Phonons in GaN-AlN Nanostructures

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Inelastic Light Scattering by Phonons in Hexagonal GaN–AlN Nanostructures

Recent advances in fabrication methods enable realization of one-dimensional hetero-structures, i. Their electronic and optical properties were studied, and a variety of possible applications utilizing the characteristics were also proposed[ 11 — 14 ]. In addition, the NWSLs are expected to yield interesting physical effects on phonons, which influence the electronic states and the transport properties via the electron-phonon interaction. These NWSLs can be regarded as wire-type phononic crystals WPCs , in which the phononic bandgaps are induced by the periodicity along the wire axis.

In a previous paper[ 15 ], we developed a numerical method to derive phonon modes in a free-standing NWSL of anisotropic material with an arbitrary shape of cross-section. Though above result revealed the important aspects of phonon modes in the NWSLs, it seems to be difficult to design WPCs with complete phononic bandgaps because in the dispersion relations of these NWSLs, many subbands are folded into the mini-Brillouin zone and the frequencies of gaps are different with phonon modes. The NWSLs with large acoustic mismatch would be suitable for designing the phonon optic devices.

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In the present work, we numerically calculate the dispersion relations and corresponding displacement fields for a circular cross-section NWSL consisting of GaN and AlN, and we determine a set of parameters which gives complete frequency gaps. The equation giving the eigenfrequencies of phonon modes in a freestanding NWSL composed of anisotropic crystals was formulated in[ 15 ]. As basis functions, we adopt the product of powers of the Cartesian coordinates in the xy plane.

The z dependence is expressed in the form of the Bloch wave:. Based on group theory[ 17 , 18 ], the phonon modes are classified symmetrically. In the present study, the constituent layers are assumed to be cubic materials, i. By considering the above symmetry, the symmetry-adapted basis function in the present system can be constructed, and the phonon dispersion relations of each mode are independently calculated. The boundary condition at the free surface of the wire requires that the wave numbers in the lateral direction are discretized. On the other hand, the wave number k in the longitudinal direction has a continuous value.

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Therefore, even the homogeneous plain nanowire has subband structure. In other words, the size of the mini-Brillouin zone and corresponding phononic bandgaps are determined by the periodicity D of the NWSL. For the nanowires with smaller R , the maximum wavelengths in the lateral directions become shorter. As a result, the subbands except for the lowest dispersion curve of each mode go up to the higher frequency region, though a lot of subbands are folded into the mini-Brillouin zone.

Changing the ratio of R and D , we can control the phonon modes in the lower frequency range. The B 2 modes exist in higher frequency range, and all the E modes are doubly degenerated. In the present frequency range, we can see four different modes, i. The dispersion curves corresponding to the B 2 modes exist in higher frequency range. The lowest dispersion curve of the A 2 mode is mostly overlapped with that of the A 1 mode. The complete frequency gaps are realized for the parameters we selected.

In this example, the broader bandgaps of the A 1 , A 2 , and E modes are nearly coincident with each other.

Here, we note that the complete bandgaps disappear if the isotropic approximation is used for each constituent layer. The lowest dispersion curves of the A 1 and A 2 modes are linear in k , i. On the other hand, the lowest frequencies of the B 1 and B 2 modes have finite values. The lowest A 1 mode has the feature of a dilatational mode, while the lowest A 2 mode shows the feature of a torsional mode. The lowest B 1 mode shows the feature of a stretching mode, i.

All dispersion curves of the E mode are doubly degenerate because the irreducible representation of the E mode is two-dimensional. This parabolic behavior is due to the fact that these modes correspond to the bending of the NWSL. Figure 2 e clearly shows that the E modes have a feature of flexural mode. We calculated their dispersion relations and phonon displacement fields.

These modes are classified into five types, i.

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We determined a set of parameters which gives complete phononic bandgaps. In the present work, we only showed the results for the NWSLs consisting of cubic materials i. The results for the NWSLs with wurtzite structure will be given and the difference will be discussed elsewhere. Page JH: Phononic crystals. Phys Stat Sol b , It should be noted that for the confined modes, the dispersion curves asymptotically approach a characteristic phonon frequency of GaN i.

This interface phonon dispersion relation gives rise to a phonon emission threshold energy in the electron—phonon scattering process that does not correspond to an energy of the characteristic phonons of either AlN or GaN. Only a few of the confined mode phonons two modes each for the asymmetric and symmetric branch with lowest quantum numbers are plotted.

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Phonon modes associated with TO phonon modes of AlN and GaN are shown in the a lower frequency region whereas those associated with LO phonon modes are in the b higher frequency region. The interface phonon resonant frequencies are shown in dotted horizontal lines and the other characteristic frequencies are shown in dashed horizontal lines.

The x -axis is shown in the dimensionless wave vector qd. Another important feature in the dispersion relation is that these interface phonon modes have a nonzero slope at the zone center. Notice that the x -axis is set to the dimensionless product of phonon wave vector q and the quantum well thickness d.

Therefore, the group velocity will increase with increasing quantum well thickness due to the stronger dispersion. To investigate the electron scattering rate with confined and interface mode phonons, we adapt the expression from ref. The upper lower signs are taken when considering electron absorption emission scattering process. The summation over the quantum number n [identical to the ones in Eqs 7 and 8 ] is included to consider scattering with all possible symmetric and asymmetric confined mode phonons.

Also, the symmetric and asymmetric confined modes must be considered separately as the dispersion relations are different from each other.

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Since the implications of the argument inside the square root of the denominator of Eq. This way, we can simply set the integral limits to the phonon frequencies where the phonon mode of interest is defined. Only the electronic ground state of the quantum well is considered and a conduction band offset of 2. In Eq.

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  • Given the complex form of the phonon wave vector q , instead of expressing these velocities in a closed form equation, we calculate the velocities numerically. The group velocity is not only a component of the integral but also a crucial factor that describes the behavior of the phonons, and furthermore, the thermal characteristics of the system. As expected by the dispersion relation shown in Fig.

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    It is easily deduced that if the full spectrum of interface phonon modes can be utilized, the phonons can carry a portion of the generated heat away along the heterointerface. This would be an additional heat transport mechanism, on top of the always existing acoustic phonon heat transport, that can help the system dissipate heat more efficiently.

    The numerically calculated group velocities of interface and confined mode phonons are shown as a function of phonon frequency. For both interface and confined modes, the solid and dashed lines indicate the asymmetric and symmetric phonon modes, respectively. This value is almost 20 times larger than the maximum group velocity of the confined mode phonons. To better understand the electron interaction with these interface and confined mode phonons, we evaluate Eq.

    For both a interface and b confined modes, the black solid line shows the total scattering rate which combines all contributions from each process indicated as dashed and dotted color lines. The general behavior of the scattering rate curves for both modes are similar. The LO emission scattering rates [ a red and b orange dashed lines] start to dominate once the electron energy exceeds the threshold energy.

    The TO absorption processes are negligible compared to the others not plotted in the figures. Due to mode mixing in wurtzites, the TO emission scattering rate is comparable to the LO absorption scattering rate. This causes the total scattering rate to show a two-step-like shape. As shown in Fig. For both modes, the total interface phonon scattering rate combining all phonon modes including the TO absorption scattering rate is shown as the black solid line. The droplines with symbols are shown to indicate the threshold energies of TO emission and LO emission scattering.

    Focusing on the interface mode LO emission scattering rate [red dashed curve in a ], we observe that the emission threshold energy is The LO phonon emission process through electron—phonon scattering can only occur when the electron has, at least, more energy than the phonon to be emitted; of course, more specifically, the condition described by Eq.

    The emission threshold energy of From Fig. The TO emission threshold energy These observations of Fig. For the confined modes in Fig. As shown in the dispersion curve in Fig. As the electron energy E k increases, the energy and momentum conservation condition Eq. Since Fig.

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    • Also notice that the interface mode scattering curve shows more than two of the step-like features. This is due to the emission threshold energy split between the symmetric and asymmetric interface modes. However, with smaller d , this is no longer the case and the condition is satisfied with phonon frequencies slightly away from the resonant frequencies. To further examine the interface phonons that are emitted in the LO emission scattering process, the integrand of Eq.

      The plotted curves represent the number of phonons produced with each phonon frequency. Only the interface phonons involved in the LO emission phonon process are plotted. The LO phonon resonant frequency is indicated as the vertical dotted line. The y -axis represents the integrand of Eq. The vertical dotted line in Fig. Phonons with frequency larger smaller than this resonant frequency are symmetric asymmetric modes. As the electron energy increases, the scattering process starts to produce more phonons with frequencies farther away from the resonant frequency.

      Returning to Fig. In order to utilize the interface phonons with larger group velocity, the electron energy must be increased. An alternative way to produce interface mode phonons with high group velocity may be increasing the GaN thickness d as was suggested from the dispersion relation shown in Fig. The group velocity in Fig. The relation between the GaN thickness d , phonon dispersion relation, interface phonon mode group velocity, and electron energy E k must all be taken into account to understand if the nonzero interface mode phonons can be exploited to effectively dissipate heat of the system.

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      Phonons in GaN-AlN Nanostructures Phonons in GaN-AlN Nanostructures
      Phonons in GaN-AlN Nanostructures Phonons in GaN-AlN Nanostructures
      Phonons in GaN-AlN Nanostructures Phonons in GaN-AlN Nanostructures
      Phonons in GaN-AlN Nanostructures Phonons in GaN-AlN Nanostructures
      Phonons in GaN-AlN Nanostructures Phonons in GaN-AlN Nanostructures

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