_{n}(k_{\perp})=\frac{\hbar^{2}k^{2}_{\perp}}{2m_{c}}+E_{n}(a)$ (2)

where $\hbar$ is the Dirac (reduced Planck) constant, $k_{\perp}$ is the
carrier’s transverse wave vector, $m_{c}$ is the carrier mass, $E_{n}(a)$ is
the $n^{\mathrm{th}}$ quantized energy level of a finite potential well of
width $a$ and depth $V_{c}$ in Fig. 1 while $V_{n}$ is a phenomenological
tunneling integral which depends on $n$. In this work, only electron processes
will be of interest and “carrier” thus refers to electron. To be concrete, we
will consider an InAs(well)/AlAsSb(barrier) superlattice, with material
parameters given in SupplementaryMaterials .

The solution of Schr odinger equation for this Hamiltonian in terms of
eigenvalues is given by

$E_{n}(\mathbf{k})=E_{n}(k_{\perp})-2V_{n}\cos k_{z}(a+b)$ (3)

giving rise to a “miniband” of width $4V_{n}$. Noting that the band width of
the $n^{\mathrm{th}}$ mini-subband is $4V_{n}$, in this work the magnitude of
$V_{n}$ in Eq.(1) is to be approximated as $1/2$ of the energy splitting
between two energy levels belonging to $n^{\mathrm{th}}$-subband in a
numerically exact Hamiltonian diagonalization calculation of a system of two
quantum wells each of thickness $a$ separated by a barrier of thickness $b$.
In the energy dispersion Eq.(3), $k_{z}$ is the wave vector of the electron
traversing the superlattice; $k_{z}\in[-\pi/(a+b),\pi/(a+b)]$.

Figure 1:  The profile of a superlattice consisting of wells of width $a$ and
barriers of width $b$ with the horizontal dashed line serving as spatial
coordinate axes.

The corresponding eigenfunction is

$\psi_{n\mathbf{k}}(z)=e^{i\mathbf{k}_{\perp}\cdot\mathbf{r}_{\perp}}\frac{1}{\sqrt{N}}\sum^{N}_{\alpha=1}e^{ik_{z}\alpha(a+b)}\phi_{n}(z-\alpha(a+b))$
(4)

where $N$ is the number of periods (one period contains one well and one
barrier) in the superlattice and $\phi_{n}(z)$ is the eigenfunction of the
$n^{\mathrm{th}}$ subband (confined electron state) of the individual well. In
the rest of this text, $n$ represents the $n^{\mathrm{th}}$ eigenstate of the
electron confined in the well whereas $\alpha$ represents the
$\alpha^{\mathrm{th}}$ quantum well.

###  II.2 Electron-Phonon Hamiltonian

For the electron-phonon Hamiltonian, we will assume Frochlich type of
Hamiltonian with 3D (bulk) phonon, with phonon band folding eventually taken
into account in the calculation. In polar semiconductors, optical phonons play
the key role in carrier thermalization. In this case, the electron-phonon
coupling Hamiltonian in second quantized form can be written as

$H_{e-phon}=\sum_{\mathbf{k},n;\mathbf{k}^{\prime},n^{\prime}}\sum_{\mathbf{q}}a^{{\dagger}}_{\mathbf{k}n}a_{\mathbf{k}^{\prime}n^{\prime}}(b_{\mathbf{q}}-b^{{\dagger}}_{-\mathbf{q}})M_{\mathbf{q}}G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})$
(5)

where $a^{{\dagger}}_{\mathbf{k}n}(a_{\mathbf{k}^{\prime}n^{\prime}})$ are the
electron creation (annihilation) operator for subband $n(n^{\prime})$ at wave
vector $\mathbf{k}(\mathbf{k}^{\prime})$ while
$b^{{\dagger}}_{-\mathbf{q}}(b_{\mathbf{q}})$ are the corresponding operators
of wave vector $\mathbf{q}$ for longitudinal optical (LO) phonon. The
electron-phonon coupling function $M_{\mathbf{q}}$ is given by

$M_{\mathbf{q}}=\sqrt{\frac{\hbar\omega_{LO}e^{2}}{8\pi
q^{2}V}\left(\frac{1}{\varepsilon_{\infty}}-\frac{1}{\varepsilon_{0}}\right)}$
(6)

where $\omega_{LO},e,V,\varepsilon_{\infty},\varepsilon_{0}$ are respectively
the LO phonon frequency, elementary charge, phonon spatial volume, infinite
frequency and static dielectric constants, while
$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})$ is the overlap
integral given by

$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})=\int
d^{3}r\psi^{*}_{n\mathbf{k}}(\mathbf{r}_{\perp},z)e^{i(q_{z}z+\mathbf{q}_{\perp}\cdot\mathbf{r}_{\perp})}\psi_{n^{\prime}\mathbf{k}^{\prime}}(\mathbf{r}_{\perp},z)$
(7)

where, in conjunction with Eq.(4), we will use

$\phi_{n}(z)=\sqrt{\frac{2}{a}}\cos\frac{n\pi z}{a}$ (8)

from the eigenstate of an infinite potential well, as an analytical wave
function approximating the actual wave function of the confined electron in
the well layer. The detailed analytical expression for the resulting overlap
integral form factor is given in SupplementaryMaterials .

In semiconductor superlattice, linear-chain model calculation shows that
optical phonon band structure is folded into a smaller Brillouin zone,
corresponding to confinement of the optical phonons into the well layer
foldingS . The resulting band dispersion within this mini Brillouin zone
consists of zig-zag dispersion for the acoustic phonon and a close ensemble of
flat bands for optical phonon foldingS KleinreviewS . The energy dispersion of
the longitudinal optical phonon becomes qualitatively the same as that of the
bulk phonon; the only difference being that the Brillouin zone is folded from
$|\mathbf{q}|\in[0,\pi/a_{\mathrm{lattice}}]$ for bulk optical phonon to
$|\mathbf{q}|\in[0,\pi/(a+b)]$ for optical phonon in superlattice. In a
semiclassical description for superlattice, one can thus use the plane-wave
function $\exp(iq_{z}z)$ but with $q_{z}\in[0,\pi/(a+b)]$, as implemented in
the present work.

###  II.3 Phonon Hamiltonian

The phonon Hamiltonian consists of the non-interacting phonon Hamiltonian in
terms of harmonic oscillators. In addition, our model also includes phonon
anharmonicity RidleyBook Klemens , that is, phonon-phonon interaction
describing the transformation of an LO phonon to two acoustic phonons (1 and
2)RidleyBook . The acoustic phonons are assumed to be always in equilibrium at
lattice temperature $T_{L}$ while the LO phonons may go out of equilibrium,
corresponding to the non-equilibrium hot phonon effect, that may play a key
role in solar cell physics Tsai . The simplest model is employed; the phonon-
phonon scattering is described within relaxation time approximation where the
scattering time is assumed to be a constant independent of wave vector

$\tau^{LO-ac}_{\mathbf{q}}=\frac{\tau^{LO-ac}_{0}}{N_{1}+N_{2}+1}$ (9)

where $\tau^{LO-ac}_{0}$ is the $T=0$K value of $\tau^{LO-ac}_{\mathbf{q}}$,
$N_{1(2)}=(\exp(E_{1(2)}/(k_{B}T_{L}))-1)^{-1}$ of the two acoustic phonons of
energies $E_{1}$ and $E_{2}$, with $k_{B}$ the Boltzmann constant, and the
lattice temperature $T_{L}$. This description for phonon-phonon scattering
time is assumed to hold even with phonon band folding, justified in a model
where phonon-phonon coupling depends only on the energies of the involved
phonons Klemens  with equal partition of the energy of an LO phonon to the two
acoustic phonons, that is, $E_{1}=E_{2}=E_{0}/2$ where
$E_{0}=\hbar\omega_{LO}$ is the LO phonon energy. The optical phonons are
effectively confined or quasi-confined in the wells, because the optical
phonons encounter and sense the abrupt change in the degree of polarizability
at the boundary between the two polar materials, which effectively serves as a
confining wall for the optical phonons. The existence of such confined or
quasi-confined optical phonon modes is well established in literature, even
for a superlattice made of similar materials (for example, two polar
semiconductors of III-V family, as considered in this work) due to the
sensitivity of the optical phonons to the difference in materials’
polarizabilities, leading to quasi-confined optical phonon modes StroscioDutta
. As a final note, while the electron Hamiltonian, electron-phonon
Hamiltonian, and phonon-phonon interaction are respectively written in first
quantized, second-quantized and phenomenological forms, their treatment is
unified in the semiclassical theory that is developed in this work.

##  III Coherent Phonons

Phonons are quantum particle excitations of lattice vibrations. As such,
phonons posses some intrinsically quantum mechanical properties, such as
bosonic quantum statistics and can also be in coherent state. Phonons are
coherent when there is a well-defined phase relationship between vibrations of
atoms separated by distance shorter than or equal to the phonon coherence
length $l_{c}$ VolzPRB . The study of phonon coherence on quantum transport in
semiconductor heterostructures has remained important research problem over
decades to recent years MahanPRL -KomirenkoPRB .

In characterizing the coherence property of phonons, one can speak of spatial
coherence and temporal coherence. The spatial coherence of phonon can be
described in terms of phonon wave packet. Indeed, intuitively speaking,
coherence can be associated with a wave-like picture of phonons, just like the
wave function description of quantum mechanics. Conversely, decoherence which
would give classical picture of phonons, can be associated with description of
phonons as (classical) particle. The atomic vibrations have well-defined phase
relationship (that is, coherent) when they are located within the envelope of
the wave packet peak. Outside this envelope, their vibrations are random, and
their average displacement (averaged over the atoms) vanish. The semiclassical
framework in terms of wave packet describes the phonon in terms of the
displacement function of the atoms in a one-dimensional atomic chain treatment
of phonons, rather than in terms of the eigenfunctions of quantized
Hamiltonian for phonons.

In addition to spatial coherence, in general, phonon also has temporal
coherence. In the literature, the phonon temporal coherence has been discussed
in terms of fully quantum mechanical description involving eigenstates of
harmonic oscillator model of phonon and the so-called phonon coherent states
StantonPRL . The phonon temporal coherence is however less relevant in the
present work because we assume Frohlich type of electron-phonon coupling
Hamiltonian that does not depend on time. In addition, our theory eventually
pertains only to steady-state situation of carrier thermalization process
where no explicit time dependence shows up. Furthermore, as shown in
LatourChalopinPRB , phonon temporal coherence corresponds to its mean free
path. Longitudinal optical (LO) phonons on the other hand have nearly-flat
energy dispersion. In fact, in our calculation, we have approximated the LO
phonons dispersion to be a constant, wave vector-independent energy dispersion
$\hbar\omega_{\mathrm{LO}}=30$ meV. As such, the effective group velocity of
LO phonons is zero; they do not propagate. As a result, in this description,
the LO phonons do not propagate and their mean-free path vanishes. Therefore,
the phonon temporal coherence length is irrelevant in this case; the LO
phonons are completely incoherent in time.

In photovoltaic research, the importance of this fundamental property of
phonon coherence has not been explored, despite the crucial role of phonons on
carrier energy loss in a solar cell. The question that we aim to answer is the
following: What is the effect of phonon coherence on carrier thermalization
via electron-phonon scattering? To this end, we develop a semiclassical
picture in terms of phonon wave packet where the atoms vibrate in phase when
they are within the length scale (the “width”) of the wave packet relative to
the center of the latter, which defines $l_{c}$ as the phonon coherence
length. In the present work, only coherence of LO phonons is considered;
acoustic phonons are assumed to remain incoherent. Furthermore, only spatial
coherence is relevant in this case, but the temporal coherence is not, because
while the former corresponds to spatial localization, the latter corresponds
to mean free path LatourChalopinPRB , which vanishes for optical phonons
because they do not propagate.

For simplicity, we will assume a one-dimensional (1D) wave packet along $z$,
described in terms of the displacement function in a 1D (atom chain) model in
the classical treatment of phonon, as given by

$u^{\prime}_{q_{z}}(z,t)=W^{\prime}_{q_{z}}e^{i(q_{z}z-\omega_{q_{z}}t)}$ (10)

where $W^{\prime}_{q_{z}}$ is the weight function of the mode $q_{z}$ of the
phonon. This weight $W^{\prime}_{q_{z}}$ depends on the initial condition
$u^{\prime}(z,0)=(1/\sqrt{\pi}q_{UV})\int^{q_{UV}}_{0}dq_{z}u^{\prime}_{q_{z}}(z,0)$
where $q_{UV}=\pi/(a+b)$. To describe a phonon wave packet of coherence length
$l_{c}$ localized at the center of each well (e.g. $z=0$ for the central well
in Fig. 1), we employ a Gaussian wave packet of half width $l_{c}$, described
by the wave function

$u^{\prime}(z,0)=\frac{1}{q_{UV}}\int^{q_{UV}}_{0}dq_{z}\tilde{W}^{\prime}_{q_{z}}\frac{a}{2}e^{iq_{z}z}=\frac{a}{2}e^{-\frac{z^{2}}{l^{2}_{c}}}$
(11)

where $\tilde{W}^{\prime}_{q_{z}}$ is a dimensionless weight factor to be
defined below, as the initial condition, chosen in such a way that the atom at
$z=0$ gets displaced by half thickness of the well layer (one may consider
normally smaller and more realistic amplitude, half of lattice spacing
$a_{\mathrm{lattice}}$ for example; the final result for weight function is
independent of this amplitude in displacement). We will work with a slightly
modified weight function given by

$W_{q_{z}}=\frac{1}{l_{c}}\int^{\frac{a}{2}}_{-\frac{a}{2}}u^{\prime}(z,0)e^{-iq_{z}z}dz=\frac{a}{2}\sqrt{\pi}\tilde{W}_{q_{z}}$
$=\frac{a}{4}e^{-\frac{1}{4}l^{2}_{c}q^{2}_{z}}\sqrt{\pi}\left(\mathrm{Erf}\left[\frac{(a-il^{2}_{c}q_{z})}{2l_{c}}\right]+\mathrm{Erf}\left[\frac{(a+il^{2}_{c}q_{z})}{2l_{c}}\right]\right)$
(12)

where $\mathrm{Erf}$ is error function and $\tilde{W}_{q_{z}}$ is defined such
that $\mathrm{limit}_{l_{c}\rightarrow 0}\tilde{W}_{q_{z}}=1$. This is
consistent with the constraint that incoherent phonon case corresponds to
$\tilde{W}_{q_{z}}=1$ while giving a generally complex-valued real space
function $u(z,0)$ given by Eq.(11) but with substitution
$\tilde{W}^{\prime}_{q_{z}}\rightarrow\tilde{W}_{q_{z}}$, as illustrated in
Fig. 2A), where only the real part of $u(z,0)$ is plotted as it corresponds to
physical ion displacement while the imaginary part of $u(z,0)$ does not. While
the $\mathrm{Re}[u(z,0)]$ no longer looks like a Gaussian function of half
width $l_{c}$, it still satisfies the physically intuitive picture that this
function approaches more closely a Dirac delta function in real space
($\delta(z)$) as $l_{c}\rightarrow 0$ as its amplitude is largest in this
limit. The immediate role of this coherence factor is that it leads to
replacement

$e^{iq_{z}z}\rightarrow\tilde{W}_{q_{z}}e^{iq_{z}z}$ (13)

in the overlap integrals in Eq.(7). The phonon coherence therefore manifests
simply by modifying the effective electron-phonon coupling in Eq.(5).

Our phenomenological theory of wave packet to describe coherent phonons can
also be deduced by comparison with a more rigorous microscopic theory Lax
-LaxPRB87  formulated to construct a phonon wave function spatially localized
in a well, properly adapted to define phonon coherence as conceptually
proposed in VolzPRB . The principal result of those studies Lax -LaxPRB87
gives the phonon wave packet operator Lax

$b_{\mathbf{q}_{\perp},q_{z}}\rightarrow\tilde{b}_{n^{\prime}n}(\mathbf{q}_{\perp})\sim\sum_{q_{z}}G_{n^{\prime}n}(\mathbf{q}_{\perp},q_{z})b_{\mathbf{q}_{\perp},q_{z}}$
(14)

where $n,n^{\prime}=1,\cdots,N_{s}$ and
$G_{n^{\prime}n}(\mathbf{q}_{\perp},q_{z})$ takes particularly simple form for
$n=n^{\prime}=1$

$G_{11}(\mathbf{q}_{\perp},q_{z})=\frac{1}{\sqrt{L}}\int^{\infty}_{-\infty}|\zeta_{1}(z)|^{2}dz\frac{\exp(iq_{z}z)}{\sqrt{q^{2}_{\perp}+q^{2}_{z}}}.$
(15)

The wave packet thus indeed manifests as an effective weight factor
($\tilde{W}_{q}=1/\left(\sqrt{L}\sqrt{q^{2}_{\perp}+q^{2}_{z}}\right)$ in the
context of phonon localization in the well of thickness $L$ Lax ) multiplying
the factor $\exp(iq_{z}z)$ or, equivalently, the Frohlich electron-phonon
coupling in Eq.(5), justifying our semiclassical picture.

The phonon spatial coherence enters our semiclassical theory in terms of the
weight factor $\tilde{W}_{q_{z}}$ given in Eq.(12). Our analysis shows that
having incoherent phonons gives weight factor equal unity independent of
$q_{z}$ while having coherent phonons gives smaller weight factor, as
illustrated in Fig. 2. This means coherent phonons gives weaker effective
electron-phonon interaction. With this formulation, the overlap integrals in
Eq.(7) correspond to the case with vanishing phonon coherence length
$l_{c}\rightarrow 0$, that is, the case with completely incoherent phonons,
which leads to constant weight factor $\tilde{W}_{q_{z}}=1$. Having coherent
phonons implies large enough $l_{c}$(relative to the system’s characteristic
length scale), leading to $q_{z}$-dependent weight factor $\tilde{W}_{q_{z}}$
with profile illustrated in Fig. 3. Comparing the blue and light yellow
regions suggests that having coherent LO phonons weakens electron-phonon
scattering and should thus reduce carrier thermalization. This is one of the
key results of our work.

Figure 2:  The profile of (A)the wave packet
$\mathrm{Re}[u(z,0)]=(1/q_{UV})\int^{q_{UV}}_{0}dq_{z}\tilde{W}_{q_{z}}(a/2)\exp(iq_{z}z)$
where $q_{UV}=\pi/(a+b)$ with $a=5$nm, $b=1$nm and (B) the corresponding
weight factor $\tilde{W}_{q_{z}}$ for the case with incoherent phonon
(red)($l_{c}=0.000001$nm), weakly coherent (blue)($l_{c}=1$nm), and strongly
coherent phonon (green) ($l_{c}=10$nm). The vertical dashed line delimits the
folded phonon Brillouin zone. Figure 3:  The contour plot of
$|\tilde{W}_{q_{z}}|$ (with $a=5$nm) dissecting regions varying from coherent
(blue) to incoherent (light yellow) phonons, indicating that coherent phonons
give smaller effective electron-phonon coupling due to smaller
$|\tilde{W}_{q_{z}}|$.

##  IV Scattering Rate Calculation

The electron thermalization rate is determined in terms of thermalization
power, given by the rate of energy transferred by the electrons to the
longitudinal optical (LO) phonons emitted by the electrons. To that end, we
compute the rate of change of the number of LO phonons,

$\frac{dN_{{\bf q}_{\perp},q_{z}}}{dt}=\frac{2\pi}{\hbar}|M_{{\bf
q}_{\perp},q_{z}}|^{2}A_{\mathbf{q}_{\perp},q_{z}}$ (16)

where

$A_{\mathbf{q}_{\perp},q_{z}}=2\sum_{\mathbf{k}_{\perp}}\sum_{k_{z},k^{\prime}_{z}}I^{2}_{\mathbf{k},\mathbf{k}-\mathbf{q}}((N_{\mathbf{q}_{\perp},q_{z}}+1)f_{\mathbf{k}_{\perp},k_{z}}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z}})$
$-N_{\mathbf{q}_{\perp},q_{z}}f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z}}(1-f_{\mathbf{k}_{\perp},k_{z}}))\delta(\Delta
E_{n\mathbf{k};n^{\prime}\mathbf{k}^{\prime},\mathbf{q}}-\hbar\omega_{\mathbf{q}_{\perp},q_{z}})$
(17)

where $\Delta
E_{n\mathbf{k};n^{\prime}\mathbf{k}^{\prime},\mathbf{q}}=E_{n}(\mathbf{k}_{\perp},k_{z})-E_{n^{\prime}}(\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z})$,
${\bf k}=({\bf k}_{\perp},k_{z})$,
$I^{2}_{\mathbf{k},\mathbf{k}-\mathbf{q}}=|G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})|^{2}|\tilde{W}_{q_{z}}|^{2}$,
$\omega_{\mathbf{q}_{\perp},q_{z}}=\omega_{LO}$, and
$f_{\zeta}=(\exp((E_{\zeta}-\mu_{c})/(k_{B}T_{c})+1)^{-1}$; the Fermi-Dirac
distribution of electron of wave vector $\zeta$ having energy $E_{\zeta}$ at
temperature $T_{c}$ and chemical potential $\mu_{c}$.

As shown in SupplementaryMaterials , in the limit of large $N$ (the number of
periods of the superlattice), the form factor asymptotically approaches a
Dirac delta function and can be approximated as

$|G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})|^{2}=C_{N}(n,n^{\prime},k_{z},q_{z})\delta(\mathbf{k}^{\prime}-\mathbf{k}+\mathbf{q})$
(18)

where the function $C_{N}(n,n^{\prime},k_{z},q_{z})$ is given in
SupplementaryMaterials . This Dirac delta function-type of form factor implies
that the Bloch wave vector is also conserved along the transport direction of
the superlattice, due to the macroscopic translational invariance arising from
the superlattice periodicity. Using the Dirac delta function form factor in
Eq.(18), the $dN_{\mathbf{q}_{\perp},q_{z}}/dt$ becomes

$\frac{dN_{\mathbf{q}_{\perp},q_{z}}}{dt}=\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}}}{\tau^{c-LO}_{\mathbf{q}}}=(N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}})$
$\times\left[2N\frac{1}{2\pi\hbar}\left(\frac{e^{2}\hbar\omega_{\mathbf{q}}}{q_{\perp}}\left[\frac{1}{\varepsilon_{\infty}}-\frac{1}{\varepsilon_{s}}\right]\frac{1}{8\pi
q^{2}V}\right)\right]\frac{2m_{c}}{\hbar^{2}}\frac{S}{(2\pi)^{2}}$
$\times(|\tilde{W}_{q_{z}}|)^{2}(a+b)^{2}\sum_{n,n^{\prime}}\int^{\frac{\pi}{(a+b)}+q_{z}}_{-\frac{\pi}{(a+b)}+q_{z}}dk_{z}C_{N}(n,n^{\prime},k_{z},q_{z})$
$\times\frac{1}{2}\frac{\sqrt{2m_{c}k_{B}T_{c}}}{\hbar}\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}(Li_{1/2}(-e^{x_{2}})-Li_{1/2}(-e^{x_{1}}))$
(19)

where
$N_{\mathbf{q}}(T_{c})=(\exp(\hbar\omega_{\mathbf{q}}/(k_{B}T_{c}))-1)^{-1}$,
$\omega_{\mathbf{q}}=\omega_{LO}$, $T_{c}$ is the carrier (electron)
temperature, $S$ is the transverse cross sectional area of the superlattice,
$\Gamma(x)$ is gamma function, and $Li_{\alpha}(x)$ is polylogarithmic
function of order $\alpha$ while $x_{1}$ and $x_{2}$ are given by

$x_{1}=-\beta\Bigl{[}\frac{\hbar^{2}}{2m_{c}}(k^{min}_{\perp})^{2}+E_{n^{\prime}}(a)-2V_{n^{\prime}}\cos(k_{z}-q_{z})(a+b)\Big{]}$
$+\beta(\mu_{c}+\hbar\omega_{LO}),$ (20)
$x_{2}=-\beta\Bigl{[}\frac{\hbar^{2}}{2m_{c}}(k^{min}_{\perp})^{2}+E_{n}(a)-2V_{n}\cos
k_{z}(a+b)-\mu_{c}\Big{]}$ (21)

where $k^{min}_{\perp}$ is the minimum transverse wave vector an electron must
posses to be able to emit an LO phonon,

$k^{\mathrm{min}}_{\perp}(n,n^{\prime})=\frac{q_{\perp}}{2}+\frac{(E_{n^{\prime}}(a)-E_{n}(a))}{2q_{\perp}}\frac{2m_{c}}{\hbar^{2}}+\frac{m_{c}\omega_{q_{\perp},q_{z}}}{\hbar
q_{\perp}}$ $-\frac{m_{c}}{\hbar^{2}q_{\perp}}\left(2V_{n^{\prime}}\cos
k^{\prime}_{z}(a+b)-2V_{n}\cos k_{z}(a+b)\right)$ (22)

and $\mu_{c}=\mu_{e}$ the electron chemical potential. One can see that the
scattering rate is proportional to $N$; the number of periods in the
superlattice. Only the scattering rate per period (consisting of one well and
one barrier) will be concerned.

##  V Thermalization Power

The scattering rate is used to compute the thermalization power, defined as
the power needed to keep the electrons at a given effective temperature
$T_{c}$. In steady state, this quantity equals the rate of energy lost by the
electrons to LO phonons, computed from $dN_{\mathbf{q}_{\perp},q_{z}}/dt$ in
Eqs.(19). The thermalization power (per well) is given by

$P_{\mathrm{th}}=a\int\frac{d^{3}q}{(2\pi)^{3}}\hbar\omega_{\mathbf{q}}\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}}(T_{L})}{\tau^{c-LO}_{\mathbf{q}}+\tau^{LO-
ac}_{\mathbf{q}}}$ (23)

in units of Watt/cm2 describing the rate of energy loss of the electrons from
emitting LO phonons within the well layer. The form of thermalization power in
Eq.(23) reflects non-equilibrium hot LO phonon effect in steady-state Tsai .

The numerical results for electron thermalization power $P_{\mathrm{th}}$ and
its dependencies on well layer and barrier layer thicknesses are presented in
Fig. 4, for incoherent and coherent phonons. Numerical simulation demonstrates
that $l_{c}$ for optical modes is dependent on the superlattice period
$d_{SL}=a+b$, with “leading order” dependence of the form $l_{c}=\xi d_{SL}$
where $\xi$ is a (numerically-computed) function of energy of the mode and
$d_{SL}$ itself LatourChalopinPRB ; we take $\xi=2$ for simple illustration
(as we fix the energy to $E_{0}$, the actual $\alpha$ should in principle
still have some dependence on $d_{SL}$). More generally, coherent phonons
correspond to $\xi\geq 1$ LatourChalopinPRB  and our results apply as long as
this constraint is satisfied. Fig. 4A) demonstrates that in a superlattice
heterostructure, the thermalization power decreases with the barrier thickness
increase. This is because with thicker barrier, the electron is more confined,
and stronger confinement reduces carrier thermalization, as is the case as
well with a quantum well PRapplied . The hot carrier effect is thus enhanced
in a superlattice structure as one increases the barrier thickness because the
power needed to sustain a given electron effective temperature is lower, at
larger barrier thickness.

Figure 4:  The theoretical dependencies of the thermalization power (in W/cm2)
per well of the superlattice: (A) $P_{\mathrm{th}}$ as a function of barrier
thickness $b$ at fixed well layer thickness $a=4$ nm (chosen to give two mini-
subbands for an InAs/AlAsSb superlattice) in terms of $P_{\mathrm{th}}$ (B)
$P_{\mathrm{th}}$ as a function of well thickness $a$ at fixed barrier
thickness $b=1$ nm, for LO phonon coherence length $l_{c}=10^{-6}$nm
(red)(incoherent phonons) and $l_{c}=2(a+b)$nm (black)(coherent phonons). All
other parameter values are given in SupplementaryMaterials . The dashed lines
serving as guides to the eye are obtained from interpolations of data points
marked by the circles.

Fig. 4A) also indicates that coherent phonons lower the thermalization power
(by as much as
$(P_{\mathrm{th}}(\mathrm{incoh})-P_{\mathrm{th}}(\mathrm{coh}))/P_{\mathrm{th}}(\mathrm{incoh})\simeq
62.8\%$ for 4 nm well layer thickness and barrier layer thickness of 1 nm in
an InAs/AlAsSb superlattice) and thus strongly enhances the hot carrier
effect. Fig. 4B) also demonstrates that thermalization power is strongly
suppressed (hot carrier effect is significantly enhanced) by coherent phonons,
especially for thin enough well layer. A critical well layer thickness can be
defined (about 4 nm in our example of InAs/AlAsSb superlattice, corresponding
to 2 mini-subbands); for thinner wells, coherent phonons can strongly reduce
the thermalization power (by one order of magnitude or more for well layer
thickness of 3 nm or smaller), while for thicker wells, phonon coherence
effect is not as strong. This strong enhancement of hot carrier effect at
small well layer thickness $a$(and thus small $d_{SL}$ for fixed $b$) is
furthermore enforced by the fact that the actual numerically-determined
$\alpha$ turns out to be larger at smaller $d_{SL}$LatourChalopinPRB ; the
black curve would therefore be even more lowered for small $a$ while raised
for large $a$. The hot carrier effect in superlattice solar cell is thus
maximally enhanced when electrons are strongly confined and the LO phonons are
coherent at the same time. This is another key result of our paper.

Figure 5:  The normalized thermalization power dependence on barrier thickness
from our theory with coherent phonons of $l_{c}=2d_{SL}=2(a+b)$ (black),
incoherent phonons ($l_{c}=10^{-6}$nm)(red) and experiment
(orange)EsmaielpourAPL . Here, $P_{\mathrm{th0}}=P_{\mathrm{th}}(b\gg a)$.

##  VI Comparison with Experiment

In this section, we provide evidence that the coherent phonons-driven hot
carrier effect predicted in the present work has in fact been observed in an
experiment on a superlattice solar cell prototype studied in EsmaielpourAPL .

Our theoretical result showing that the thermalization power at fixed well
layer thickness decreases with the increase of barrier thickness, as shown in
Fig. 4A), agrees with experimental observation EsmaielpourAPL . However, the
amount of the decrease of $P_{\mathrm{th}}$ with $b$ at fixed $a$ for the case
with incoherent phonon, as shown by the red curve in Fig. 4A), while in accord
with expectation, is very weak, being at least one order of magnitude smaller
than its “principal value”, and also weaker than that observed in experiment
EsmaielpourAPL . Noting from Eq.(12) that the phonon coherence-induced weight
factor $\tilde{W}_{q_{z}}$ has no explicit dependence on barrier thickness
$b$, this simply implies that electron confinement effect alone cannot account
for the decrease in thermalization power with barrier thickness and confirms
that the phonon coherence length $l_{c}$ must indeed depend on the barrier
thickness. Our result thus verifies existing theoretical works that show that
the phonon coherence length $l_{c}$ in a superlattice is dependent on its
period $d_{SL}=a+b$ VolzPRB LatourChalopinPRB ; more precisely, for optical
mode, $l_{c}$ is dependent on the superlattice period $l_{c}\gtrsim
d_{SL}=(a+b)$ LatourChalopinPRB .

To make a more quantitative comparison with the experiment, the thermalization
power $P_{\mathrm{th}}$ is computed at a fixed well layer thickness $a=2$nm
and three barrier layer thicknesses $b$’s used in EsmaielpourAPL . Just like
in previous section, $l_{c}=2(a+b)$ is used for illustration, corresponding to
$\xi=2$, indicating rather strongly coherent phonons. A quantity called
“normalized thermalization power” is defined, given by
$P_{\mathrm{th}}/P_{\mathrm{th0}}-1$, where
$P_{\mathrm{th0}}=P_{\mathrm{th}}(b\gg a)$, to compare theory and experiment.
The reason of doing this is at least two folds; first our theory is derived
for infinitely long superlattice $N\rightarrow\infty$ while available
experiments clearly studied finite-length superlattice ($N$ is finite).
Second, our theory only considers the so-called “‘volumic” contribution to
thermalization power whereas a full description of experimental system should
cover both volumic and surfacial contributions. Therefore, we cannot compare
the absolute values of thermalization between theory and experiment. Instead,
the normalized thermalization power as defined earlier better captures the
dependence of thermalization power on barrier layer thickness, excluding the
surface contribution, which is normally additive GiteauJAP  and minimizing
finite-size($N$) effect.

As demonstrated in Fig. 5, the decrease in thermalization power with barrier
thickness observed in EsmaielpourAPL  cannot be reproduced (in the strength of
its dependence on barrier thickness) by assuming incoherent phonons, while
assuming coherent phonons (with $l_{c}=2(a+b)$ for illustration) does reflect
the experimental data reasonably well, with residual discrepancy that may
originate from simplifications involved our theory. In other words, our theory
suggests that the hot carrier effect observed in the superlattice
heterostructure for solar cell in EsmaielpourAPL  is largely driven by
coherent phonons, which are activated by the laser irradiation. Altogether,
this result indicates that, while both electron confinement and phonon
coherence leads to reduction of thermalization power, and thus enhancement of
hot carrier effect, the contribution of phonon coherence can in fact be
significant or even dominant, under steady-state irradiation situation, as
implemented in EsmaielpourAPL  and as is also the standard condition in solar
cell operation.

##  VII Discussion and Conclusions

Our work presents a minimal model to describe carrier thermalization in a
superlattice solar cell with electron-phonon scattering as the main channel of
electron energy loss. The role of an inherently quantum mechanical property of
phonons; the quantum coherence, is unraveled, and its impact on the carrier
thermalization, a process that crucially determines the solar cell efficiency,
is investigated. Our results provide a proof of principle that having coherent
phonons would enhance hot carrier effect in superlattice solar cell and,
prospectively, its efficiency, complementing also the use of photon quantum
coherence ScullyPRL  to achieve the same objective. From the phonon’s
perspective, our finding that longer phonon coherence length gives lower
thermalization power implies longer phonon life time, a conclusion that fully
agrees with existing works GargChen AdvancesinPhysics . Our analysis suggests
that the hot carrier effect observed in EsmaielpourAPL  is largely driven by
coherent phonons, indicating that phonon coherence can have dominant
contribution in suppressing carrier thermalization under steady-state
irradiation as in standard solar cell operation.

Our calculations implicitly assumed that the LO phonons are confined (or
localized) in the wells, periodically arranged along the superlattice. This is
justified by the fact that the difference in the polarizabilities of the
semiconductor materials that constitute the well and barrier layers acts as
phononic confining potential for the optical phonons. While this assumption
simplifies the calculations, the critical final results of coherent phonons-
driven hot carrier effect does not depend on this confined LO phonons
assumption. In fact, an alternative scenario with extended LO phonons picture
would still preserve the coherent phonons-driven hot effect, as can be deduced
by making comparison to the study on a quantum well solar cell PRapplied . The
acoustic phonons, which are indeed not polar and therefore not sensitive to
the difference in polarizabilities, remain extended, that is, not localized.
Since optical phonons have vanishing group velocity and thus do not carry heat
while acoustic phonons travel at sound velocity (at small wave vector or,
equivalently, long distance regime) and thus capable of carrying heat over a
long distance, this means the superlattice still maintains high thermal
conductivity even within our localized optical phonons picture. In other
words, our theory implies that the electrons (and holes) remain energetic
(hot) because they do not lose much energy as their scatterings with optical
phonons are reduced by the coherence of the latter, without reducing the
overall thermal conductivity of the superlattice, thus preserving the capacity
to remove excess heat. Indeed, any lost kinetic energy of the electrons due to
the scattering with optical phonons and any excess phononic thermal energy due
to temperature gradient can still be transferred via phonon-phonon scattering
to acoustic phonons, who then efficiently transport the heat across the
superlattice, because the acoustic phonons remain extended. Our results thus
offer a new approach to minimize carrier energy loss while at the same time
maintaining high thermal conductivity for effective excess heat removal
process; two qualities that are desirable for high efficiency solar cell
systems.

Ab-initio calculation LatourChalopinPRB  suggests that the coherence length
$l_{c}$ depends on the superlattice period and energy of the phonon mode. As
such, $l_{c}$ is only weakly or indirectly dependent on the material choice,
through the dependence of the phonon energy of the material parameters. Our
work pertains principally to polar semiconductor materials because in such
materials, the optical phonon plays the direct role in the electron-phonon
scattering that is responsible for the carrier thermalization. Now,
LatourChalopinPRB  shows that $l_{c}/d_{SL}$ in general decreases with energy.
This means coherence length $l_{c}$ for the optical phonon is maximized if the
optical phonon energy is minimized. This provides a guideline for the material
choice. The energy or frequency of the optical phonon mode must be minimized.
This condition can be investigated using the linear diatomic chain model of
phonon. In such model, suitable to describe diatomic III-V or II-VI
semiconductors such as GaAs, InP, … the optical phonon energy (at $q_{z}=0$)
is given by $E_{0}=2\hbar K(1/m_{1}+1/m_{2})$ where $K$ is an effective spring
constant of the force between the ions while $m_{1}$ and $m_{2}$ are atomic
masses of the two ions of the elements constituting the compound
semiconductor. While the effective spring constant $K$ may have delicate
dependence on the atomic configuration and potential energy, the dependence on
the masses clearly suggests that the optical phonon energy is minimized when
using elements of largest ion masses $m_{1}$ and $m_{2}$. For III-V family,
this would be InSb compound, while for II-VI family, this would be SrTe
(excluding the heavier elements in periodic table in these two groups which
are not normally used for semiconductor materials as they are radioactive).
Certainly, these choices of materials must be consolidated with the electronic
band gap requirement needed to form superlattice and the final choice of
materials may involve trade off or compromise between these different
requirements and can also necessitate the use of alloys of different compound
semiconductors.

As a final note, while our theory is formulated in terms of superlattice due
to the resulting analytical simplicity, the conclusions are expected to apply
as well to multi-quantum wells heterostructure, as the former can be taken as
the extension of the latter to infinitely long and perfectly periodic
structure; both structures are extensively used for solar cell design. Our
present work should therefore motivate further theoretical and experimental
studies in cultivating quantum mechanical properties of phonons to improve the
performance of photovoltaic devices.

##  VIII Acknowledgements

I.M. acknowledges funding from the French Agence Nationale de la Recherche
(ANR) through project ANR-ICEMAN under Grant Number:19-CE05-0019-01 during his
work at IM2NP Marseille. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 899987 (H.E.).

## References

  * (1) L. Esaki and R. Tsu, IBM Journal of Research and Development 14: 61–65 (1970).
  * (2) L. Esaki and L. L. Chang, Phys. Rev. Lett. 33, 495 (1974).
  * (3) D. L. Smith and C. Mailhiot, Rev. Mod. Phys. 62, 173 (1990).
  * (4) R. Tsu, Superlattice to Nanoelectronics (Elsevier Science, Amsterdam, Netherlands (2005)).
  * (5) G. Conibeer, N. Ekins-Daukes, J-F. Guillemoles, D. König, E-C. Cho, C-W. Jiang, S. Shrestha, and M. Green, Solar Energy Materials and Solar Cells 93 (2009) 713-719.
  * (6) G. Conibeer, R. Patterson, L. Huang, J-F. Guillemoles, D. König, S. Shrestha, and M. A. Green, Solar Energy Materials and Solar Cells 94 (2010) 1516-1521.
  * (7) J. Shah, Hot Carriers in Semiconductor Nanostructures (Academic Press, Inc., San Diego, 1992).
  * (8) I. Makhfudz, N. Cavassilas, M. Giteau, H. Esmaielpour, D. Suchet, A-M. Daré, and F. Michelini, Journal of Physics D: Applied Physics 55 (47), 5102 (2022).
  * (9) N. Cavassilas, I. Makhfudz, A-M. Daré, M. Lannoo, G. Dangoisse, M. Bescond, and F. Michelini, Phys. Rev. Applied 17, 064001 (2022).
  * (10) K. W. J. Barnham and G. Duggan, J. Appl. Phys. 67, 3490 (1990).
  * (11) N. Sawaki and I. Akasaki, Physica 134B (1985) 494-498.
  * (12) N. Sawaki, Surface Science 170 (1986) 537-541.
  * (13) N. Sawaki, J. Phys. C: Solid State Phys. 19 (1986) 4965-4975.
  * (14) M. Courel, J. C. Rimada, and L. Hernández, Appl. Phys. Lett. 100, 073508 (2012).
  * (15) H. Esmaielpour, B. K. Durant, K. R. Dorman, V. R. Whiteside, J. Garg, T. D. Mishima, M. B. Santos, I. R. Sellers, J-F. Guillemoles, and D. Suchet, Appl. Phys. Lett. 118, 213902 (2021).
  * (16) C. Colvard, T. A. Gant, M. V. Klein, R. Merlin, R. Fischer, H. Morkoc, and A. C. Gossard, Phys. Rev. B 31, 2080 (1985).
  * (17) M. V. Klein, IEEE J. Quantum Electron. 22, 1760 (1986).
  * (18) R. T. Ross and A. J. Nozik, J. Appl. Phys. 53, 3813 (1982).
  * (19) W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510 (1961).
  * (20) Please see the Supplementary Materials.
  * (21) C. Colvard, T. A. Gant, M. V. Klein, R. Merlin, R. Fischer, H. Morkoc, and A. C. Gossard, Phys. Rev. B 31, 2080 (1985).
  * (22) M. V. Klein, IEEE J. Quantum Electron. 22, 1760 (1986).
  * (23) B. K. Ridley, Electrons and Phonons in Semiconductor Multilayers, 2nd Edition (Cambridge University Press, Cambridge, UK (2009)).
  * (24) P. G. Klemens, Phys. Rev. 148, 845 (1966).
  * (25) C-Y. Tsai, Progress in Photovoltaics Res. Appl. 26: 10, 808 (2018).
  * (26) M. A. Stroscio and M. Dutta, Phonons in Nanostructures(Cambridge University Press, Cambridge, UK, 2004) and references therein.
  * (27) B. Latour, S. Volz, and Y. Chalopin, Phys. Rev. B 90, 014307 (2014).
  * (28) M. V. Simkin and G. D. Mahan, Phys. Rev. Lett. 84, 927 (2000).
  * (29) G. C. Cho, W. K utt, and H. Kurz, Phys. Rev. Lett. 65, 764 (1990).
  * (30) T. Pfeifer, W. K utt, H. Kurz, and R. Scholz, Phys. Rev. Lett. 69, 3248 (1992).
  * (31) W. Albrecht, Th. Kruse, and H. Kurz, Phys. Rev. Lett. 69, 1451 (19920.
  * (32) K. J. Yee, Y. S. Lim, T. Dekorsy, and D. S. Kim, Phys. Rev. Lett. 86, 1630 (2001).
  * (33) W. A. Kutt, W. Albrecht, and H. Kurz, IEEE J. Quantum Electron. QE-28, 2434 (1992).
  * (34) H. J. Zeiger, J. Vidal, T. K. Cheng, E. P. Ippen, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 45, 768 (1992).
  * (35) A. V. Kuznetsov and C. J. Stanton, Phys. Rev. Lett. 73, 3243 (1994).
  * (36) W. Grill and O. Weis, Phys. Rev. Lett. 35, 588 (1975).
  * (37) C-K. Sun, J-C. Liang, and X-Y. Yu, Phys. Rev. Lett. 84, 179 (2000).
  * (38) S. M. Komirenko, K. W. Kim, V. A. Kochelap, I. Fedorov, and M. A. Stroscio, Phys. Rev. B 63, 165308 (2001).
  * (39) B. Latour and Y. Chalopin, Phys. Rev. B 95, 214310 (2017).
  * (40) M. Lax, W. Cai, P. Hu, T. F. Zheng, B. Yudanin, and M. C. Marchetti, AIP Conference Proceeding 213, 195 (1990).
  * (41) W. Cai, M. C. Marchetti, and M. Lax, Phys. Rev. B 34, 8573 (1986).
  * (42) W. Cai, M. C. Marchetti, and M. Lax, Phys. Rev. B 35, 1369 (1987).
  * (43) M. Giteau, E. de Moustier, D. Suchet, H. Esmaielpour, H. Sodabanlu, K. Watanabe, S. Collin, J-F. Guillemoles, and Y. Okada, Journal of Applied Physics 128, 193102 (2020).
  * (44) M. O. Scully, Phys. Rev. Lett. 104, 207701 (2010).
  * (45) J. Garg and G. Chen, Phys. Rev. B 87, 140302(R) (2013).
  * (46) G. Xie, D. Ding, and G. Zhang, Advances in Physics: X, 3:1, 719 (2018).

Supplementary Materials: Coherent Phonons-Driven Hot Carrier Effect in
Superlattice Solar Cell
I. Makhfudz1, N. Cavassilas1, Y. Hajati2, H. Esmaielpour3, and F. Michelini1

1IM2NP, UMR CNRS 7334, Aix-Marseille Université, 13013 Marseille, France
2Department of Physics, Faculty of Science, Shahid Chamran University of
Ahvaz, 6135743135 Ahvaz, Iran
3Walter Schottky Institut, Technische Universität München, Am Coulombwall 4,
D-85748 Garching, Germany

In these Supplementary Materials, we provide additional details supplementing
the main text. First, we present a derivation of the electron density of
states, useful for calculation of electron density given its chemical
potential. Next, we give the details of the form factor appearing in the
electron-phonon Hamiltonian. Third, we describe the derivation of the
electron-phonon scattering rate. The file ends with details on material and
operational parameters used in the calculations.

##  IX Electron Density of States in Superlattice

Figure 6:  The profile of the electron density of states in a superlattice
consisting of wells of width $a$ and barriers of width $b$. The dashed curves
represent the mini-subbands.

From the energy dispersion Eq.(3) in the main text, one can compute the
density of states given by (not including the spin degree of freedom)

$\rho(E)=\sum^{N_{s}}_{n=1}\int\frac{d^{3}k}{(2\pi)^{3}}\delta(E-E_{n}(\mathbf{k}))$
(24)

where $N_{s}$ is the number of mini-subbands. Using the energy dispersion

$E_{n}(\mathbf{k})=E_{n}(k_{\perp})-2V_{n}\cos k_{z}(a+b)$ (25)

tedious but otherwise straightforward evaluation of the integral in Eq.(24)
gives

$\rho(E)=\frac{dN(E)}{dE}=\begin{cases}\frac{m_{c}}{2\pi^{2}\hbar^{2}}\sum_{n}\left(k^{\mathrm{max}}_{z}(n,E)-k^{\mathrm{min}}_{z}(n,E)\right)&\text{if
$0<k^{\mathrm{max}}_{z}(n,E),k^{\mathrm{min}}_{z}(n,E)<\frac{\pi}{(a+b)}$}\\\
\frac{m_{c}}{2\pi^{2}\hbar^{2}}\sum_{n}k^{\mathrm{max}}_{z}(n,E)&\text{if
$k^{\mathrm{min}}_{z}(n,E)<0<k^{\mathrm{max}}_{z}(n,E)<\frac{\pi}{(a+b)}$}\\\
\frac{m_{c}}{2\pi^{2}\hbar^{2}}\sum_{n}\frac{\pi}{(a+b)}&\text{if
$k^{\mathrm{min}}_{z}(n,E)<0<\frac{\pi}{(a+b)}<k^{\mathrm{max}}_{z}(n,E)$}\end{cases}$
(26)

where

$k^{\mathrm{min}}_{z}(n,E)=\frac{1}{(a+b)}\arccos\left[\frac{\hbar^{2}}{4m_{c}V_{n}}\left(\frac{n\pi}{a}\right)^{2}-\frac{E}{2V_{n}}\right],$
(27)
$k^{\mathrm{max}}_{z}(n,E)=\frac{1}{(a+b)}\arccos\left[\frac{\hbar^{2}\left({k^{\mathrm{max}}_{\perp}}^{2}+\left(\frac{n\pi}{a}\right)^{2}\right)}{4m_{c}V_{n}}-\frac{E}{2V_{n}}\right]$
(28)

noting that $\rho(E)$ involves a sum over densities of states of all mini-
subbands. In the above expression, $k^{\mathrm{max}}_{\perp}$ is the upper
(ultraviolet UV) cutoff on the transverse wave vector of the electron (that
is, the 2D wave vector defined in the plane of the layer of the
heterostructure). We will take

$k^{\mathrm{max}}_{\perp}=\Lambda_{\perp}=\frac{\pi}{a_{\mathrm{lattice}}}$
(29)

where $a_{\mathrm{lattice}}$ is the lattice spacing of the crystal of the
semiconductor quantum well layer.

The first case in Eq.(26) is equivalent to zero density of states because
$\left(k^{\mathrm{max}}_{z}(n)-k^{\mathrm{min}}_{z}(n)\right)<0$ while a
density of states can never be negative. The remaining (second and third)
possibilities give a compact expression for the density of states as

$\rho(E)=\frac{m_{c}}{2\pi^{2}\hbar^{2}}\sum_{n}\left(\frac{\pi}{(a+b)}\Theta(E-(E_{n}(\Lambda_{\perp})+2V_{n}))+k^{\mathrm{max}}_{z}(n,E)(\Theta(E-(E_{n}(\Lambda_{\perp})-2V_{n}))-\Theta(E-(E_{n}(\Lambda_{\perp})+2V_{n})))\right)$
(30)

with a profile shown in Fig. above. The density of states of holes is obtained
in similar manner. These densities of states are then used in the calculation
of chemical potentials $\mu_{e}(\mu_{h})$ for electron (hole) respectively,
related by quasi-Fermi level splitting $\Delta\mu=\mu_{e}-\mu_{h}$, using the
charge neutrality condition $n_{d}=p_{d}$, where $n_{d},p_{d}$ are
respectively the electron and (heavy) hole densities MakhfudzJPDAP , that
describes the intrinsic semiconductor (InAs in our example) that acts as the
well layer. The details of the computation of chemical potential from
densities of states and charge neutrality condition follow closely that
presented in MakhfudzJPDAP  and will not be repeated here.

##  X Evaluation of Overlap Integral Form Factor

Evaluating the overlap integral form factor in Eq.(7) in the main text gives

$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})=\frac{\delta(\mathbf{k}^{\prime}_{\perp}-\mathbf{k}_{\perp}+\mathbf{q}_{\perp})}{N}$
$\times\sum^{N}_{\alpha,\alpha^{\prime}=1}e^{i(k^{\prime}_{z}z_{\alpha^{\prime}}-k_{z}z_{\alpha})}G(n\alpha
n^{\prime}\alpha^{\prime}q_{z})$ (31)

and

$G(n\alpha n^{\prime}\alpha^{\prime}q_{z})=\int
dz\phi^{*}_{n}(z-z_{\alpha})e^{iq_{z}z}\phi_{n^{\prime}}(z-z_{\alpha^{\prime}})$
(32)

where $z_{\alpha}=\alpha(a+b),z^{\prime}_{\alpha}=\alpha^{\prime}(a+b)$. As
stated in Eq.(8) in the main text, we have employed analytical wave function;
the eigenfunction of an infinite potential well, to compute the overlap
integral form factor, as it permits exposition of analytical properties of
electron-phonon interaction in superlattice problem, especially those emerging
from the spatial periodicity in a superlattice. This approximation is good
enough for relatively deep well (or tall barrier), as is the case for the
InAs/AlAsSb studied in present work. While the eigenfunction Eq.(8) in the
main text has zero overlap between two adjacent wells, the effect of wave
function overlap in the barrier region is contained in the tunneling energy
$V_{n}$. Within this analytical approximate formulation, only
$\alpha=\alpha^{\prime}$ contribute to the sum in Eq.(31). In this case, the
interval of integration over $z$ is $z\in(\alpha(a+b)-a/2,\alpha(a+b)+a/2)$.
Within our model, $V=Sa$ where $S$ is the transverse cross sectional area of
the superlattice.

The Dirac delta in Eq.(31) imposes conservation of transverse wave vector
$\mathbf{k}^{\prime}_{\perp}=\mathbf{k}_{\perp}-\mathbf{q}_{\perp}$.
Evaluating the integral, we obtain

$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})=\frac{\delta(\mathbf{k}^{\prime}_{\perp}-\mathbf{k}_{\perp}+\mathbf{q}_{\perp})}{N}\mathcal{G}(k_{z}nk^{\prime}_{z}n^{\prime}q_{z})$

where $\mathcal{G}(k_{z}nk^{\prime}_{z}n^{\prime}q_{z})$ is a dimensionless
complex function

$\mathcal{G}(k_{z}nk^{\prime}_{z}n^{\prime}q_{z})=\sum^{N}_{\alpha,\alpha^{\prime}=1}e^{i(k^{\prime}_{z}\alpha^{\prime}-k_{z}\alpha)(a+b)}\left(-e^{i(a(-(\frac{1}{2})+\alpha)+\alpha
b)q_{z}}\left(\frac{(iaq_{z}\cos[\frac{1}{2}(n-n^{\prime})\pi]+(-n+n^{\prime})\pi\sin[\frac{1}{2}(n-n^{\prime})\pi])}{((n\pi-n^{\prime}\pi-
aq_{z})(n\pi-n^{\prime}\pi+aq_{z}))}\right)\right)$
$+\sum^{N}_{\alpha,\alpha^{\prime}=1}e^{i(k^{\prime}_{z}\alpha^{\prime}-k_{z}\alpha)(a+b)}\left(-e^{i(a(-(\frac{1}{2})+\alpha)+\alpha
b)q_{z}}\left(\frac{(iaq_{z}\cos[\frac{1}{2}(n+n^{\prime})\pi]-(n+n^{\prime})\pi\sin[\frac{1}{2}(n+n^{\prime})\pi])}{((n\pi+n^{\prime}\pi-
aq_{z})(n\pi+n^{\prime}\pi+aq_{z}))}\right)\right)$
$+\sum^{N}_{\alpha,\alpha^{\prime}=1}e^{i(k^{\prime}_{z}\alpha^{\prime}-k_{z}\alpha)(a+b)}\left(e^{i(a(\frac{1}{2}+\alpha)+\alpha
b)q_{z}}\left(\frac{(iaq_{z}\cos[\frac{1}{2}(n-n^{\prime})\pi]+(n-n^{\prime})\pi\sin[\frac{1}{2}(n-n^{\prime})\pi])}{((n\pi-n^{\prime}\pi-
aq_{z})(n\pi-n^{\prime}\pi+aq_{z}))}\right)\right)$
$+\sum^{N}_{\alpha,\alpha^{\prime}=1}e^{i(k^{\prime}_{z}\alpha^{\prime}-k_{z}\alpha)(a+b)}\left(e^{i(a(\frac{1}{2}+\alpha)+\alpha
b)q_{z}}\left(\frac{(iaq_{z}\cos[\frac{1}{2}(n+n^{\prime})\pi]+(n+n^{\prime})\pi\sin[\frac{1}{2}(n+n^{\prime})\pi])}{((n\pi+n^{\prime}\pi-
aq_{z})(n\pi+n^{\prime}\pi+aq_{z}))}\right)\right).$ (33)

Evaluating the sum over $\alpha$, we obtain

$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})=G_{\perp}F_{z}G_{z}$
(34)

where

$G_{\perp}=\frac{\delta(\mathbf{k}^{\prime}_{\perp}-\mathbf{k}_{\perp}+\mathbf{q}_{\perp})}{N}$
(35)

$F_{z}(k_{z},k^{\prime}_{z},q_{z})=\frac{((e^{\frac{1}{2}i(a(2k^{\prime}_{z}-2k_{z}N+q_{z})+2b(k^{\prime}_{z}-k_{z}N+q_{z}))})(e^{i(a+b)k_{z}N}-e^{i(a+b)N(k^{\prime}_{z}+q_{z})})}{((e^{i(a+b)k_{z}}-e^{i(a+b)(k^{\prime}_{z}+q_{z})})}$
(36)
$G_{z}(n,n^{\prime},q_{z})=\frac{(A_{nn^{\prime}}-B_{nn^{\prime}})}{(-\Sigma_{n,n^{\prime}}\pi+aq_{z})(-\Delta_{n,n^{\prime}}\pi+aq_{z})(\Delta_{n,n^{\prime}}\pi+aq_{z})(\Sigma_{n,n^{\prime}}\pi+aq_{z}))}$
(37)
$A_{nn^{\prime}}=2n\pi\sin\left[\frac{n\pi}{2}\right]\left(-(1+e^{iaq_{z}})((n^{\prime
2}-n^{2})\pi^{2}+a^{2}q_{z}^{2})\cos\left[\frac{n^{\prime}\pi}{2}\right]+2ia(-1+e^{iaq_{z}})n^{\prime}\pi
q_{z}\sin\left[\frac{n^{\prime}\pi}{2}\right]\right)$ (38)
$B_{nn^{\prime}}=2\cos\left[\frac{n\pi}{2}\right]\left(-ia(-1+e^{iaq_{z}})q_{z}((n^{2}+n^{\prime
2})\pi^{2}-a^{2}q^{2}_{z})\cos\left[\frac{n^{\prime}\pi}{2}\right]+(1+e^{iaq_{z}})n^{\prime}\pi((n^{2}-n^{\prime
2})\pi^{2}+a^{2}q_{z}^{2})\sin\left[\frac{n^{\prime}\pi}{2}\right]\right)$
(39)

where $\Sigma_{n,n^{\prime}}=n+n^{\prime},\Delta_{n,n^{\prime}}=n^{\prime}-n$.
For intra-miniband scattering ($n^{\prime}=n$), the expression for
$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})$ simplifies to

$G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})=\frac{\delta(\mathbf{k}^{\prime}_{\perp}-\mathbf{k}_{\perp}+\mathbf{q}_{\perp})}{N}$
$\times\frac{-((ie^{\frac{1}{2}i(a(2k^{\prime}_{z}-2k_{z}N+q_{z})+2b(k^{\prime}_{z}-k_{z}N+q_{z}))})(-1+e^{iaq_{z}})(e^{i(a+b)k_{z}N}-e^{i(a+b)N(k^{\prime}_{z}+q_{z})})(-4n^{2}\pi^{2}+(1+(-1)^{n})a^{2}q_{z}^{2})}{((e^{i(a+b)k_{z}}-e^{i(a+b)(k^{\prime}_{z}+q_{z})})(q_{z}a)(-2n\pi+aq_{z})(2n\pi+aq_{z}))}.$
(40)

Figure 7:  The profile of the form factor of electron-phonon interaction in a
superlattice consisting of wells of width $a$ and barriers of width $b$. The
peak is located at $k^{\prime}_{z}=k_{z}-q_{z}$.

The profile of $|G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})|^{2}$
is numerically computed and illustrated in Fig. above in this Supplementary
Materials, plotted as function of $k^{\prime}_{z}$ while all other parameters
fixed
($n=1,n^{\prime}=1,N=10^{6},a=5\mathrm{nm},b=2\mathrm{nm},k_{z}=4.48\times
10^{8}m^{-1},q_{z}=2.24\times
10^{8}m^{-1},k^{\prime}_{\perp}=k_{\perp}=q_{\perp}=0$). It is clear that the
profile takes the form of Dirac delta function, centered at
$k^{\prime}_{z}=k_{z}-q_{z}$. As such, combining with the Dirac delta function
in Eq.(35), we can write

$|G(\mathbf{k}n\mathbf{k}^{\prime}n^{\prime}\mathbf{q})|^{2}=C_{N}(n,n^{\prime},k_{z},q_{z})\delta(\mathbf{k}^{\prime}-\mathbf{k}+\mathbf{q})$
(41)

where the coefficient $C_{N}(n,n^{\prime},k_{z},q_{z})$ is to be determined
from the normalization condition, giving

$C_{N}(n,n^{\prime},k_{z},q_{z})=\frac{2\pi}{(a+b)}\tilde{F}_{N}(k_{z},q_{z})|G_{z}(n,n^{\prime},q_{z})|^{2}.$
(42)

The real function $\tilde{F}_{N}(k_{z},q_{z})$ in the above equation is
determined from the relation

$\frac{1}{N^{2}}\int^{\frac{\pi}{(a+b)}}_{-\frac{\pi}{(a+b)}}dk^{\prime}_{z}|F_{z}(k_{z},k^{\prime}_{z},q_{z})|^{2}=\frac{2\pi}{(a+b)}\tilde{F}_{N}(k_{z},q_{z}).$
(43)

It turns out that $\tilde{F}_{N}(k_{z},q_{z})$ converges to $1/N$; where $N$
is the number of periods of the superlattice.

##  XI Derivation of Electron-Phonon Scattering Rate

The change in the electron energy (denoted by $\Delta
E_{n\mathbf{k};n^{\prime}\mathbf{k}^{\prime},\mathbf{q}}$ in Eq. (15) in the
main text) upon electron-phonon scattering reads

$E_{n}(\mathbf{k}_{\perp},k_{z})-E_{n^{\prime}}(\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z})=\frac{\hbar^{2}}{2m_{c}}\left[(k^{2}_{nz}-{k_{n^{\prime}z}}^{2})-q^{2}_{\perp}+2k_{\perp}q_{\perp}\cos\phi\right]+2V_{n^{\prime}}\cos
k^{\prime}_{z}(a+b)-2V_{n}\cos k_{z}(a+b)\ ,$ (44)

where

$E_{n}(a)=\frac{\hbar^{2}k^{2}_{nz}}{2m_{c}}$ (45)

and $\phi$ is the angle between ${\bf k_{\perp}}$ and ${\bf q}_{\perp}$. The
distinction between $k_{nz}$ and $k_{z}$ is to be noted; the former gives the
energy level of a confined electron state (the center of the mini-subband)
while the latter is a quasi-momentum of the electron in the superlattice
direction due to the periodicity of the structure. Within infinite potential
well approximation, $k_{nz}=n\pi/a$ while within finite potential well picture
using the depth of the well $V_{c}$ in Fig. 1 in the main text, it is the
energy level $E_{n}(a)$ that is determined from the solution of transcendental
equation from solving Schrodinger equation for a finite potential well. The
following derivation uses infinite potential well approximation. The
corresponding finite potential well results are obtained simply by the
following replacement

$k^{2}_{nz}=\frac{n^{2}\pi^{2}}{a^{2}}\rightarrow\frac{2m_{c}E_{n}(a)}{\hbar^{2}}$
(46)

to get the final results of this section.

Using

$\sum_{\mathbf{k}_{\perp}}\rightarrow\frac{S}{(2\pi)^{2}}\int_{0}^{2\pi}d\phi\int_{0}^{+\infty}d{k}_{\perp}k_{\perp}\
,$ (47)

we get

$A_{\mathbf{q}_{\perp},q_{z}}=\frac{2m_{c}S}{(2\pi)^{2}\hbar^{2}q_{\perp}}\sum_{n,n^{\prime};k_{z},k^{\prime}_{z}}\int_{0}^{2\pi}d\phi\int_{0}^{+\infty}d{k}_{\perp}I^{2}_{\mathbf{k}_{\perp},n,\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}\left[(N_{\mathbf{q}_{\perp},q_{z}}+1)f_{\mathbf{k}_{\perp},n}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}})-N_{\mathbf{q}_{\perp},q_{z}}f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}(1-f_{\mathbf{k}_{\perp},n})\right]$
$\times\delta\left(\cos\phi+\frac{(n^{2}-{n^{\prime}}^{2})}{2k_{\perp}q_{\perp}}\frac{\pi^{2}}{a^{2}}-\frac{q_{\perp}}{2k_{\perp}}-\frac{m_{c}\omega_{q_{\perp},q_{z}}}{\hbar
k_{\perp}q_{\perp}}+\frac{m_{c}}{\hbar^{2}k_{\perp}q_{\perp}}\left(2V_{n^{\prime}}\cos
k^{\prime}_{z}(a+b)-2V_{n}\cos k_{z}(a+b)\right)\right)$ (48)

where $\sum_{n,n^{\prime}}$ represents the sum over mini-subbands while
$I^{2}_{\mathbf{k}_{\perp},n,\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}=|G(\mathbf{k}_{\perp},k_{z},n;\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z},n^{\prime};\mathbf{q}_{\perp},q_{z})|^{2}|\tilde{W}_{q_{z}}|^{2}$.
The argument of the Dirac delta function can be written as
$\delta(\cos\phi-\cos\phi_{0})$.

Since $|\cos\phi|\leq 1$, we obtain a constraint on the quantity following the
$\cos\phi$ within the Dirac delta function in Eq.(56) above, resulting in
$k_{\perp}\geq k^{\mathrm{min}}_{\perp}(n,n^{\prime})$ where

$k^{\mathrm{min}}_{\perp}(n,n^{\prime})=\frac{q_{\perp}}{2}+\frac{({n^{\prime}}^{2}-n^{2})}{2q_{\perp}}\frac{\pi^{2}}{a^{2}}+\frac{m_{c}\omega_{q_{\perp},q_{z}}}{\hbar
q_{\perp}}-\frac{m_{c}}{\hbar^{2}q_{\perp}}\left(2V_{n^{\prime}}\cos
k^{\prime}_{z}(a+b)-2V_{n}\cos k_{z}(a+b)\right)$ (49)

giving the minimum transverse wave vector that an electron must have in order
to be able to emit an LO phonon. Integrating over $\phi$, we have

$\cos\phi_{0}=\pm\frac{k^{\mathrm{min}}_{\perp}(n,n^{\prime})}{k_{\perp}}$
(50)

giving

$\int^{2\pi}_{0}d\phi\delta(\cos\phi-\cos\phi_{0})=\frac{2k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp}(n,n^{\prime}))^{2}}}.$
(51)

Applying the above result to Eq.(56) gives

$A_{\mathbf{q}_{\perp},q_{z}}=\frac{2m_{c}S}{(2\pi)^{2}\hbar^{2}q_{\perp}}\sum_{n,n^{\prime};k_{z},k^{\prime}_{z}}\int_{k^{\mathrm{min}}_{\perp}}^{+\infty}d{k}_{\perp}I^{2}_{\mathbf{k}_{\perp},n,\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}\left[(N_{\mathbf{q}_{\perp},q_{z}}+1)f_{\mathbf{k}_{\perp},n}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}})-N_{\mathbf{q}_{\perp},q_{z}}f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}(1-f_{\mathbf{k}_{\perp},n})\right]$
$\times\frac{2k_{\perp}}{\sqrt{k^{2}_{\perp}-\left(\frac{(n^{2}-{n^{\prime}}^{2})}{2q_{\perp}}\frac{\pi^{2}}{a^{2}}-\frac{q_{\perp}}{2}-\frac{m_{c}\omega_{q_{\perp},q_{z}}}{\hbar
q_{\perp}}+\frac{m_{c}}{\hbar^{2}q_{\perp}}\left(2V_{n^{\prime}}\cos
k^{\prime}_{z}(a+b)-2V_{n}\cos k_{z}(a+b)\right)\right)^{2}}}.$ (52)

The final expression for the rate of change of number of LO phonons is given
by

$\frac{dN_{\mathbf{q}_{\perp},q_{z}}}{dt}=(N_{\mathbf{q}_{\perp},m}(T_{c})-N_{\mathbf{q}_{\perp},m})$
$\times\left[2\frac{2\pi}{\hbar}\left(\frac{e^{2}\hbar\omega_{q_{\perp},m}}{q_{\perp}}\left[\frac{1}{\varepsilon_{\infty}}-\frac{1}{\varepsilon_{s}}\right]\frac{|\tilde{W}_{q_{z}}|^{2}}{8\pi
q^{2}V}\right)\right]\frac{2m_{c}}{\hbar^{2}}\frac{S}{(2\pi)^{2}}$
$\times\sum_{n,n^{\prime};k_{z},k^{\prime}_{z}}|G_{n,n^{\prime}}(k_{z},k^{\prime}_{z},q_{z})|^{2}\int^{+\infty}_{k^{\mathrm{min}}_{\perp}}dk_{\perp}\left[f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}-f_{\mathbf{k}_{\perp},n}\right]$
$\times\frac{k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp})^{2}}},$
(53)

where we have made use of a nice identity Wurfel ,

$f_{\mathbf{k}_{\perp},n}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}})=N_{\mathbf{q}}(T_{c})(f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}-f_{\mathbf{k}_{\perp},n})\
,$ (54)

where $N_{\mathbf{q}}(T_{c})=1/(\exp(\hbar\omega_{LO}/(k_{B}T_{c}))-1)$ is the
equilibrium phonon distribution at the carrier temperature $T_{c}$.

To arrive at the final result given in Eq.(17) in the main text, we replace
the sums over $k_{z},k^{\prime}_{z}$ with integrals over
$k_{z},k^{\prime}_{z}$,

$\sum_{k_{z}}h(k_{z})\rightarrow$
$\mathrm{lim}_{N\rightarrow\infty}\sum^{\frac{N}{2}}_{j=-\frac{N}{2}}h(k_{z}(j))\rightarrow
N(a+b)\int^{\frac{\pi}{(a+b)}}_{-\frac{\pi}{(a+b)}}\frac{dk_{z}}{2\pi}h(k_{z})$
(55)

where $k_{z}(j)=2\pi j/(N(a+b))$ and similarly for $k^{\prime}_{z}$.
Evaluating the integral over $k^{\prime}_{z}$ gives

$\frac{dN_{\mathbf{q}_{\perp},q_{z}}}{dt}=\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}}}{\tilde{\tau}^{c-LO}_{\mathbf{q}}}=(N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}})\left[2N\frac{2\pi}{\hbar}\left(\frac{e^{2}\hbar\omega_{\mathbf{q}}}{q_{\perp}}\left[\frac{1}{\varepsilon_{\infty}}-\frac{1}{\varepsilon_{s}}\right]\frac{|\tilde{W}_{q_{z}}|^{2}}{8\pi
q^{2}V}\right)\right]\frac{2m_{c}}{\hbar^{2}}\frac{S}{(2\pi)^{2}}\sum_{n,n^{\prime}}\int^{\frac{\pi}{(a+b)}}_{-\frac{\pi}{(a+b)}}\frac{dk_{z}}{(2\pi)^{2}}$
$\times\Theta\left(\frac{\pi}{(a+b)}-k_{z}+q_{z}\right)\Theta\left(k_{z}-q_{z}+\frac{\pi}{(a+b)}\right)C_{N}(n,n^{\prime},k_{z},q_{z})(a+b)^{2}\frac{1}{2}\frac{\sqrt{2m_{c}k_{B}T_{c}}}{\hbar}\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}(Li_{1/2}(-e^{x_{2}})-Li_{1/2}(-e^{x_{1}}))$
(56)

where
$N_{\mathbf{q}}(T_{c})=(\exp(\hbar\omega_{\mathbf{q}}/(k_{B}T_{c}))-1)^{-1}$,
$\Theta(\cdots)$ is Heaviside theta function, $\Gamma$ is gamma function and
$Li_{\alpha}(x)$ is polylogarithmic function of order $\alpha$ while
$x_{1},x_{2},k^{\mathrm{min}}_{\perp}$ are given in the following paragraph.
It is to be noted that the limits of integration over $k_{z}$ imposed by the
product of two Heaviside functions in Eq.(56) indicates the inclusion of
Umklapp scattering of electron by LO phonon for any finite wave vector $q_{z}$
of the latter. We have included the phonon spatial coherence weight factor
squared $|\tilde{W}_{q_{z}}|^{2}$ that is discussed in the following section.

To obtain the last line of Eq. (56), we rewrite and evaluate the integral in
Eq.(53) as follows,

$I=\int^{\infty}_{k^{\mathrm{min}}_{\perp}}dk_{\perp}\frac{k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp})^{2}}}(f_{1}-f_{2})\
,$ (57)

where $f_{i}=\frac{1}{e^{\tilde{x}_{i}}+1}$, with respectively

$\tilde{x}_{1}=\beta(E_{n^{\prime}}(k_{\perp})-2V_{n^{\prime}}\cos(k_{z}-q_{z})(a+b)-\mu_{c})-\beta\hbar\omega_{LO}$
(58)

and

$\tilde{x}_{2}=\beta(E_{n}(k_{\perp})-2V_{n}\cos k_{z}(a+b)-\mu_{c}),$ (59)

with $\mu_{c}$ the carrier chemical potential and $\beta=1/k_{B}T_{c}$. An
obvious change of variable leads to

$I=\frac{1}{2}\frac{\sqrt{2m_{c}k_{B}T_{c}}}{\hbar}\int_{0}^{+\infty}dx\frac{1}{\sqrt{x}}\Bigl{[}\frac{1}{e^{(x-x_{1})}+1}-\frac{1}{e^{(x-x_{2})}+1}\Bigr{]}\
,$ (60)

with

$x_{1}=-\beta\Bigl{[}\frac{\hbar^{2}}{2m_{c}}(k^{min}_{\perp})^{2}+E_{n^{\prime}}(a)-2V_{n^{\prime}}\cos(k_{z}-q_{z})(a+b)\Big{]}$
$+\beta(\mu_{c}+\hbar\omega_{LO}),$ (61)
$x_{2}=-\beta\Bigl{[}\frac{\hbar^{2}}{2m_{c}}(k^{min}_{\perp})^{2}+E_{n}(a)-2V_{n}\cos
k_{z}(a+b)-\mu_{c}\Big{]}$ (62)

where $k^{min}_{\perp}$ is the minimum transverse wave vector an electron must
posses to be able to emit an LO phonon,

$k^{\mathrm{min}}_{\perp}(n,n^{\prime})=\frac{q_{\perp}}{2}+\frac{(E_{n^{\prime}}(a)-E_{n}(a))}{2q_{\perp}}\frac{2m_{c}}{\hbar^{2}}+\frac{m_{c}\omega_{q_{\perp},q_{z}}}{\hbar
q_{\perp}}$ $-\frac{m_{c}}{\hbar^{2}q_{\perp}}\left(2V_{n^{\prime}}\cos
k^{\prime}_{z}(a+b)-2V_{n}\cos k_{z}(a+b)\right)$ (63)

and $\mu_{c}=\mu_{e}$ the electron chemical potential. Equation (60) can be
expressed in terms of a polylogarithmic function using Ref. nist, ,

$\int_{0}^{+\infty}\frac{dt}{\sqrt{t}}\frac{1}{e^{t-u}+1}=-\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}Li_{1/2}(-e^{u})\
.$ (64)

This leads to

$I=\frac{1}{2}\frac{\sqrt{2m_{c}k_{B}T_{c}}}{\hbar}\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}(Li_{1/2}(-e^{x_{2}})-Li_{1/2}(-e^{x_{1}}))\,,$
(65)

which gives Eq.(56) when substituted into Eq.(53).

Material Parameters (InAs(well) AlAsSb(barrier))
---
Quantity |  Symbol |  Value
LO phonon energy |  $E_{0}$ |  $30$ meV
Conduction band well depth |  $V_{c}$ |  1.92 eV
Energy gap (InAs) |  $E_{g}$ |  0.354 eV
Valence band well depth |  $V_{h}$ |  0.05 eV
Energy gap (AlAsSb) |  $E_{g}$ |  2.22 eV
Electron effective mass (InAs) |  $m_{c}$ |  0.023 $m_{0}$
Heavy hole effective mass (InAs) |  $m_{hh}$ |  0.410 $m_{0}$
Electron effective mass (AlAsSb) |  $m_{c}$ |  0.1369 $m_{0}$
Heavy hole effective mass (AlAsSb) |  $m_{hh}$ |  0.98 $m_{0}$
Sound velocity (InAs) |  $v_{s}$ |  $4.4\times 10^{3}$ m/s
Infinite-frequency susceptibility (InAs) |  $\varepsilon_{\infty}$ |  15.15
Static susceptibility (InAs) |  $\varepsilon_{s}$ |  12.30
Gruneisen parameter (InAs) |  $\gamma$ |  0.28SparksSwenson
Zero temperature LO phonon decay time (InAs) |  $\tau^{LO-ac}_{0}$ |  $4.363\times 10^{-10}$ s
Lattice constant (InAs) |  $a$ |  6.06 Å
Mass density (InAs) |  $\rho$ |  5680 kg/m3
Table 1: The parameters of the semiconductor materials for the superlattice
used in our calculations. The masses are expressed in terms of free electron
mass $m_{0}=9.11\times 10^{-31}$kg.

##  XII Material and Operational Parameters

In this section of the Supplementary Materials, we provide the material and
operational parameters that we used to produce the figures given in the main
text. The superlattice materials are taken to be InAs for the well layer and
AlAsSb for the barrier layer. The corresponding carrier (electron) can be
computed from the density of states given in the Section I of the Supplementar
Materials. For the calculation of electron thermalization power of interest,
only the parameters of electron in the well layer (InAs) enter directly in the
equations in the main text. The carrier parameters for the holes in the well
layer (InAs) and those of the electrons and the holes for the barrier layer
(AlAsSb) however enter indirectly in the determination of the tunneling energy
$V_{n}$ appearing in Eq.(1) in the main text, and so they are listed as well
in Table I. The zero temperature LO phonon decay time is determined from using
MakhfudzJPDAP

$\tau^{LO-ac}_{0}=\frac{32\pi\rho
v^{3}_{s}}{\Gamma^{2}\hbar\omega^{3}_{\mathrm{LO}}}.$ (66)

where RidleyBookS

$\Gamma=\sqrt{\frac{4}{3}}\frac{\gamma\omega_{\mathrm{LO}}}{v_{s}}\ ,$ (67)

with $E_{0}=\hbar\omega_{\mathrm{LO}}$. In addition to the material parameters
given in the Table, we have to assume the numerical values for the parameters
representing the operating condition. These include the lattice temperature
$T_{L}$, carrier temperature $T_{c}$ and the quasi-Fermi level splitting
$\Delta\mu=\mu_{e}-\mu_{h}$ where $\mu_{e}(\mu_{h})$ are respectively the
electron (hole) chemical potential.

Normally, $T_{L}$ can be fixed to be an appropriate ambient temperature. We
take $T_{L}=300$K to describe solar cell operation at ambient room
temperature. On the other hand, the carrier parameters $T_{c}$ and $\Delta\mu$
are ought to be either determined from a detailed balance equation involving a
given absorption of radiation and recombination process, or be taken from
available experimental data. In this work, we do not pursue the former, but
adopt the later instead, partially, where we fix $T_{c}$ and assume some
appropriate values for $\Delta\mu$, which is empirically constrained to be
$0\leq\Delta\mu\leq E_{g}$, where $\Delta\mu=0$ corresponds to absence of
irradiation, with $E_{g}$ the band gap of the well layer. In producing Fig. 3
in the main text, we have taken $T_{c}=450$K, and $\Delta\mu=0.354$eV, taken
to be equal to $E_{g}$ for InAs, which are relatively high $T_{c}$ and large
$\Delta\mu$ but are generally consistent with empirical observation that
$T_{c}$ is higher when $\Delta\mu$ is larger. In producing Fig. 4 in previous
section on the other hand, to make direct comparison with the data presented
in EsmaielpourAPLs , we have set $T_{c}=345$K and assumed $\Delta\mu=0.2$eV,
while setting $a=2$nm. The fact that $P_{\mathrm{th}}$ in our result is very
small (in the order of 10-13W/cm2) for such small thickness, since only one
mini-subband exists while phonon-driven electron thermalization mainly comes
from inter-mini-subband scattering, poses no problem because our theory
strictly speaking assumes an infinitely long $N\rightarrow\infty$
superlattice; the total thermalization power will be finite as one considers
$N$ times the $P_{\mathrm{th}}$ per period of superlattice that we compute. It
is for this reason that in Fig. 4, we only present the normalized
$P_{\mathrm{th}}$ relative to its large barrier thickness limit
$P_{\mathrm{th0}}$ as the quantity that is appropriately compared with
experiment EsmaielpourAPLs .

For carrier photogeneration by monochromatic irradiation (with laser, for
example), given the thermalization power, the necessary absorbed power (with
incoherent phonons) can be estimated using the following relation GiteauJAPs

$P_{\mathrm{th}}=\frac{E_{\mathrm{laser}}-(E_{g}+3k_{B}T_{L})}{E_{\mathrm{laser}}}P_{\mathrm{abs}}$
(68)

where $E_{\mathbf{laser}}=hc/\lambda$ where $h$ is Planck constant, $c$ the
speed of light, and $\lambda$ the (monochromatic) laser wave length. Taking
$\lambda=532$nm, the absorbed power of about $P_{\mathrm{abs}}=14.8$ W/cm2 for
the upper subfigure of Fig. 3 in the main text, as an example.

## References

  * (1) I. Makhfudz, N. Cavassilas, M. Giteau, H. Esmaielpour, D. Suchet, A-M. Daré, and F. Michelini, Journal of Physics D: Applied Physics 55 (47), 5102 (2022).
  * (2) P. Wurfel, J. Phys. C: Solid State Phys. 15 3967 (1982).
  * (3) See for example https://dlmf.nist.gov/25.12.
  * (4) M. Lax, W. Cai, P. Hu, T. F. Zheng, B. Yudanin, and M. C. Marchetti, AIP Conference Proceeding 213, 195 (1990).
  * (5) W. Cai, M. C. Marchetti, and M. Lax, Phys. Rev. B 34, 8573 (1986).
  * (6) W. Cai, M. C. Marchetti, and M. Lax, Phys. Rev. B 35, 1369 (1987).
  * (7) B. Latour, S. Volz, and Y. Chalopin, Phys. Rev. B 90, 014307 (2014).
  * (8) A. V. Kuznetsov and C. J. Stanton, Phys. Rev. Lett. 73, 3243 (1994).
  * (9) B. Latour and Y. Chalopin, Phys. Rev. B 95, 214310 (2017).
  * (10) H. Esmaielpour, B. K. Durant, K. R. Dorman, V. R. Whiteside, J. Garg, T. D. Mishima, M. B. Santos, I. R. Sellers, J-F. Guillemoles, and D. Suchet, Appl. Phys. Lett. 118, 213902 (2021).
  * (11) M. Giteau, E. de Moustier, D. Suchet, H. Esmaielpour, H. Sodabanlu, K. Watanabe, S. Collin, J-F. Guillemoles, and Y. Okada, Journal of Applied Physics 128, 193102 (2020).
  * (12) P. W. Sparks and C. A. Swenson, Phys. Rev. B 163, 779 (1967).
  * (13) B. K. Ridley, Quantum Processes in Semiconductors, 3rd ed. (Clarendon Press, Oxford, 1993).