2.1.1.

Wide phononic gaps in III-Vs and analogues

For some compounds in which there is a large difference in masses of the constituent elements, there exists a large gap in the phonon dispersion between high-lying optical phonon energies and low-lying acoustic phonon energies. If large enough this “phononic band gap” can prevent Klemens decay of optical phonons, because no allowed states at half the LO phonon energy exist. Indium nitride (InN) is an example of such a material with a very large phonon gap.

This prevention of the Klemens mechanism forces optical phonon decay via the next most likely, Ridley mechanism, of emission of one TO and one low energy LA phonon. Such a mechanism only has appreciable energy loss (although still much less than Klemens’ decay) if there is a wide range of optical phonon energies at zone centre. This is only the case for lower symmetry structures such as hexagonal. For a high symmetry cubic structure, LO and TO modes are close to degenerate at zone centre with a very flat dispersion and the Ridley mechanism is severely restricted or forbidden. Unfortunately cubic InN is very difficult to fabricate precisely because of the large difference in masses that give it its interesting phononic dispersion. In addition to the small dispersion in cubic structures dispersionless structures can be achieved by engineering nanostructures.

Slowed cooling has been observed in some III-V compounds in which there is a large difference in atomic mass. This has been shown in slowed carrier cooling for InN ¹⁰ as plotted in Fig. 2, and for slowed carrier cooling in InP compared to the small mass ratio GaAs,¹¹ as discussed below in section 4. Analogues of InN with abundant elements, but also with narrow E_g are discussed below in Section 3.1.

Figure 2:

Effective carrier temperature as a function of carrier lifetime for bulk GaAs as compared to GaAs/AlGaAs MQWs: time resolved transient absorption data for different injection levels, from Rosenwaks,¹² recalculated by Guillemoles.¹³ Also shown are temperatures from transient absorption data for InN from Chen ¹⁰ and indicative carrier temperatures based on long phonon lifetimes for AlSb and InP from Barman.²⁶

2.2.1.

Slowed carrier cooling in MQWs

Low dimensional multiple quantum well (MQW) systems have also been shown to have lower carrier cooling rates. Comparison of bulk and MQW materials has shown significantly slower carrier cooling in the latter. Figure shows data for bulk GaAs as compared to MQW GaAs/AlGaAs materials as measured using time resolved transient absorption by Rosenwaks et. al. ¹² recalculated to show effective carrier temperature as a function of carrier lifetime by Guillemoles.¹³ This clearly shows that the carriers stay hotter for significantly longer times in the MQW samples, particularly at the higher injection levels by 1½ orders of magnitude. This is due to an enhanced ‘phonon bottleneck’ in the MQWs allowing the threshold intensity at which a certain ratio of LO phonon re-absorption to emission is reached which allows maintenance of a hot carrier population, to be reached at a much lower illumination level. More recent work on strain balanced InGaAs/GaAsP MQWs by Hirst ¹⁴ has also shown carrier temperatures significantly above ambient, as measured by photoluminescence. Increase in In content to make the wells deeper and to reduce the degree of confinement is seen to increase the effective carrier temperatures.¹⁵ And extraction of a small portion of this nonequilibrium hot carrier population as a very small external current at low temperatures is demonstrated for a single InGaAs QW by Hirst.¹⁶

The mechanisms for the reduced carrier cooling rate in these MQW systems are not yet clear. However there are at least three possible effects that are likely to contribute, more than one of which could well occur in parallel. The first of these effects is that in bulk material photogenerated hot carriers are free to diffuse deeper into the material and hence to reduce the hot carrier concentration at a given depth. This will also decrease the density of LO phonons emitted by hot carriers as they cool and make a phonon bottleneck more difficult to achieve at a given illumination intensity. Whereas in a MQW there are physical barriers to the diffusion of hot carriers generated in a well and hence a much greater local concentration of carriers and therefore also of emitted optical phonons. Thus the phonon bottleneck condition is achieved at lower intensity.

The second effect is that for the materials systems which show this slowed cooling, there is very little or no overlap between the optical phonon energies of the well and barrier materials. For instance the optical phonon energy ranges for the GaAs wells and AlGaAs barriers used in ¹² at 210-285cm^-1 and 280-350cm^-1, respectively, exhibit very little overlap in energy, with zero overlap for the zone centre LO phonon energies of 285 and 350cm^-1.¹⁷ Consequently the predominantly zone centre LO phonons emitted by carriers cooling in the wells will be reflected from the interfaces and will remain confined in the wells, thus enhancing the phonon bottleneck at a given illumination intensity.

Thirdly, if there is a coherent spacing between the nano-wells (as there is for these MQW or superlattice systems) a coherent Bragg reflection of phonon modes can be established which blocks certain phonon energies perpendicular to the wells, opening up one dimensional phononic band gaps (analogous to photonic band gaps in modulated refractive index structures. For specific ranges of nano-well and barrier thickness these forbidden energies can be at just those energies required for phonon decay. This coherent Bragg reflection should have an even stronger effect than the incoherent scattering of the second mechanism above at preventing emission of phonons and phonon decay in the direction perpendicular to the nano-wells.

It is likely that all three of these effects will reduce carrier cooling rates. None depend on electronic quantum confinement and hence should be exhibited in wells that are not thin enough to be quantized but are still quite thin (perhaps termed ‘nano-wells’). In fact it may well be that the effects are enhanced in such nano-wells as compared to full QWs due to the former’s greater density of states and in particular their greater ratio of density of electronic to phonon states which will enhance the phonon bottleneck for emitted phonons. The fact that the deeper and hence less confined wells in ¹⁴ show higher carrier temperatures is tentative evidence to support the hypothesis that nano-wells without quantum confinement are all that are required. Whilst several other effects might well be present in these MQW systems, further work on variation of nano-well and barrier width and comparison between material systems, will distinguish which of these reduced carrier diffusion, phonon confinement or phonon folding mechanisms might be dominant.

3.1.1.

II-IV-Vs:ZnSnN;ZnPbN;HgSnN;HgPbN

With reference to Figure 3, it can be seen that replacement of In on the III sub-lattice with II-IV compounds is analogous and is now quite widely being investigated in the Cu₂ZnSnS₄ analogue to CuInS₂.¹⁸

Figure 3:

Use of the periodic table to analyse possible analogue compounds of InN based on atomic mass combination and electronegativity.

ZnGeN can be fabricated ^19,20 and is most directly analogous with Si and GaAs. However, its electronic band gap is large at 1.9eV. It also has a small calculated phononic band gap.²¹ ZnSnN has a smaller electronic gap (1 eV) and larger calculated phononic gap. ^20,21 It is however difficult to fabricate, and also its phononic gap is not as large as the acoustic phonon energy making it difficult to block Klemens decay completely. HgSnN or HgPbN should both have smaller E_g and larger phononic gaps. These materials have not yet been fabricated and have the problem that Hg has about half the abundance of In and is toxic.²²

3.1.2.

Large mass anion:

Bi and Sb are heavy elements and hence their compounds with B and Al have large predicted phononic gaps.²³ BiB has the largest phononic gap but AlBi, Bi₂S₅, Bi₂O₃ (bismuthine) are also attractive. AlSb has a calculated and measured phononic gap the same size as its acoustic phonon energy.²⁴ It also has a very flat optical phonon dispersion that should suppress Ridely as well as there being almost no chance of Klemens decay.²⁵ ASb has a very high calculated phonon lifetime probably because of its large phononic gap, similar to InP,²⁶ which give the predicted large carrier temperatures shown in Fig. 2. The electronic band gap of AlSb is 1.5eV which, although indirect, makes it marginally suitable as an absorber. These phonon properties, and the fact that it is of reasonably widespread use and can be made of reasonable quality, make AlSb reasonably attractive as a hot carrier absorber material. BSb has a larger elemental mass difference giving it a very large phonon band gap more than twice the acoustic phonon energy, as calculated in 3D using an ab-initio DFT approach,²⁷ sufficient to block Klemens decay completely if the material quality is high enough. It also can be fabricated by co-evaporation and has a measured E_g of 0.59eV.²⁸ Its band gap is indirect and hence not ideal, but the absorption coefficient remains high even out to 1700nm. Bi is a rare element with an earth abundance about twice that of In, with Sb about five times In’s abundance.²² Nonetheless their large phonon gaps make particularly BSb and AlSb interesting to pursue further for proof of concept. A quantum well or quantum dot nanostructure of AlSb/BSb would be particularly interesting as it could combine phononic band gap and MQW slowed carrier cooling effects. Test structures in related GaInAsSb/AlGaAsSb MQW systems have indeed shown potential for reduced cooling due to reflection of phonons at interfaces.²⁹

3.1.3.

Transition metal nitrides:

The Group IIIA nitrides LaN, YN and ScN have large elemental mass differences and would have been thought to have large phononic gaps, but as they form in a NaCl structure rather than the zinc-blende structure of the Group IIIB nitrides, they actually have zero phonon band gap.^30,31 However the IVA nitrides of Hafnium and Zirconium have large predicted and measured phononic gaps,³⁰ both bigger than their acoustic phonon energies. The phonon gap for TaN is reasonable but not as large as its acoustic phonon energy. Hf and Ta are about the same abundance as Ge and As, with Zr 100x more abundant and all are widely used metals.²² Hf₃N₄ also shows a large phonon band gap (although still too small to block Klemens decay completely) ³² but also a narrow optical phonon dispersion – important to block secondary Ridley decay. These nitrides can also readily be fabricated and have many existing applications, although the lack of stability of HfN in moist air would need to be managed in a real application, probably by encapsulation. Interestingly both Hf₁N₁ and Zr₁N₁ have zero electronic band gap, leading to the surprising conclusion that these semi-metals may be good absorbers for a hot carrier cell.

The Lanthanides can also form III-nitrides. ErN and other rare earth nitrides can be grown by MBE. The phononic band gaps of the Er compounds are predicted to be large, because of the heavy Er cation, but its discrete energy levels make it not useful as an absorber, although the combination of properties in a nanostructure could be advantageous.

3.1.4.

Group IV alloy/compounds:

All of the combinations Si/Sn, Ge/C, Sn/C or Sn/Si look attractive with large phonon gaps predicted in an approach using perturbation DFT,²⁵ with that for SnSi particularly large and with a very flat dispersion, which should enable it to suppress most phonon decay mechanisms. The predominantly covalent bonding, due to all the elements being group IV, results in very little splitting of optical modes and hence flat dispersions, but also in relatively weak bonds that make true stoichiometric compounds difficult to form. Unfortunately SiC, whether 3C, 4H or 6H, has too narrow a phononic gap. Nonetheless GeC and SnSi are of significant interest and the former has been grown as a compound, although stoichiometric quality is as yet not high.^33,34

The IV-VI PbS also has a large mass ratio and is interesting, but its measured phonon dispersion shows an additional TO branch that fills the phonon gap,²⁴ thus likely making it unsuitable for an absorber in bulk form. But initial modelling indicates that phonon folding in a nanostructure could reflect these modes and give PbS nano-crystals a large phonon band gap and hence potentially be highly suitable as an absorber.

There are several other inherent advantages of group IV compounds/alloys all of which are associated with the four valence electrons of the group IVs that result in predominantly covalent bonding:

3.1.5.

Nanostructures:

The phonon dispersion of quantum dot (QD) nanostructures can be calculated in a similar way to compounds.⁶ Their phononic properties can be estimated from consideration of their combination force constants. Hence it is possible to ‘engineer’ phononic properties in a wider range of nanostructure combinations. Of the materials discussed above the Group IVs lend themselves most readily to formation of nanostructures instead of compounds due to their predominantly covalent bonding, which allows variation in the coordination number. Therefore the nanostructure approaches of ⁶ are consistent with a similar description as analogues of InN, whether it be III-V QDs, colloidally dispersed QDs or for core shell QDs.