Phonon Contribution in Thermodynamic Properties of Single Quantum Dot D.I

Vol. 136 (2019) ACTA PHYSICA POLONICA A No. 1

Phonon Contribution in Thermodynamic Properties of Single Quantum Dot D.I. Ilića,∗, J.P.Šetrajčićb,c and I.J.Šetrajčićb aUniversity of Novi Sad, Faculty of Technical Sciences, Vojvodina, Serbia bAcademy of Sciences and Arts of the Republic of Srpska, Banja Luka, Bosnia and Herzegovina cUniversity “Union — Nikola Tesla”, Faculty of Sport, New Belgrade, Vojvodina, Serbia (Received January 17, 2018; revised version July 17, 2019; in final form July 23, 2019) In this paper, we have theoretically and analytically examined the impact of the quantum size effect on mechanical and thermodynamic properties of the single crystalline quantum dot. Spectra of allowed phonon energies, as well as thermodynamic characteristics, are analyzed using the method of two-time dependent Green’s functions. The internal energy of the system as well as the thermal capacitance of the quantum dot in a low- temperature region are found. The behaviour of quantum dot’s specific heat with temperature is compared to that of corresponding bulk structures, as well as those of thin films, superlattices, and quantum wires. It has been shown that at extremely low temperatures thermal capacitance of the quantum dot is significantly lower than the thermal capacitance listed for above structures. Consequences of this fact are discussed in detail and its influence on kinetic and thermodynamic properties of materials are estimated. DOI: 10.12693/APhysPolA.136.49 PACS/topics: phonons, quantum dots, Green’s function, thermodynamics, specific heat

1. Introduction conductivity is becoming extremely important for heat removal and device design and reliability. It is also fore- seen that the use of nanostructured components may in- Determining the basic physical properties underlying crease the sensitivity of measuring instruments, which in the nanostructures is essential for their implementation. turn leads to new experimental results. As a consequence of several various factors, nanostructures are characterized by a variety of diverse features, Quantum dots are confined crystalline struc- such as superconductivity, transport of carriers, heat- tures [11–13] in which conditions on boundary surfaces insulation, acoustic properties, and many other. Most, differ from those inside the dots, i.e. translational sym- if not all of these factors, are directly related to the exis- metry is disrupted alongside all three crystallographic tence of the boundaries of the structure. The main goal directions (x, y, and z, Fig. 1). Providing that there of this paper, which can be considered as a continuation is no disturbance of the crystalline structure inside of our previous research, is to investigate how dimen- the dot (between its boundary surfaces), we assume sional phonon confinement in low-dimensional structures that the quantum dot is ideal. On the contrary, if (quantum dots) influences its thermodynamic properties, there are impurities, vacancies, and the like in the which forms the basis for the concept of phonon engi- crystalline lattice, the quantum dot is deformed. The neering (nanophononics) [1–5]. Research of this type has scope of our study in this paper is the ideal quantum dot become extremely important in recent years, due to the great commercializing potential of novel two-dimensional materials such as graphene, being the lightest, strongest, and most conductive material known today. Examining the influence of the phonon subsystem on the properties of the graphene is mandatory in order to tailor graphene’s mechanical properties [6–10]. Phonons are collective excitations of atoms (molecules) and represent the most important subsystem in condensed matter. Without them, it is almost impossible to examine and describe the acoustical characteristics, as well as the thermal, conductive, and superconductive properties of solids. Thermal properties of nanostructures have attracted a lot of attention lately: the influence of size effects on thermal

∗corresponding author; e-mail: [email protected] Fig. 1. Quantum dot model.

(49) 50 D.I. Ilić, J.P. Šetrajčić, I.J. Šetrajčić of a simple cubic crystalline structure consisting of between atoms in boundary layers of the quan- identical atoms, with one atom per elementary cell. At tum dot and external areas, disregarding that first glance, this fact may seem like a big limitation in along x, y, and z directions outside boundary sur- terms of the applicability of the described model, but faces there are no atoms belonging to the quan- this is not the case: in accordance with the method tum dot. However, boundary atoms are coupled of achievement, statistical and dynamical equivalence through changed Hooke’s forces with the atoms of between rectangular and structures with lower symme- the external environment [14–16]. In accordance try [5], it is applicable to a large number of crystalline with these conditions, elastic constants which de- structures. In the case of monoclinic structures, for scribe interactions between atoms of boundary sur- instance, the method of equivalence is applicable with faces and external environment are modified with no restrictions. appropriate coefficients ε, γ, σ, ϕ, ϑ, and φ. These The basic crystallographic parameters of the chosen perturbations of boundary surfaces do not dis- model structure, in the nearest neighbours approxima- turb the macroscopic geometry of the structure tion, are and only redefine the small movements of atoms. A simpler but more pragmatic approach implies ax = ay = az = a, Nx,y,z ∼ 10, that the quantum dot atoms are surrounded by Cα,α = Cα ≡ Cα = atoms/molecules belonging to only two different en- n,m n,n±λ nxny nz ;nx±1,ny nz vironments (σ, ϕ, ϑ, φ 7→ ε). Cα = Cα ≡ Cα, nxny nz ;nxny ±1,nz nxny nz ;nxny nz ±1 With respect to described model and regarding the fact Cα = Cα = (1+ε)Cα, nx,ny ,0;nx,ny ,−1 nx,ny ,−1;nx,ny ,0 that layers with nx ≤ −1 and nx ≥ Nx + 1, ny ≤ −1 and n ≥ N + 1 and also n ≤ −1, and n ≥ N + 1 are not Cα = Cα = (1+γ)Cα, y y z z z nx,ny ,Nz ;nx,ny ,Nz +1 nx,ny ,Nz +1;nx,ny ,Nz present, we have to take into account the following: Cα = Cα = (1+σ)Cα, u = 0, −1 ≥ n ∧ n ≥ N + 1, nx,0,nz ;nx,−1,nz nx,−1,nz ;nx,0,nz α;nx,ny ,nz x,y,z x,y,z x,y,z Cα = Cα = (1+ϕ)Cα, (n 6∈ [0,N ]). nx,Ny ,nz ;nx,Ny +1,nz nx,Ny +1,nz ;nx,Ny ,nz x,y,z x,y,z Cα = Cα = (1+ϑ)Cα, These boundary conditions are in accordance with the 0,ny ,nz ;−1,ny ,nz −1,ny ,nz ;0,ny ,nz free surface model, but at this point, it should be noted α α α that there are also other approaches. One of them, which CN ,n ,n ;N +1,n ,n = CN +1,n ,n ;N ,n ,n = (1+φ)C , x y z x y z x y z x y z is often found in literature, is the rigid walls model (or (ε, γ, σ, ϕ, ϑ, φ) ≥ −1, (1) frozen surfaces model), where boundary conditions with where a is the lattice constant, Nx,y,z are the num- zero atomic displacements are required at the boundary bers of atoms along x, y, and z directions, Cα is the surfaces. In this paper we have chosen flexible bound- straining Hooke elastic constant in direction α, nx,y,z ∈ ary surfaces, considering that this is closer to the real (0, 1, 2, ··· ,Nx,y,z) is the atom site counter along x, y, situation in which the quantum dot can “breathe”. On and z directions, and vector λ associates atom in place the other hand, the rigid walls model presumes the ap- n with its nearest neighbors. pearance of phonon standing waves with nodes at the Respecting above facts, we are able to say the following boundaries. about the described model structure: 1. Quantum dots have six boundary surfaces: two of 2. Theoretical analysis them are parallel to the xy planes (for z = 0 and z = Lz = Nza), two to xz planes (for y = 0 and Starting point of our theoretical approach is the y = Ly = Nya), and two to yz planes (for x = 0 standard Hamiltonian of the phonon subsystem for and x = Lx = Nxa). Thus, these structures are bulk structures [16–18], written in the harmonic as confined along x, y, and z directions. Along x-axis, well as in the nearest neighbors approximations, which there are Nx +1 atoms, along y-axis, Ny +1 atoms, is adapted to the model-structure of quantum dot and along z-axis, Nz + 1 atoms. presented in Fig. 1 αβ 2 2. Torsion Hooke’s elastic constants C are negligi- X pα;n H = T + V ,T = , (2) ble relative to the straining constants Cα. Besides eff 2M that, it is considered that there is an interaction α;n X Cα 2 2 2 V = u −u + u −u + u −u eff 4 α;nx+1,ny ,nz α;nx,ny ,nz α;nx−1,ny ,nz α;nx,ny ,nz α;nx,ny +1,nz α;nx,ny ,nz α;n ,n ,n x y z 2 2 2 + uα;nx,ny −1,nz −uα;nx,ny ,nz + uα;nx,ny ,nz +1−uα;nx,ny ,nz + uα;nx,ny ,nz −1−uα;nx,ny ,nz , (3) Phonon Contribution in Thermodynamic Properties of Single Quantum Dot 51 where uα;n are the small movements of atom in position and undetermined Green’s functions from Eq. (3) can be n ≡ (nx, ny, nz) from its equilibrium position in direction expressed as follows: α, and p are the corresponding momentum, and M is α;n Dnx,ny ,nz the mass of the atoms. Gnx,ny ,nz = , (9) DN +1,N +1,N +1 We are looking for the phonon dispersion law with x y z where D is the determinant of the variable and the help of the phonon two-time commutator Green’s nx,ny ,nz D is the three-dimensional determinant function [18–20]: Nx+1,Ny +1,Nz +1 of the system. Poles of Green’s functions by which the α 0 0 Gn,m(t − t ) ≡ hhuα;n(t)|uα;m(t )ii = phonon dispersion law is determined can be obtained on 0 0 condition that the determinant of the system is equal to Θ(t − t )h[uα;n(t), uα;m(t )]i0, (4) zero which satisfies the equation of motion D (%) = 0. (10) 2 α Nx+1,Ny +1,Nz +1 −Mω Gn,m(ω) = Determinant DNx+1,Ny +1,Nz +1 can be expressed through i~ 1 Chebyshev’s polynomials of second order, by which the − δn,m + hh[pα;n,H] |uα;miiω. (5) phonon dispersion law can be obtained in form 2π i~ By calculating corresponding commutators in the Green r 2 akx(χ) 2 aky(µ) 2 akz(ν) function which appears in Eq. (5), we obtain the system Eα = 2 sin + sin + sin , k 2 2 2 of (Nx + 1) × (Ny + 1) × (Nz + 1) nonhomogeneous algebraic-difference equations with the same number of χ = 1, 2,...Nx + 1, µ = 1, 2,...Ny + 1, undetermined Green’s functions [10–13]: ν = 1, 2,...Nz + 1 , (11) G + G + G nx−1,ny ,nz nx,ny −1,nz nx,ny ,nz −1 similar to that of the bulk structures. The main differ- +℘G + G + G ence is, however, that phonon quasimomentum in quan- nx,ny ,nz nx,ny ,nz +1 nx,ny +1,nz tum dots takes discrete values in all three directions. It can also be seen that minimum phonon energy in quan- +Gnx+1,ny ,nz = Knxny nz , (6) where tum dots differs from zero, and is given by α h ∆min = E min min min = kx ,ky ,kz ℘ = % − ε δnx,0 + δnx,Nx + δny ,0 + δny ,Ny + δnz ,0 i ω2 C s 2 α akmin(χ) akmin(µ) akmin(ν) −γδnz ,Nz ,% = 2 − 6, Ωα = , 2 x 2 y 2 z Ωα M 2 sin + sin + sin , i 2 2 2 K = ~ δ . (7) nx,ny nz 2πC nx,ny nz ,mxmy mz π 1 π 1 α kmin(χ = 1) = , kmin(µ = 1) = , x a N + 2 y a N + 2 In order to find the spectra of the allowed phonon en- x y ergies amounts, we must determine the zeroes of the de- min π 1 kz (ν = 1) = , (12) terminant of the system of Eq. (6). This task, in general, a Nz + 2 is not analytically solvable (it can be solved numerically while the corresponding minimum phonon frequency is with given parameters ε, γ, Nx,Ny, and Nz). Hereafter, min v 2 π 2 π we will give our attention to the model of the loose sur- ω = 2 sin + sin α a 2 (N + 2) 2 (N + 2) faces [16, 17], when surface perturbations are negligible, x y i.e. ε = γ = 0. In this model, an elastic interaction π 1/2 + sin2 . (13) of the quantum dot surface atoms with atoms/molecules 2 (N + 2) of surrounding environments is of the same nature and z Although in crystals with a simple elementary cell only strength. We call it “a model of the ideal quantum dot acoustic phonon branches can occur, the previous analy- with free surfaces”. The reason for choosing this model is sis has shown that due to the influence of the quantum that there is an analytical solution for the phonon disper- size effect there is an energy gap, which is a characteristic sion law and other physical properties. Besides that, in of optical phonon modes. It can, therefore, be concluded determining the micro- and macroscopic physical prop- that in quantum dots, as in other nanostructures, acous- erties of the sample this choice favours the quantum size tic phonons of the optical type appear. effect [1,9], while the contributions of all other confinement effects (shape, etc.) are negligible and can only slightly affect changes caused by the size effect [2, 3, 11– 3. Phonon thermodynamics of quantum dots 13]. In that case, the system of Eq. (6) reduces to In order to determine thermodynamic properties of G + G + G nx−1,ny ,nz nx,ny −1,nz nx,ny ,nz −1 quantum dots, it is necessary to find corresponding values for Debye’s wave vector kD and Debye frequency ωD. +%Gn ,n ,n + Gn ,n ,n +1 + Gn ,n +1,n x y z x y z x y z We assume that phonon wave vectors of quantum dot lie in the sphere of radius kD. Since the translational +Gnx+1,ny ,nz = Knx,ny nz , (8) symmetry of the quantum dot is interrupted alongside 52 D.I. Ilić, J.P. Šetrajčić, I.J. Šetrajčić

3 3 all three directions, possible values of wave vectors along where V = Na = (Nx + 1)(Ny + 1)(Nz + 1)a , ω ωmin ωD TD x, y, and z directions are: x = ~ , xmin = ~ , and xD = ~ = . In a low kBT kBT kBT T 1 π Nx/y/z + 1 π temperature region, when xD → ∞, the last expression kx/y/z ∈ , becomes Nx/y/z + 2 a Nx/y/z + 2 a ∞ 3 3 Nx/y/z π T Z x ⇒ ∆k = , x/y/z U = 9NkBT x dx = Nx/y/z + 2 a TD e − 1 xmin and we obtain  ∞ x  ( 3 min 4 d 3 T Z x3 Z x3 π akD 3 9NkBT dx − dx = V = 3 ⇒  x x  a ∆kx∆ky∆kz TD e − 1 e − 1 0 0 s r 2  xmin  3 3π N N N 3 4 3 d 3 x y z T π Z x kD = 3 ⇒ 4a Nx + 2 Ny + 2 Nz + 2 9NkBT  − x dx . (17) TD 15 e − 1 s 0 N N N −1 ∞ d b 3 x y z On the basis of expansion (et − 1) = P e−jt the kD = kD , j=1 2 (Nx + 2) 2 (Ny + 2) 2 (Nz + 2) last equation becomes

√  xmin  where kb = 3 6π2/a is the Debye wave vector in corre- 3 4 ∞ Z D T π X 3 −jx sponding unbounded (bulk) structure. For the number U = 9NkBT  − x e dx . TD 15 of allowed values of k per volume unit of k-space, next j=1 0 adjusted general expression is applicable Integral from this expression is analyzed by multiple π 2π kD 3N N N a3 Z Z Z partial integration D(ω)= x y z sin θdθ dϕ k2 dkδ(ω−vk) =  (2π)3 T 4 π4 ∞ 1 X −jxmin 0 0 kmin U = 9NkB + e 3 2 3  N N N a ω TD 15 j x y z (14) j=1 2π2 v3 ∞  and by applying the normalization condition (according 3 6 6 X 6 × x3 + x2 + x + − . (18) to fact that the total number of phonon states is equal min j min j2 min j3 j4  to the number of atoms) j=1 ω Z D D(ω)dω = N ⇒ To find the expression for the thermal capacitance per a unit cell (here: per atom), the standard defini- ωmin ∗ 1 ∂U ωD tional [18, 20] form is used: C = . In accordance 3 Z N ∂T NxNyNza 2 with that we obtain 2 3 ω dω = (Nx + 1) (Ny + 1) (Nz + 1) , 2π v 4 3 3 ωmin ∗ 12π T T C = kB + 9kB we obtain an expression for Debye frequency in quantum 5 TD TD dot in the forms ∞ X 1 1 3 N + 1 N + 1 N + 1 4 π × e−jxmin x4 + 1 + x3 b x y z 2 j j min j2 min ωD = ωD + 2 sin j=1 Nx Ny Nz 3π 2 (Nx + 2) 6 1 6 1 24 π π 3/21/3 + 1 + x2 + 3 + x + , + sin2 + sin2 , (15) j j2 min j2 j2 min j3 2 (N + 2) 2 (N + 2) y z or where ωb = kb v is the Debye frequency in the corre- D D 4 3 sponding bulk-structure. It can be seen that the Debye ∗ 12π T 1 C = kB 3 + 9kB 3 frequency has somewhat greater value in the quantum 5 f (Nx,Ny,Nz) f (Nx,Ny,Nz) dot than in an unbounded structure. ∞ 4 X 1 −j ∆ 1 ∆ 3 3 6 1 2 Internal energy of the quantum dot is calculated in × e T + 1 + ∆ + 1 + ∆ T j j T j2 j j2 terms of standard definition form [18, 20]: j=1 ωD ωD Z Z ω2V 6 1 24 U = dωD(ω)hn(ω, T )i ω = dω + 3 + ∆T 2 + T 3 , (19) ~ 2π2v3 j2 j2 j3 ω ω min min b b b b xD where T ≡ T/TD, ∆ ≡ ωmin/ωD,(TD, and ωD are the ω T 3 Z x3 × ~ = 9Nk T dx, (16) Debye temperature and frequency for bulk structure, re- ~ω B x e kBT − 1 TD e − 1 spectively) and relative Debye’s frequency related to bulk xmin ones is Phonon Contribution in Thermodynamic Properties of Single Quantum Dot 53

ωD Nx + 1 Ny + 1 Nz + 1 4 2 π It can be seen that phonon contribution to the thermal b = + 2 sin ωD Nx Ny Nz 3π 2 (Nx + 2) capacitance of crystalline quantum dot is lowest, compared to that of more massive specimens. On this basis, π π 3/21/3 + sin2 + sin2 it can be concluded that the quantum dots are the weak- 2 (Ny + 2) 2 (Nz + 2) est thermal as well as electrical conductors.

≡ f(Nx,Ny,Nz). 4. Conclusions It is well known that the phonon part in the thermal capacitance of the system is described with cubic temper- Application of nanostructures requires a knowledge of ature dependence. For comparison of these dependences their fundamental physical properties. Lately, a great for the bulk-structure and the quantum dot, the last ex- 4 interest has been induced by the thermodynamic as- 12π kB pression is divided by the constant C0 = 5 , whose pect associated with phonon movements through the dimension is equal to the dimension of thermal capaci- nanometer-sized samples. Spatial confinement of acous- tance tic and optical phonons in nanostructures unavoidably T 3 15 1 changes their properties in comparison with bulk mate- C = 3 + 4 3 f (Nx,Ny,Nz) 4π f (Nx,Ny,Nz) rials. Their interactions are altered by the effects of dimensional constraint on the phonon modes in nanostruc- ∞ 4 X 1 −j ∆ 1 ∆ 3 3 6 1 2 tures. Phonon confinement in low-dimensional structures × e T + 1+ ∆ + 1+ ∆ T j j T j2 j j2 leads to the emergence of the quantized energy sub-bands j=1 with the corresponding alteration of the phonon density 6 1 24 of states. The changes in the phonon dispersion law + 3+ ∆T 2+ T 3 . (20) j2 j2 j3 lead to a modification in the electron–phonon scattering Figure 2 shows the relative (non-dimensional) thermal ca- rates, optical properties of the nanostructured materials, pacitances of bulk structure, superlattice, ultrathin film, and phonon scattering on defects, boundaries, and other quantum wire, and quantum dot subject to the relative phonons. In this paper, we applied a strict theoretical ap- temperature T in the low-temperature region. proach in order to examine the influence of the quantum size effect on the thermodynamic properties of a quantum dot that is surrounded by different materials from every side. In this respect, desirable properties of the structure can be modified by changing the lattice constant, i.e. dimensions of the quantum dot, by inserting atoms of different kinds and by changing the parameters ε, γ, σ, ϕ, ϑ, and φ. However, considering the technology of fabricating quantum dots, there is no reason to assume that it will be on all sides surrounded by different materials. A simpler but more pragmatic approach implies that the quantum dot atoms are surrounded by atoms/molecules belonging to only two different environments (σ, ϕ, ϑ, φ 7→ ε). With respect to all of the above we come to the following conclusions: since phonons with the Debye fre- quencies are responsible for electrically and thermically induced transport properties of materials, it follows that the quantum dot will be inferior electrical and thermal conductor in contrast to the relative massive structures, providing there are no chemical and structural differences Fig. 2. Low-temperature behavior of thermal capaci- between them. On the other hand, it is a well known tance for bulk and quantum dot. fact that the more inferior electrical conductor material is (under normal conditions), the better superconductor Determining the phonon impact on the physical, and it becomes. Due to that, the experimental fact can be primarily thermodynamic properties of low-dimensional concluded and justified, that in very spatially confined structures (ultrathin films, superlattices, and ultranar- structures more qualitative superconductive properties row wires) has been the subject of research of our team have been achieved. for many years and the results of these studies have been These facts point out that the key role in high TC su- published [4–6, 16, 17, 21–26] in the previous period. perconductors comes from the low dimension of the ob- Here, we will use some of these results in order to com- served structure. More detailed answer to this question pare them with the results obtained in this paper for will be obtained by examination of the electronic subsys- quantum dots. tem in quantum dots. 54 D.I. Ilić, J.P. Šetrajčić, I.J. Šetrajčić

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