Analysis on carrier energy relaxation in superlattices and its implications on the design of hot...

Analysis on carrier energy relaxation in superlattices and its

implications on the design of hot carrier solar cell absorbers

Y. Fenga, P. Aliberti, R.J. Patterson, B.P. Veettil, S. Lin, H. Xia, S. Shrestha, M.A. Green,G.J. Conibeer

Center of Photovoltaics, UNSW, Sydney, NSW, Australia

ABSTRACT

Hot carrier solar cell (HCSC) requires a slow cooling rate of carriers in the absorber, which can potentially befulfilled by semiconductor superlattices. In this paper the energy relaxation time of electrons in InN InxGa1-xNsuperlattices are computed with Monte Carlo simulations considering the multi-stage energy loss of electrons.As a result the e↵ect of each stage in the relaxation process is revealed for superlattice absorbers. The energyrelaxation rate figures are obtained for di↵erent material systems of the absorber, i.e. for di↵erent combinationsof Indium compositions and the thicknesses of well and barrier layers in the superlattices. The optimum materialsystem for the absorber has been suggested, with the potential to realize HCSCs with high e�ciency.

Keywords: hot carrier solar cell, multiple quantum-well superlattice, wurtzite-structured nitrides, carrier energyrelaxation, Frohlich interaction, three-phonon process, hot phonon e↵ect

1. INTRODUCTION

The hot carrier solar cell is an advanced approach of solar-electricity converters to realize high conversion ef-ficiency, i.e. exceeding Shockley-Quessier limit.1 The underlying principle of this approach is to minimize thecarrier cooling rate and maintain the energies of photo-generated carriers well above the band edge.2 The hotcarriers are then extracted through selective energy contacts to minimize the entropy increase during the extrac-tion.3,4 The thermal energy of the hot distribution is converted into electricity, leading to e�ciency much higherthan that of the conventional solar cells.5

The crucial criteria to obtain high e�ciency from the hot carrier e↵ect is to minimize the energy relaxationrates of carriers in the absorber layer, where most of the electron-hole pairs are generated from absorbing pho-tons.5 The energy relaxation of carriers occurs with several sequential processes. For polar materials carriers lossenergy by emitting polar optical phonons.6,7 The increased occupations of polar phonon modes feed energy backto carriers, unless the occupations are relaxed by phonon decay and phonon di↵usion. Due to the anharmonicityof the crystal potential three-phonon interaction or even higher-order phonon-coupling occurs, resulting in anenergy relaxation of polar modes.8,9 Phonon di↵usion, on the other hand, comes from the spatial non-uniformityand continually relax the occupations of phonon modes generated from hot electrons or hot phonons.

The electron-phonon interaction lifetimes7,10 and the lifetimes of the polar optical modes11,12 had beencomputed in bulk III - V semiconductors, while in superlattice materials the carrier cooling processes are evennot completely understood. Experimental evidence demonstrates a reduced cooling rate in some superlatticesystem.13 The explanation partly lies on the non-equilibrium distribution of hot electrons and high-lying opticalphonons. The detailed computation, based on the balance at steady state of the flow of phonon emission, decayand di↵usion, have not been well studied.

Here InN/Inx

Ga1�x

N multiple quantum-well superlattices (MQW-SL) with wurtzite crystal structure arestudied as the absorber of the hot carrier solar cell, due to the significant hot-phonon-bottleneck e↵ects of theircomponent materials.14 The reason mainly comes from the large contrast of the atomic masses, and hence a largeband-gap between high-lying and low-lying phonon modes.12,15 The phonon band-gap e↵ectively stop Klemensdecay, i.e. one high-lying phonon decaying into two low-lying phonons,8 and produce hot population on the polarmodes, feeding energy back to electrons.16 InN and In

x

Ga1�x

N have very similar lattice structures, with almost

aE-mail: [email protected]

Next Generation (Nano) Photonic and Cell Technologies for Solar Energy Conversion III, edited by Loucas Tsakalakos, Proc. of SPIE Vol. 8471, 847103 · © 2012 SPIE

CCC code: 0277-786/12/$18 · doi: 10.1117/12.928459

Proc. of SPIE Vol. 8471 847103-1

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the same lattice constant. This benefits the solar energy conversion, in terms of both the carrier transport andthe reduction of recombination sites. The 1-dimensional superlattice structure ensures a continuous electronicenergy spectrum and hence a broad-band absorption. The absorption is further enhanced by the small electronicband-gap of InN.

2. THE MODEL

2.1 phononic model

The dispersion relations of phonon modes in superlattices have been computed with a 1-dimensional atomic-planemodel. Since the epitaxial growth is intended to along the high-symmetrical � � A direction for both InN andIn

x

Ga1�x

N layers, the periodicity of MQW-SL and hence the zone-folding is along this direction. For simplicitywe assumed that the mode frequency of each zone-folded mini-band only depends on the vertical component ofthe wave-vector (i.e. along �� A). The reason for the assumption partly comes from the high concentration ofelectron-emitted polar phonons around the zone-center, where the dispersions are relatively flat compared to themini-gaps. In wurtzite structure periodic atomic planes with alternative elements align perpendicular to the ��Adirection. If only considering the vertical component of wave-vectors, all atoms in one atomic planes vibrate inphase and can be treated as one uniform displacement. The phononic model adopted here treats a 1-dimensionalchain with an equal length of the superlattice periodicity, taking into account the plane-to-plane force constants.The plane-to-plane force constants are calculated by adopting the conventional Keating potentials for all bonds.(Eq. (1))

Uk

=3↵

16r20

4X

j=1

(r2ij

� r20)2 +

3�

8r20

4X

j=1

4X

k>j

(rij

r

ik

+r203)2 (1)

where ↵ and � are the bond-stretching and bond-bending force constants. rij

is the vector from a givenatom i to one of its four nearest neighbors j. The two types of atoms are modeled as positive-charged andnegative-charged points respectively to take into account the long-range Coulumb interatomic forces . Ab-initiocalculated Born e↵ective charges,17 as well as the bond-stretching and bond-bending force constants,18,19 areadopted for InN and GaN. Under harmonic approximation the planar interaction model generates the phonondispersions along ��A direction for InN and GaN. The calculated zone-center LO and TO energies are around9.1eV and 6.8eV for GaN, and 7.2eV and 5.5eV for InN.

For the Inx

Ga1�x

N layer we have atomic planes which consist of both Indium and Gallium. The forceconstants and atomic mass for this planes are regarded as weighted averages of these parameters of pure InN andGaN. The reason comes from the large number of atoms in one atomic plane, where Indium and Gallium atomsare mixed uniformly, following the mole ratio of x : 1 � x. The computed phonon dispersions of superlatticesshow confined optical modes and zone-folded acoustic branches. The confined nature of the optical modes comesfrom the large contrast between the atomic masses and the force constants of two component materials, i.e. InNand In

x

Ga1�x

N.

2.2 electronic structure

The electronic structures, involving the wavefuntions and energies of all the states, can be calculated by usingthe Kronig-Penney model for superlattices.20 By doing that the band structure of each component, i.e. InNor In

x

Ga1�x

N, is assumed as parabolic, and is characterized by two parameters: the bandgap and the carriere↵ective mass. The theorectical bandgap of In

x

Ga1�x

N has been expressed as a function of x,21 while there is nosolid expression for the variation of the e↵ective mass and thus the e↵ective mass of GaN has been adopted forthe barrier layer. In equilibrium conditions space charges occur in the well layers (negative) and in the barrierlayers (positive) due to the di↵erence of their electron a�nities. The electron a�nity of GaN is significantlylower than that of InN , giving a larger o↵-set of the conduction band edge rather than that of the valence bandedge. In the superlattices consisting of InN and In

x

Ga1�x

N, similar band shifts are expected. The conductionband o↵-set V

B

, or the barrier height of electrons, has been taken into Eq. (2) for the dispersion relation.



cos(kz

(LW

+ LB

)� cos(kW

LW

)cos(kB

LB

)

=1

2(kW

LB

kB

LW

+kB

LW

kW

LB

)sin(kW

LW

)sin(kB

LB

)

kW

=p

2mE(kz

) (2)

kW

=p

2m(E(kz

)� VB

) (3)

where kz

is the Bloch wave-vector of electrons in superlattice and E(kz

) is its corresponding energy, withthe reference level of the conduction band edge of the well material. L

W

and LB

are the well thickness and thebarrier thickness. Apart from the dispersion relations the electron envelop wavefunction for each state has beencalculated, as the evaluation of the electron-phonon coupling strength requires calculation of the overlap integral.Coupling of electronic waves in di↵erent layers generates widely-spread wavefunctions with mixed parities.20

2.3 electron-phonon interaction

For polar semiconductors the energy relaxation of electrons is dominantly through polar interactions with opti-cal modes rather than through electron-phonon coupling due to the deformation potential.7 In bulk wurtzite-structured materials only one branch of optical modes are infrared-active.10 Free carriers are able to response tothe high-frequency polarization field generated by these polar modes. In the MQW-SL consisting of two compo-nent wurtzite-structured layers, the polar modes sit in the similar positions. For InN/In

x

Ga1�x

N superlatticesthese modes lie in the high-lying optical branches, separated from the low-lying modes by a significant energygap. Therefore the generated polar modes stays in high-lying branches with a long relaxation time in terms ofdecaying into low-lying modes, leading to a non-equilibrium phonon distributions. This phonon-bottleneck e↵ecthas been investigated during the recent decades for both bulk materials14,22 and nano-structured materials.13,23

The calculation of the rate of polar interaction between hot electrons (here emission by holes is out of ourconcern) and polar phonons are based on the Frohlich-type Hamiltonian and the 1-st order perturbation theory.In 1-dimensional superlattices the elements of the Frohlich-type Hamiltonian can be represented in eq4, withan overlap integral between the envelop wavefunction of electrons and the e↵ective electric potential generatedby the lattice mode. The e↵ective potential can be calculated by taking the divergence of the polarizationfield of each mode. For confined modes the potential decays exponentially when being away from the layer ofconfinement.24 The e↵ect of static screening from the populated carrier density has been taken into account byadopting the Debye screening length, which is proportional to the carrier density N .25 Recent report has provedthe significant e↵ect of screening on the polar interactions in heavily-doped InN.26

H2k

0k

=e2}!8V

✓1

✏1� 1

✏

◆n(q,!) +

1

2⌥ 1

2

�q4

(q2 + q2D

)2

⇥ �(k0 � k± q)[Al

Zdz�⇤

j

(z)�j

0(z)Il

(z)]2 (4)

where ! is the angular frequency of the lattice mode of interest and q is its wavevector. V is the volumeof the superlattice. ✏ and ✏1 are the static permittivity and the optical permittivity respectively. n(q,!) isthe occupation number of the mode (q,!); the term n(q,!) + 1 corresponds to the process of phonon emissionof the mode while n(q,!) corresponds to phonon absorption. q

D

is the inverse screening length calculatedfrom the classic Debye-Huckel model: q

D

= e2N/✏1kB

TC

, where TC

refers to the temperature of the hotelectrons. The delta function �(k0 � k ± q) indicates the conservation of the crystal momentum. �

j

(z) is theone-dimensional electron envelop function and I

l

(z) is the e↵ective potential of the l-th confinement mode, ofwhich the corresponding modulation function requires a normalization factor A

l

.27



2.4 phonon-phonon interaction

The interactions between phonons are due to the anharmonicity of the crystal potential. The dominant anhar-monicity is of the 3rd-order and leads to a three-particle processes.8 Fission of one phonon into two can happenas well as fusion of two phonons. The potential of anharmonicity is taken as the interaction Hamiltonian, andideally a tensor coe�cient presents the interaction strengths between di↵erent polarizations of the three modesinvolved. However the tensor of the anharmonicity coe�cient is hard to determined experimentally and it is stillnot clear for some nitrides. Therefore we have adopted a mode-average coe�cient instead which can be easilydetermined from experiment F = �/c, where � is the temperature dependent Gruneisen’s constant and c is theaverage acoustic speed. The reported values for InN/In

x

Ga1�x

N have been adopted for the lifetime calculationof two types of confined modes respectively.12

c(!,!0,!00) = � ipN

2Mp3F!!0!00 (5)

where c is the three-phonon coupling strength. M is the average atomic mass and N is the atom density. !,!0 and !00 are the respective angular frequencies of the three modes involved in the interaction. The elementsof the interaction Hamiltonian due to the 3rd order anharmonicity contain an overlap integral of modulationfunctions of all three modes.28

H2qq

0q

00 =}3

MM 0M 00V 2

|c(!,!0,!00)I(k,k0,k00)|2

!!0!00

⇥ [(n(n0 + 1)(n00 + 1)� (n+ 1)n0n00] (6)

2.5 Energy relaxation time of systems in equilibrium

To compare the relaxation properties of hot electrons and hot polar phonons under constant illumination (forsimulating the operation of solar cells) between di↵erent superlattice structures, the energy relaxation timesare computed for each material structure. The energy relaxation time of electrons due to the emission of polarphonons is defined as �

e

= EeP✏pwp

, where Ee

is the total energy of the electron system and the summation in the

denominator is over the polar modes, with ✏p

and wp

referring to the energy and emission rate of each mode. Togive indicative and comparable relaxation time values for all MQW-SL structures the distributions of electronsand phonons are pre-set. In the case of phonon emission the electrons are set as be in equilibrium at T

C

=1000K,while the reservoir including all phonons maintains at the room temperature (300K).

For the decays of excess polar phonons, all phonons can no longer be assumed to be in equilibrium, in orderto to avoid zero net transition rates. Here we assume the high-lying modes are in equilibrium at 1000K whilethe low-lying modes are at the room temperature. The setting is physical since the energies of the emitted polarphonons are quickly re-distributed into other high-lying modes due to the phonon-phonon interaction (throughRidley channel or Barman-Srivastava channel). Therefore the high-lying branches are populated very fast whichcan be regarded as a particle system with high-level interactions. On the other hand, the low-lying branches canhardly be populated partly due to the large phononic gap. Another reason is that the energies of the decayedlow-lying phonons are immediately re-distributed into the whole system of the low-lying phonons, because theirinteractions are very fast due to the quasi-continuous phonon density of states and their higher occupancies.Besides, the excess phonons in acoustic branches will quickly di↵use out, being relaxed into an equilibriumoccupancy at the room temperature. Therefore the low-lying phonons can be regarded as a reservoir at the roomtemperature. The energy relaxation time of the hot phonons is defined as referring to the energy flow out ofthe high-lying LO phonon system, i.e. �

ph

= EhphP✏lphwlph

, where Ehph

is the total energy of the high-lying LO

phonons and the summation in the denominator is over the modes of high-lying LO phonons, with ✏lph

and wp

indicating the energy and annihilation rate of each mode.



w =2⇡

} |Hi

|2�(�E) (7)

The transition rates are calculated from the 1st order perturbation theory shown in Eq. (7), where Hi

is theinteraction hamiltonian given by Eq. (4) and Eq. (6) and �E is the energy change during the transition, i.e. thedi↵erence of the total energy of the final states and that of the initial states. The 1st order perturbation theory isonly valid for the case that the interaction energy much weaker than the self-energy of particles involved, and theinteraction should last for su�cient long time. For electron-phonon interaction, although there is some deviationresulted from the further decay of the emitted phonons, the approximation is still accurate enough and has beenadopted in most of the relevant works.

2.6 Particle dynamics at steady state with non-equilibrium occupancy

In reality the high-lying phonon modes have occupancies which deviate from the equilibrium, for they interactwith two reservoirs with very di↵erent statistics: in some branches of the high-lying region (i.e. polar opticalmodes) phonons are quickly generated by interacting with the hot electrons, while each high-lying phononthen decays into a low-lying phonon and a high-lying phonon with relatively lower frequency (Ridley channel orBarman-Srivastava channel). The hot electron system is regarded as a reservoir in equilibrium, with temperaturehigher than the room temperature, because of their quick mutual interactions and continuous photon energysupply when the device operates at steady state. The low-lying phonons are assumed to be in equilibrium atthe room temperature. Only the high-lying phonons are assumed as being in non-equilibrium, of which theoccupancies are to be calculated.

In this work the non-equilibrium occupancies of high-lying modes have been computed under the steady-statecondition, with the assumptions that the emission of polar phonons are merely due to hot electrons (assumedas 1000K). In fact holes obtain less energy from photo-generation and are more easy to be thermalised partlydue to their large e↵ective mass. The computational strategy is based on the time-domain iteration and MonteCarlo simulation. Initially a guessed occupancy function is adopted for high-lying modes and the fixed step-time is set for the iteration. Each iteration starts from the generation of a random set of electrons accordingto the Fermi statistics, with information of their energies and momentums. For each electron the probabilityof emitting phonons for each polar mode is calculated; after one iteration the occupancies of the polar modesincrease accordingly. Then the transition rates of all possible three-phonon processes are calculated within thesame time interval, leading to a re-normalization of all phonon modes. The computation process is iterated untilsaturation on the mode occupancies is reached, resulting in the steady-state occupancies of all high-lying modes.

In order to simulate the dynamics of particle interactions the phonon di↵usion is necessary to be considered, asan alternative relaxation path of hot phonons. In fact both Ridley and Barman-Srivastava channels conserve thephonon number in high-lying branches, while at steady state the emission of optical phonons from hot electronsmust be balanced by some relaxation path. In ideal InN/In

x

Ga1�x

N SL the relaxation is from phonon di↵usionmost probably. A simplified model has been suggested, where an average product of mean free path and groupvelocity is adopted for all optical modes. In fact the wave-vector-dependent di↵usion rates are smeared, for thewave-vector of a phonon will be randomized after several successive three-phonon processes.

@nq,w

@t= Bq,w �Aq,wnq,w + ⌧v2

@2nq,w

@x2(8)

where nq,! is the occupation number at mode (q,!). G(q,!,x, t) is the generation rates of polar phonons,emitted by hot electrons. With a linear approximation, Aq,w is defined as its averaged occupancy-derivative

Aq,w = �@G(q,!)/@nq,w. At steady state Eq. (8) gives the solution Eq. (9) if canceling the dependance onBq,w.



6-

4-

2-

2 4 6

Barrier Thickness (nm)8

zemss.,(PS)0.04000

0.1600

0.2800

0.4000

0.5200

0.6400

0.7600

0.8800

1.000

1.120

1.240

6-

4-

2-

2 4 6


emss Jps)0.08500

0.2050

0.3250

0.4450

0.5650

0.6850

0.8050

0.9250

1.045

1.165

⌧v2@2nq,w

@x2

��x=0

=Aq,w

cosh(W/2p↵q,w)� 1

⇥nq,w � neq

q,w(Trt

)⇤

(9)

where ↵q,w = ⌧v2/Aq,w, neq

q,w(Trt

) is the occupation number of each mode, in equilibrium at the roomtemperature T

rt

. W is the dimension along the direction of phonon di↵usion, and the superlattice slab is placedfrom �W/2 to W/2. The equation provides the occupation annihilation rate at the middle of the system, i.e.x = 0.

3. RESULTS AND DISCUSSION

3.1 Polar phonon emission

The energy relaxation times referring to the polar phonon emission are illustrated in Fig. (1), for all the MQW-SLconfigurations, i.e. for di↵erent well and barrier thicknesses and di↵erent barrier materials. 18⇥18 combinationsof the thicknesses are sampled for representing all the possible structures from 2nm to 9nm. The number ofcombinations comes from the fact that a complete well/barrier layer should include a integer number of unitcells. Fig. (1) is generated by interpolating the relaxation time data of the sampling combinations.

Fig 1. Energy relaxation times of the hot electron system (1000K), in the reservoir of phonons (300K), with superlatticestructures of di↵erent well/barrier thicknesses and In

x

Ga1�x

N compositions (Left: x=0 Right: x=0.2)

From Fig. (1) the general trend indicates the thinner barrier layer gives the longer energy relaxation timeof the electrons. There are several possible reasons to explain the observation. The screening of polar modesstrongly depends on the carrier concentration, which favors a wide well layer and a thinner barrier layer. Infact the MQW-SL structure with a wider well layer has lower confined energy levels, leading to a higher electronoccupancy, while in SL with a thinner barrier layer the average potential of electrons is reduced resulting inlower energy levels of non-bound states. The high carrier concentration significantly screens the potential ofpolar modes, and hence demonstrates a reduced cooling rate of electrons. It is also possible that the larger well-to-barrier ratio increases the relaxation time because InN has flatter dispersions than that of GaN. For MQW-SLthe zone-folding of flat dispersions results in very narrow mini-bands, which slows down the rates of intra-bandtransitions, while the inter-band transitions are prevented by the mini-gaps. Another reason of the trend resultsfrom the multiple energy levels of the confined polar modes in the MQW-SL. If the periodic dimension of the SLbecomes larger then the increased number of these energy levels could increase the number of allowed transitions,and hence the cooling rate. This e↵ect favors a thinner well layer as well, which balances o↵ its benefits due to



8-

6

4-

2-

2 4 6


tieeceY (Ps)

0.000

36.00

72.00

108.0

144.0

180.0

216.0

252.0

288.0

324.0

8-

6

4-

2-i

2 4 6


tiaecay (PS)

6.000

32.00

58.00

84.00

110.0

136.0

162.0

188.0

214.0

240.0

266.0

the first two reasons. Therefore the preferable SL structure involves a thinner barrier layer, while having littledependance on the well thickness, as shown in Fig. (1).

As to the e↵ect of di↵erent Indium content x in the barrier material Inx

Ga1�x

N, the contrast between theminimum and maximum relaxation times becomes smaller for a larger x, if comparing the two figures in Fig. (1).In fact with a larger Indium content, the electronic and phononic properties of In

x

Ga1�x

N are more close to thatof bulk InN, and hence the relaxation time values will shift towards the value of bulk InN. Hence the optimalstructure with the longest relaxation time (in terms of the hot phonon emission) exist in the case of x=0, whileon the other hand the high potential barrier is detrimental to the carrier transport. In terms of designing hotcarrier solar cells, a comprehensive consideration is necessary to take all these properties into account.

3.2 Three-phonon process

The energy relaxation times of high-lying longitude optical phonons are demonstrated in Fig. (2), for di↵erentcombinations of well and barrier thicknesses. For Indium mole fraction x=0 (left figure), the contrast of relaxationtimes are larger than that of the case for x=0.2 (right figure); this is reasonable for more Indium content inthe barrier layer would make the structure closer to bulk InN. Both figures show the same trend with di↵erentbarrier/well thicknesses: the relaxation time increases with thicker well layers and thinner barrier layers, in spiteof some irregular local variations. The regular variation mainly results from the change of numbers of InN-likemodes and InGaN-like modes.

Fig 2. Energy relaxation times of the high-lying LO phonon system (1000K), in the reservoir of 1000K high-lying phononsand 300K low-lying phonons, with superlattice structures of di↵erent well/barrier thicknesses and In

x

Ga1�x

N compositions(Left: x=0 Right: x=0.2)

To explain the variation, we need to first examine the phonon dispersions of the MQW-SL. Taking x=0 asan example, the left figure in Fig. (3) shows the dispersions of InN/GaN SL structure, with 6 layers of nitrogenatoms inside each layer (same for the InN layer and the GaN layer). From the number of atomic layers involvedwe could expect the same amount of high-lying optical modes which are InN-like and GaN-like respectively.The GaN-like optical modes (blue color: solid line for LO, dash line for TO) have much higher frequencies thanthe InN-like optical modes (green color), for the bond force constants of GaN is larger and the atomic mass ofgallium is smaller. There are also four branches (orange color: two of LO and two of TO) corresponding to theinterfacial modes (IF), with frequencies between GaN-like modes and InN-like modes. From the right figure ofFig. (3), the vibrations are almost completely confined in the respective type of layers, for InN-like and GaN-likeoptical modes. For IF most of the vibrational energy is within the 1st and 2nd nitrogen atoms from the interface,and hence has little overlap with InN-like or GaN-like modes.



0.10 -

1

0.05 -

c0.00 .= -

1

-9.63x108 -3.23 3.17x108 9.57x108

Wave Vector (m1)

- GaN /InN mixedGaN- confined

- GaN -like LO- IF (LO)-InN -like LO- - GaN /InN mixed

GaN -confinedGaN -like TO

- - IF (TO)- - InN -like TO

\,/47\,/

" InN GaN

0.0 1.0x109 2.0x109 3.0x109

Positions of Nitrogen Atoms (nm)

-t-GaN/InN mixedt GaN-confined-A- GaN-like LO-y- IF (LO)-4- InN-like LO

Considering all three-phonon processes, only the Ridley channel and the S-B channel (decaying into a high-lying optical phonon and a low-lying optical phonon) are allowed due to the energy conservation law. As inwurtzite-structured SL, the di↵erence between the acoustic branches and the low-lying optical branches becomesunimportant, and the low-lying branches can be separated into two categories: GaN-confined modes (red lines)and GaN-InN mixed modes (gray lines). In fact the InN-like low-lying modes sit within the allowed frequencyband of GaN, which also excite vibrations in GaN layers. Therefore no InN-confined modes exist in the low-lyingbranches. Among all allowed three-phonon processes, GaN-like optical modes can decay into both types of low-lying modes (See the solid arrow and the dash arrow in Fig. (3)), while InN-like optical modes can only decayinto the mixed modes for it only overlaps with the mixed modes. Besides, the energy gap between GaN-like LOmodes and GaN-like TO modes is relatively large; hence the resulting low-lying phonons have relatively highenergies, compared to the decayed low-lying phonons from the InN-like modes. Since the low-lying brancheswith high energies are generally flatter, leading to a larger joint density of states of transition, the decay ratesof GaN-like LO modes could be enhanced further. Due to the two reasons explained above the GaN-like LOmodes decay faster than the InN-like LO modes. Therefore with a thicker well layer (or a thinner barrier layer),the energy relaxation time of the high-lying LO phonon system becomes longer, for it introduces more InN-likemodes (or less InGaN-like modes).

The energy relaxation time of 1000K high-lying LO phonon system (in the reservoir of 1000K high-lyingphonons and 300K low-lying phonons) varies from several picoseconds to several hundreds of picoseconds. Theresulting order of magnitude matches reported experimental observations. The irregular local variation occursdue to the formation of mini-gaps in the phonon energy spectrum. Some decays is blocked if resulting inphonon energies within the mini-gaps. Therefore by finely engineering the InN/In

x

Ga1�x

N SL structure (i.e.well/barrier thicknesses and Indium composition), the e↵ect of mini-gaps may significantly reduce the phonondecay rate, although practically this technique may be too ideal to be reliable. In reality the spatial non-uniformity of fabricated well/barrier layers (including defects and uneven Indium distribution in In

x

Ga1�x

N)is hard to control, and could significantly a↵ect the energy-dependent density of states of phonon modes. Itsenergy spectrum also strongly depends on the interfacial conditions. In the calculation we assume the InN/GaNinterface strictly lies on the same atomic plane without any interfacial mixture of Indium and Gallium atoms.Finally even with ideal fabrication conditions there must be several layers of stressed atoms due to the latticemismatch, whose equilibrium positions are shifted as well as their potentials in the crystal.

Fig 3. The right figure shows the phonon modulation function of a representative mode from each category of modes,computed from the eigenfunctions of the lattice dynamic equation. The transverse modes have similar modulationfunctions and hence only that of longitude modes are shown.



1.20E-013 -

NE

0

m 1.10E-013-TIKm

-01.00E-013 -

- M-x=0t x=0.2

Thermalised phonon limit

0.1 10

Diffusion Parameter v2 (mv /s)

0.09 -

0.08 -404.v.,,k.,,00,..4004"*4

0.07

0.06

0.05

InN -like polar modes

GaN -like polar modes

0.04-9.40E +008 -4.70E +008 0.00E +000 4.70E +008 9.40E +008

Wave Vector (m')

3.3 Non-equilibrium distributions of high-lying modes

The non-equilibrium occupancies of all high-lying modes have been computed considering the processes of polarphonon generation, three-phonon interaction and hot phonon di↵usion. Due to the limited resources of com-putation, only a fixed InN/In

x

Ga1�x

N MQW-SL structure has been taken into account, with three layers ofunit cells stacking in each layer (layer thickness ⇡ 2nm). The slab is assumed to have a thickness of 100µm(around 25,000 bi-layers of InN and In

x

Ga1�x

N). For simplicity we only focus on the statistics at the middle ofthe slab, and hence Eq. (9) can be used to treat the phonon di↵usion. The left figure in Fig. (4) demonstratesthe steady-state energy relaxation times of electrons for both cases of x=0 and x=0.2. With a larger di↵usionparameter ⌧v2, the hot high-lying phonons di↵use away faster creating a less occupied steady-state distribution.For ⌧v2 > 1m2/s the hot distribution of high-lying phonons no longer exist (thermalised phonon limit), andhence the relaxation time reaches the lower limit which is merely determined by the electron-polar phonon in-teraction. On the other hand due to the lack of Klemens decay the relaxation time does not have a upper limit,as the number of high-lying phonons can only be relaxed by hot phonon di↵usion.

Fig 4. The left figure is the variation of energy relaxation times of electrons in a MQW-SL with three unit cells in eachlayer, on di↵erent di↵usion parameters ⌧v2; the right figure shows the non-equilibrium occupation numbers of polar modes,for ⌧v2 = 1m2/s and x=0.

The right figure in Fig. (4) shows the occupation number of all the six polar mode branches in the samestructure (x=0). Value of the di↵usion parameter adopted here is ⌧v2 = 1m2/s. The three upper branchescorrespond to the InN-like polar modes and the lower three are for the GaN-like polar modes; a branch witha higher occupancy corresponds to a phonon branch with a lower frequency. From the figure the polar branchwith the lowest energy has the steady-state distribution which deviates most from the equilibrium distribution,the reason of which lies in the overlap integral between the electronic state and the polar mode involved inthe interaction. The first InN-like polar mode has a modulation function with zero node, i.e without changeof sign within one InN layer; therefore it interact most strongly with the lowest confined electronic state whichalso involves zero node. The other InN-like polar modes have weaker (odd number of nodes) or no interaction(even number of nodes) with the first confined electronic state due to their symmetry properties. As to theGaN-like polar modes, they are hard to be populated because of the small electron concentration in the barrierlayers. Generally the occupation number gradually decreases from the zone center to the zone edge, while inthe region closely around the zone center the occupancy drops because of the carrier screening e↵ect on polarinteractions. In fact by taking the screening of polar modes into consideration, the electrons can only interactwith the ionic dipoles within the region characterized by the screening length, instead of the larger region betweentwo neighboring vibrational nodes. This reduces the interaction strength most for phonons located at the zonecenter.



3.4 Implication on the design of hot carrier solar cells

The nature of confinement in InN/Inx

Ga1�x

N MQW-SLs (with small x) enables di↵erent carrier relaxationproperties from that in bulk materials, and hence provide possibilities to realize a long cooling time of electrons.In terms of the emission of polar phonons, a InN/In

x

Ga1�x

N MQW-SL structure with a thin barrier layer hasa long relaxation time, while the phonon decay favors a SL structure with a thin barrier layer and a thick welllayer. On the other hand, in terms of blocking the hot phonon di↵usion, a thinner bi-layer of InN and In

x

Ga1�x

Nbenefits more, for in this structure phonon modes have a reduced group velocity along the vertical direction.With very thin barrier layers (less than two unit cells) the In

x

Ga1�x

N-like modes can no longer be completelyconfined and are mixed with the interfacial modes; in conclusion a SL structure with a reasonably thin barrierlayer (around 2nm) and a reasonably thick well layer (around 10nm) has the potential benefit of reducing theoverall carrier cooling rate. Increase of the Indium content in the barrier layer could shift the material propertiestowards that of the bulk, which is not favorable in terms of the relaxation properties. But the design of hot carriersolar cells require a combined benefits of di↵erent material properties: a high barrier height is impropriate forthe carrier transport, and hence practically some amount of Indium in the barrier layer may benefit the overallperformance of the cell.

4. CONCLUSION

In this work the energy relaxation processes of photo-generated electrons are analyzed in the MQW-SL structureconsisting of bi-layers of wurtzite-structured InN and In

x

Ga1�x

N. The energy relaxation times are chosen asthe indicator to characterize the electron cooling properties of each SL structure. The phononic structures andelectronic structures of the MQW-SLs are modeled and computed as well as the transition rates of relevantprocesses. In terms of the polar phonon emission, SLs with thinner barrier layers show longer relaxation times,and a smaller Indium content in the barriers gives a more di↵erent relaxation characters from the bulk. Therelaxation of polar phonons is the fastest in a SL structure with a thick well layer and a thin barrier layer, despiteof the local fluctuation due to the open-up of phononic mini-gaps. The steady-state non-equilibrium distributionof high-lying phonons has also been analyzed including the e↵ect of hot phonon di↵usion. As a result, the mostappropriate InN/In

x

Ga1�x

N MQW-SL structure should contains a thin barrier layer and a thick well layer, whilethe necessaries of the confined modes requires that the barrier layer cannot be too thin (less than two unit cells).

5. ACKNOWLEDGMENTS

This project has been supported by the Australian Government through the Australian Solar Institute (ASI),part of the Clean Energy Initiative. The Australian Government, through the ASI, is supporting Australianresearch and development in solar photovoltaic and solar thermal technologies to help solar power become costcompetitive with other energy sources.

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