Crossover in the electron-phonon heat exchange in layered nanostructures
We study theoretically the effect of the effective dimensionality of the phonon gas distribution on the heat exchange between electrons and phonons in layered nanostructures. If we denote the electrons temperature by and the phonons temperature by , then the total heat power is proportional–in general–to , the exponent being dependent on the effective dimensionality of the phonon gas distribution. If we vary the temperature in a wide enough range, the effective dimensionality of the phonon gas distribution changes going through a crossover around some temperature, . These changes are reflected by a change in . On one hand, in a temperature range well below a crossover temperature only the lowest branches of the phonon modes are excited. They form a (quasi) two-dimensional gas, with . On the other hand, well above , the phonon gas distribution is quasi three-dimensional and one would expect to recover the three dimensional results, with . But this is not the case in our layered structure. The exponent has a complicated, non-monotonous dependence on temperature forming a “plateau region” just after the crossover temperature range, with between 4.5 and 5. After the plateau region, decreases, reaching values between 3.5 and 4 at the highest temperature used in our numerical calculations, which is more than 40 times higher than .
pacs:73.50.Lw, 85.80.Fi, 85.25.-j, 73.23.-b
Continuous efforts in miniaturizing devices require better understanding of their quantum behaviors at nanoscale lengths and sub-Kelvin temperatures. Such devices are, e.g., microrefrigerators, microbolometers, and microcalorimeters. Giazotto et al. (2006) The microrefrigerators have been proven to be able to reduce the temperature of the electron gas of a normal metal island down to 30 mK, starting from a bath temperature of 150 mK. Nguyen et al. (2014) Such experimental achievement allows the use of these devices in refrigerating microbolometers, microcalorimeters, qubits, etc.
The devices are usually layered structures consisting of thin metal films (for example, Cu, Al, etc.) deposited on free standing dielectric membranes, typically made of silicon nitride (SiN). The membrane is itself supported by a bulk substrate, which may be considered the thermostat for the whole device. A central issue in the understanding of how the devices function or how they respond to the absorption of electromagnetic radiation is the interaction and the heat exchange between the electron system and the phonon system in the normal metal films. Our study is based on the model presented in Refs. Anghel and Cojocaru, 2016, 2017, in which a Cu layer of thickness is deposited on a SiN membrane of thickness , such that the total thickness of the system is . The temperature range in which the microcalometers operate is several hundreds of mK. Nguyen et al. (2014); Giazotto et al. (2006) For typical values of , which is of the order of 100 nm, and the sound velocities of the SiN, the dominant phonon wavelength of a three-dimensional (3D) phonon gas model is comparable to at temperatures of the order of 100 mK. As the temperature decreases, the 3D dominant phonon wavelength may become much bigger than and excitations of phonon modes perpendicular to the membrane surfaces become very unlikely. In such a case, we can consider the phonon gas as being two-dimensional (2D). In the other extreme, at high enough temperatures, the dominant phonon wavelength may become much smaller than and the phonon gas is practically 3D. Anghel et al. (1998); Anghel and Manninen (1999); Kühn et al. (2004, 2007)
At temperatures of the order of 1 K, due to the depopulation of the phonon modes and the degeneracy of the electrons system, the two subsystems of a normal metal–the electrons and the phonons–become almost isolated from each other(see, e.g., the discussion in Sec. II of Ref. Giazotto et al., 2006). For this reason, they can “equillibrate” independently, that is, they may reach equilibrium distributions over the quasiparticle states at some effective temperatures that we shall denote by (for electrons) and (for phonons).111 Note that we don’t aim at calculation of the temperatures and , which may differ from the temperature of bulk substrate (the thermostat). Therefore we do not need to include phonon-electron scattering explicitly. Using the effective temperatures, the heat exchange between the electrons and the phonons may be written in most cases as Wellstood et al. (1994); Viljas and Heikkilä (2010); Cojocaru and Anghel (2016); Hekking et al. (2008)
|where is the volume and is a constant of material. The exponent varies, depending on the model and the dimensionalities of the electrons and the phonon gases. For 3D models, , Wellstood et al. (1994) for quasi-2D ones, , Viljas and Heikkilä (2010) whereas for a 1D phonon gas coupled to a 3D electrons system, .Hekking et al. (2008) Apparently, this leads to the conjecture|
where is the dimensionality of the phonon gas. Apart from Eq. (1b), it was observed that may take also fractional values: if the phonon gas is 2D and the electron gas distribution is a superposition of 2D conduction bands, then . Anghel and Cojocaru (2016, 2017)
The exponent depends on the effective dimensionality of the phonon gas, but the latter may change with temperature, as we discussed above. Anghel et al. (1998); Anghel and Manninen (1999); Kühn et al. (2004, 2007) Therefore, in a temperature range where the dimensionality of the phonon gas changes, the exponent may also change, showing a crossover between the values corresponding to different dimensionalities. This issue, which, to the best of our knowledge, has not been investigated will be explored here, as a continuation of previous studies. Anghel and Cojocaru (2016, 2017)
As in Refs. Anghel and Cojocaru, 2016, 2017, we consider a metallic film (for example, Cu) deposited on a free-standing dielectric membrane (for example silicon nitride) and perfectly glued to it. Anghel and Kuzmin (2003); Anghel et al. (2001) For the mathematical description, we choose a Cartesian system of coordinates, such that all the surfaces of the membrane and of the film are parallel to the plane. In the direction, the membrane extends in the interval , whereas the metal occupies the interval . In the and directions, the system is considered to be very large (in comparison with the wavelengths of the quasiparticles involved) and has the area ; we assume that .
ii.1 Description of the electrons
The metal contains free electrons, which interact with the phonons that propagate in the whole system. The electrons’ wave functions propagate in the plane and are confined along the direction. We denote the electron’s wavevector by , where and are the components of perpendicular and parallel to the axis, respectively. For , we impose periodic boundary conditions in the plane, whereas for we impose Dirichlet boundary conditions at and . This leads to a density of (allowed quantum) states (DOS) in the variable equal to and to the quantization condition , where is an integer. Then, the electron’s wavefunction is of the form
where and the electron’s energy is
where is the free electron mass. In these notations, the electron annihilation and creation field operators are
respectively, where and are the electron annihilation and creation operators on the state .
Since , has a quasi-continuous spectrum, whereas will be considered discreet. States of constant (or ) and any value of form the 2D conduction bands. We denote by the number of occupied 2D bands at K,
where is the biggest integer smaller or equal to and is the Fermi energy. If the normal metal is Cu, eV and for nm, , whereas for nm, .
ii.2 Description of the phonons
The phonons are quanta of elastic vibrations of the system. They propagate in the plane and are stationary waves along . At the surfaces of the system, at , the phonons satisfy free boundary conditions. To simplify the calculation of the phonon modes, Anghel and Cojocaru (2016, 2017) we consider that the metal and the dielectric membrane have identical mechanical properties, namely they have the same mass density and identical elastic properties. Although this seems to be a crude simplification, it gives us a method to study the characteristics of the electron-phonon interaction in general.
The phonon modes are eigenstates of the elastic dynamic equations and are grouped in three polarizations: horizontal shear (), symmetric () and antisymmetric () modes Auld (1990). While the modes are pure transversal () vibrations, the and modes are superpositions of transversal and longitudinal () vibrations Auld (1990)–in a longitudinal vibration, the displacement field is in the propagation direction, whereas in a transversal vibration the displacement field and the propagation direction are perpendicular to each other.
An mode is characterized by a single wavevector, of components , perpendicular and parallel to the axis, respectively. The component satisfy periodic boundary conditions in the plane and , where . States of constant and variable define 2D phonon branches.
An mode–and, similarly, an mode–is characterized by two wavevectors, corresponding to the and vibrations. The wavevector corresponding to the longitudinal vibration has the components and , whereas the wavevector of the transversal vibration has the components and . The component , perpendicular to the axis, is the same for both, and vibrations, whereas the components and , parallel to the axis, are, in general, different. As in the case of electrons, satisfies periodic boundary conditions on the area and has a DOS of . The free boundary conditions imposed at lead to the equation Auld (1990); Anghel and Cojocaru (2017)
|for the components and , where . The exponents and on the right hand side (rhs) of Eq. (4a) correspond to the polarizations and , respectively Auld (1990), whereas is a doublet, containing the polarization and the branch number (as in the case of the polarization), as we shall specify below. If we denote by and the longitudinal sound velocity and the transversal sound velocity, respectively, then the angular frequency, common to both, longitudinal and transversal vibrations, is given by|
The system (4) has a countable, infinite number of solutions, for each and . These solutions are denoted by and form the branches, when plotted as functions of . In general, and may take both, (positive) real and imaginary values, as explained for example in Auld (1990) (see also Anghel and Kühn (2007)), whereas takes only real positive values, by definition. If is the mass density of the system (considered homogeneous in the whole volume), then the sound velocities may be expressed in terms of the Lamé coefficients and Auld (1990),
and we define .
with the normalization constants
Then the displacement field operator is
where and are the phonon creation and annihilation operators.
In the following we shall write instead of and instead of , when this does not lead to confusions.
The electron-phonon interaction is calculated in the deformation potential model, employing the interaction Hamiltonian, Ziman (1976)
Starting from Eq. (9) and applying the Fermi golden rule to calculate the transition rates between the initial state , of energy , and the final state , of energy , the heat power was calculated in Refs. Anghel and Cojocaru, 2016, 2017 to be
where the normalization constants are given in Eqs. (7).
|Although in Eq. (1a) we have only one exponent , in Eq. (12) we introduce both, and , to check the consistency of the description.|
|If does not depend on (as we shall see further) and since does not depend on , then we may write Eqs. (12a) as|
For concreteness, we consider systems of total thickness nm, of which, the thickness of the metallic film is around 10 nm. The elastic properties of the entire system correspond to the silicon nitride, which is a common material for the construction of the supporting membrane of nano-detectors. Giazotto et al. (2006) For this material, the density is , whereas the longitudinal and transversal sound velocities are m/s and m/s, respectively. The crossover temperature is defined as mK, Kühn et al. (2004) and in the low temperature limit () only the lowest and phonon branches contribute to the electron-phonon interaction, since the upper branches are depopulated. Anghel and Cojocaru (2016, 2017) As the temperature is increasing, the phonon gas changes dimensionality in a temperature interval around and upper and branches gradually start to be populated and play a role in the interaction.
In the electron system, all the bands, from 1 to , are populated at any temperature and contribute – with the electrons close to the Fermi energy – to the electron-phonon interaction. To estimate the difference between the energies of electrons of the same , but belonging to different 2D conduction bands, for a metallic layer of thickness nm, we calculate K and K. We observe that this difference is very large in comparison with the energy of the thermal phonons, so scattering of electrons between different conduction bands are very rare and therefore they do not contribute to the electron-phonon heat exchange in a temperature range below 10 K (see also Anghel and Cojocaru (2016, 2017)).
In Fig. 1 (a) we plot vs and . We observe the formation of the narrow crests, separated by valleys, as in Refs. Anghel and Cojocaru, 2016, 2017. Both, and are plotted in logarithmic scale and we observe a change of the slope of vs , indicating a dimensionality crossover of the phonon gas distribution. Apparently, the slope is increasing with temperature, but more details are seen in Fig. 2, where we show a larger temperature interval. In Fig. 1 (b) we plot separately the contribution of the symmetric phonon modes (lower values in the low temperature limit) and of the antisymmetric phonon modes (higher values in the low temperature limit).
Similarly, we calculated , , and . In the valleys (between crests) and in the temperature range investigated by us, both, and are independent of , whereas their dependence on is the same with that of and on , within the numerical accuracy. Introducing a notation, , for both, and , and a constant value , we may write, formally,
For this reason we do not plot also , or vs and . Numerically, these plot would be identical with the plots in Fig. 1.
In Fig. 2(a) we plot the exponent of the temperature dependence of , denoted simply as –since from Eqs. (12) and (13) we now know that . At low , (see Anghel and Cojocaru (2016, 2017)), then increases with temperature to a value between 4.5 and 5 (not to 5, as expected from Wellstood et al. (1994)) and varies little in an interval , forming the “plateau region.” Quite surprisingly, at temperatures above above the plateau ( K) the exponent decreases again, reaching a value between 3.5 and 4, at K. This is surprising because K is more than 1 order of magnitude above the crossover temperature and we would expect that at such temperatures the value of should correspond to a 3D phonon gas , Wellstood et al. (1994) see Eq. (1b) and the discussion around it.
In Fig. 2(b) we show the exponents of the temperature dependence of both, and ,
in a temperature range between 0 and 2 K. This is a higher resolution image, where we can notice also the crests. Nevertheless, in the crests regions the numerical errors are big, due to the high gradients and we shall not take them into account in our discussion. In the low temperature limit, and . We notice that increases from the low temperature value to a value between 4.5 and 5 in the plateau region (see above), after passing a crossover region around .
The variation of is more complicated, as it can be seen in Fig. 2(b). In the low temperature limit, and then oscillates in the crossover region, until it reaches the plateau, where it becomes practically equal to . Notice that in the crossover region increases above 5 (which is supposed to be the 3D value). At the plateau and after it, , , and become gradually (as temperature increases) numerically indistinguishable. For this reason, there was no point in plotting separately and at temperatures above 2 K.
We studied the effect of the dimensionality of the phonon gas distribution on the heat exchange between electrons and phonons in a layered structure, consisting of a metallic layer (in our case, Cu) of thickness around 10 nm deposited on an insulating free standing membrane. The total thickness of the system (metal plus supporting membrane) is nm. From a mechanical point of view (elastic properties and mass density) the system is considered homogeneous–the elastic waves propagate through a homogeneous material, making no difference between the metal and the dielectric membrane. The longitudinal sound velocity, the transversal sound velocity, and the density of the system are considered to be those of the silicon nitride. With these parameters, the crossover temperature around which the phonon gas distribution changes from two-dimensional (at lower temperatures) to three dimensional (at higher temperatures) is mK. Kühn et al. (2004)
We denoted the electron temperature by and the phonon temperature by The heat power is expected to be described by the ansatz (1). It was already noticed that the ansatz does not apply to our type of layered system and in the low temperature limit, the exponent corresponding to the 2D phonon gas distribution is , not 4. Anghel and Cojocaru (2016, 2017) We recovered here the results of Refs. Anghel and Cojocaru, 2016, 2017 and we expected that, as the temperature increases and the phonon gas distribution changes from 2D to 3D, should change continuously from 3.5 to the value corresponding to 3D systems, which is 5. Wellstood et al. (1994) Instead of this, we observed that does not have a monotonic behavior and totally disobeys the ansatz. It starts from at low temperatures, it increases through the crossover region of temperatures around and reaches a “plateau region” in which varies in a small interval which lies between 4.5 and 5. The plateau region lies in a temperature range roughly between 0.5 and 5 K. After this, the exponent decreases steadily reaching values between 3.5 and 4 at the upper limit of the temperature range investigated by us, which is 10 K.
The results did not confirm the behavior of the heat power in the corresponding limits, as expected from the literature. Nevertheless, the exponent should convey valuable information regarding the dispersion relations and the dimensionalities of the phonon system and of the electrons system. More analytical work is needed to clarify these connections.
Acknowledgements.We are grateful to Dr. Sergiu Cojocaru for useful discussions and for his comments on the manuscript. This work was supported in part by the Romania-JINR collaboration projects, positions 22, 23, 24, 27, Order 322/21.05.2018 (IFIN-HH), and position 24, Order 323/21.05.2018 (JINR). DVA was supported by the ANCS project PN18090101/2018.
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