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Electron–phonon heat exchange in layered nano-systems
Solid State Communications 227 (2016) 56–61
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Electron–phonon heat exchange in layered nano-systems D.V. Anghel n, S. Cojocaru Institutul Naţional de Fizică şi Inginerie Nucleară Horia Hulubei, RO-077125, Măgurele, Romania
art ic l e i nf o
a b s t r a c t
Article history: Received 23 September 2015 Received in revised form 16 November 2015 Accepted 30 November 2015 by E.L. Ivchenko Available online 8 December 2015
We analyze the heat power P from electrons to phonons in thin metallic films deposited on free-standing dielectric membranes in a temperature range in which the phonon gas has a quasi two-dimensional distribution. The quantization of the electrons wavenumbers in the direction perpendicular to the film surfaces lead to the formation of quasi two-dimensional electronic sub-bands. The electron–phonon coupling is treated in the deformation potential model. If we denote by Te the electrons temperature and by Tph the phonons temperature, we find that P P ð0Þ ðT e Þ P ð1Þ ðT e ; T ph Þ. Due to the quantization of the electronic states, both P ð0Þ and P ð1Þ , plotted vs ðT e ; dÞ show very strong oscillations with d, forming sharp crests almost parallel to Te. From valley to crest, both P ð0Þ and P ð1Þ 3:5 increase by more than one order of magnitude. In the valleys between the crests, P p T 3:5 e T ph in the low temperature limit, whereas on the crests P does not have a simple power law dependence on temperature. The strong modulation of P with the thickness of the film may provide a way to control the electron-phonon heat power and the power dissipation in thin metallic films. On the other hand, the surface imperfections of the metallic films can smoothen these modulations. & 2015 Elsevier Ltd. All rights reserved.
Keywords: D. Electron-phonon interaction A. Nano-systems and devices C. Lower dimensional systems
1. Introduction In a recent paper Nquyen et al. [1] reported remarkable cooling properties of normal metal-insulator-superconductor (NIS) tunnel junctions refrigerators, by reaching electronic temperatures of 30 mK or below, from a bath temperature of 150 mK, at a cooling power of 40 pW. Such micro-refrigerators have great potential for applications, since they can be mounted directly on chips for cooling qubits or ultra-sensitive detectors, like micro-bolometers or micro-calorimeters. The principle of operation of NIS micro-refrigerators has been explained in several publications (e.g. [2–7]) and consists basically in cooling of a normal metal island by evacuating the “hot” electrons (from above the Fermi sea) into a superconductor while injecting “cold” electrons (below the Fermi sea) from another superconductor using a pair of symmetrically biased NIS tunnel junctions. If the normal metal island is deposited on a chip, then it can serve as a refrigerator by cooling the chip through electron– phonon interaction. The efficiency of the electron cooling process is controlled by the bias voltages of the NIS junctions, whereas the success of the chip refrigeration strongly depends on the electron– phonon coupling. Moreover, due to the strong temperature dependence of the electric current through the junctions at fixed n
Corresponding author. E-mail address: [email protected] (D.V. Anghel).
http://dx.doi.org/10.1016/j.ssc.2015.11.019 0038-1098/& 2015 Elsevier Ltd. All rights reserved.
bias voltage (or the strong variation of voltage with temperature at fixed current) the NIS junctions can also serve as thermometers. Because of this, if the normal metal island absorbs radiation, the device turns into a very sensitive radiation detector [5,8]. When it works as a detector, the normal metal island can be kept at the nominal working temperature either by cooling it directly through NIS junctions (eventually through the thermometer junctions), or indirectly through electron–phonon coupling to a cold substrate [9–11]. Therefore in any situation, i.e. when the NIS junctions work as coolers, thermometers, or radiation detectors, the electron–phonon coupling plays a central role in the functionality of the device. A typical experimental setup [9,5] is depicted in Fig. 1 and consists of a Cu film of thickness of the order of 10 nm deposited on a dielectric silicon-nitride (SiNx) membrane of thickness of the order of 100 nm. The Cu film is the normal metal island and is connected to superconducting Al leads through NIS tunnel junctions. When it is used as a radiation detector, to reach the sensitivity required by astronomical observations, the working temperature of the device should be in the range of hundreds of mK or below [1,10,11]. At such temperatures the phonon gas in the layered structure formed by the normal metal island and the supporting membrane undergoes a dimensionality cross-over from a three-dimensional (3D) gas (at higher temperatures) to a quasi two-dimensional (2D) gas (at lower temperatures) [12–14].
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
z
x
discretization of the components of the electrons wavenumbers perpendicular to the membrane's surfaces. We work in a temperature range T o200 mK in which the phonon gas is quasi-2D and we observe that the electron–phonon heat flux P cannot be simply described by a single power-law dependence, P p T xe T xph .
A
L/2 L/2−d O
57
y
3:5 While in some ranges of d we have P ð0Þ p T 3:5 e T ph , in general the heat flux has a very strong oscillatory behavior as a function of d. At certain regular intervals the power flux increases sharply with the film thickness by at least one order of magnitude and in the regions of increased heat flux the exponent of the temperature dependence is not well defined.
2. Electron–phonon interaction
−L/2 Fig. 1. (Color online). The schematic model of the system. On a dielectric membrane with parallel surfaces perpendicular on the z axis is deposited an uniform metallic film. The surfaces of the membrane cut the z axis at L=2 and L=2 d, whereas the upper surface of the film cuts the z axis at L=2. In the (x,y) plane the area of the device is A ð⪢L2 Þ. The whole system is treated like a homogeneous elastic continuum, with the elastic properties and the density of silicon nitride (SiNx).
In the stationary regime, one may assume that the electrons in the metallic layer have a Fermi distribution characterized by an effective temperature Te, whereas the phonons have a Bose distribution of effective temperature Tph. In 3D bulk systems the heat flux between the electrons and phonons varies as T 5e T 5ph at low temperatures (see e.g. [15]). Such a model is not justified for our devices and finite size effects [16] have to be taken into account. Phenomenologically, a number of experimental studies have been interpreted by assuming that the heat flux has a temperature dependence proportional to T xe T xph , where x o 5 [17–19]. On the other hand, a theoretical investigation of the surface effects for a thin metallic film deposited on a half-space (semi-infinite) dielectric showed that the value of x is actually larger than 5 [20]. Qualitatively, the growth of the exponent x in the lowest part of the measured temperature range has been later found in some experiments when metallic films were deposited on bulky substrates [18]. The electron scattering rate caused by interaction with 2D (Lamb) phonon modes in a semiconductor quantum well (QW) has been studied in Refs. [21,16] and in a double heterostructure QW including the piezoelectric coupling in Ref. [22]. The electron–phonon heat transfer in monolayer and bilayer graphene was studied in Ref. [23] and a temperature dependence of the form T 4e T 4ph was found in the low temperature regime. For a quasi one-dimensional geometry (metallic nanowire) the electron–phonon power flux was studied theoretically in [24] and a T3 dependence was obtained. It is argued that a general temperature dependence of the form T se þ 2 T sphþ 2 should be valid, where s is the smaller of dimensions of the electron and phonon system [23]. In this context it is interesting to analyze theoretically the temperature dependence of the heat transferred between electrons and phonons in the typical experimental setup of Fig. 1. We assume that the metallic layer is made of Cu and the supporting membrane is silicon nitride (SiNx). We first observe that the dimensionality crossover of the phonon gas in a 100 nm thick SiNx membrane occurs around a scaling temperature T C cl ℏ=ð2kB LÞ 237 mK [14]. Below this temperature one may expect a quasi-2D behavior of the phonon system (rigorously, for T⪡T C ), whereas above it (rigorously, at T⪢T C ) a 3D model would suffice. We carry out our analysis employing a QW picture of the metallic film [21,16,22,25] by taking into account the
2.1. The electron gas The electron system is described as a gas of free fermions confined in the metallic film (Fig. 1). The electron wavevector will be denoted by k ðk J ; kz Þ, where k J and kz are the components of k perpendicular and parallel to z, respectively. The wavefunctions satisfy the Dirichlet boundary conditions on the film surfaces (at L=2 d and L=2) and periodic boundary pffiffiffi conditions in the ðxyÞ plane, ψ k J ;n ðr; tÞ ¼ ϕn ðzÞeiðk J r J ϵk J ;n t=ℏÞ = A, where n ¼ 1; 2; …, pffiffiffiffiffiffiffiffi kz ¼ π n=d, ϕn ðzÞ ¼ 2=d sin z þ d 2L kz , and A is the area of the device surface. The electron energy is 2 2
ℏ π 2 ϵk ¼ ℏ2mke ¼ ℏ2mk eJ þ 2m 2 n ϵk J ;n , where me is the electron mass, d 2 2
2
2
e
k j kj , and k J j k J j . We denote the highestelectronic sub-band pffiffiffiffiffiffiffiffiffiffi 2me EF populated at 0 K by ϵnF , where nF π ℏ d (⌊xc is the biggest integer smaller than x). 2.2. The phonon gas For simplicity, we treat the whole system – supporting membrane and metallic film (Fig. 1) – as a single elastic continuum of thickness L, area A, and volume V ph ¼ LA. The elastic modes of such pffiffiffi a system [16,14,26,27] have the form wq J ξ ðzÞeiðq J r J ωq J ξ tÞ = A, where q J and r J are the components of the wavevector and position vector, respectively, parallel to the (x,y) plane. The functions wq J ξ ðzÞ are normalized on the interval z A ½ L=2; L=2, R L=2 namely L=2 wq J ξ ðzÞ† wq J ξ0 ðzÞdz ¼ δξ;ξ0 . The elastic modes are divided into three main categories: horizontal shear (h), dilatational or symmetric (s) with respect to the mid-plane, and flexural or antisymmetric (a) with respect to the mid-plane. The quantization of the elastic modes in the z direction leads to the formation of phonon branches (or subbands), such that a phonon mode is identified by its symmetry α ¼ h; s; a, sub-band number ν ¼ 1; 2; …, and q J ; ξ represents the pair ðα; νÞ. The h modes are simple transversal modes. The free boundary conditions imposed at the upper and lower surfaces of the membrane quantize the z-component of the wavevector at the values qthn ¼ nπ =L, n ¼ 0; 1; …, whereas qlhn 0. If weqdenote by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q J j q J j , then the phonon frequency is ωq J ;h;ν ¼ ct
q2J þ q2t;h;ν ,
where by ct and cl we denote the transversal and longitudinal sound velocities, respectively. The sound velocities are determined by the Lamé coefficients λ and μ, and the density ρ of SiNx: ct ¼ μ=ρ, and cl ¼ ðλ þ 2μÞ=ρ. The s and a modes are superpositions of longitudinal and transversal modes, oscillating in a plane perpendicular to the surfaces. The quantization relations for qlα and qtα – which are the z components of the wavevectors corresponding to the
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D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
longitudinal and transversal oscillations, respectively – are [27]
4q2J qlα qt α tan ðqtα L=2Þ 7 1 ¼ ; ð1Þ tan ðqlα L=2Þ ðq2J q2tα Þ2 where þ1 and 1 exponents on the right hand side correspond to the α ¼ s and α ¼ a, respectively. The closure equation is secured by the Snell's law, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωq J ;ξ ¼ cl q2l;ξ þ q2J ¼ ct q2t;ξ þ q2J ; ð2Þ and the branch number ν is determined from Eqs. (1) and (2) in the limit q J -0. We are interested in the low temperatures limit, so q J is small (rigorously it should be Lq J ⪡1) in the lowest phonon sub-band ðν ¼ 0Þ. Under these conditions, qt;s;0 qt;s is real, whereas ql;s;0 ipl;s , qt;a;0 ¼ ipt;a , and ql;a;0 ¼ ipl;a are imaginary. The component qt;h;ν is always real as stated above. 2.3. The interaction Electron–phonon interaction in the volume of the film is described by the deformation potential Hamiltonian [28], Z 2 † 3 H def ¼ EF d r Ψ ðrÞΨ ðrÞ∇ uðrÞ; ð3Þ 3 V el ¼ Ad where EF is the Fermi energy. From Eq. (3) we notice that all the transversal components of the h, s, and a modes are not contributing to the electron–phonon interaction. In the second quantization Eq. (3) reads
0 n X n0 ;n † H def ¼ g ξ;q ck J þ q ;n0 ck J ;n aq J ξ þ g nξ;q;n c†k J q n0 ck J ;n a†q ξ : k J ;q J ;ξ;n;n0
J
J
J
J
and phonons, respectively. Using the identity f ðxÞ½1 f ðx yÞ ¼ nðyÞ ½f ðx yÞ f ðxÞ and taking into account the δ functions in Eq. (7) we can write ab Γ em ξ;n;n0 ðk J ; k J q J ; q J Þ Γ ξ;n;n0 ðk J q J ; k J ; q J Þ 2π n0 ;n 2 j g ξ;q j ½f ðβ e ϵk J q J ;n Þ f ðβ e ϵk J ;n0 Þ ½nðβ e ϵq J Þ nðβph ϵq J Þ ¼ J
ℏ
ð8Þ
and the power flux becomes P ¼ P ð0Þ ðT e Þ P ð1Þ ðT e ; T ph Þ: 2.4. Long wavelength approximation We find the relevant low temperature asymptotic expressions by expanding all the quantities in a Taylor series in q J . From Eqs. (1) and (2) we derive expressions for qts, pls, pta, and pla, which we then use to calculate all the other quantities. For the symmetric modes it is sufficient to express qt;s and pl;s to the lowest order in q J , whereas for the antisymmetric modes we have to express pt;a and pl;a up to the third order. In this way we obtain pffiffiffiffiffiffiffiffiffiffiffiffi qt;s q J 3 4J ; ð10aÞ pl;s q J ð1 2JÞ pffiffiffiffiffiffiffiffiffiffi
( pl;a q J 1
Because the h modes do not contribute to the interaction and we are interested in the low temperature limit, in the following we 0 shall take into account in the expression of g nξ;q;n (5) only the J modes with ξ ¼ ðs; 0Þ and ða; 0Þ. The energy transferred from electrons to phonons in a unit of time (heat flux) is h i X 0 0 em ab P¼2 ℏωq J ;ξ Γ ξ;n;n0 ðk J ; k J ; q J Þ Γ ξ;n;n0 ðk J ; k J ; q J Þ ; ð6Þ 0
k J ;k J ;q J ;ξ
where the factor 2 comes from the electrons spin degeneracy. We introduce the notations β e 1=ðkB T e Þ, β ph 1=ðkB T ph Þ, and ϵq J ;ξ ℏωq J ;α;0 (ξ may be omitted). From Eqs. (3) and (5), applying the Fermi's golden rule, we obtain the emission and absorption rates Γ em and Γ abs , i 2π n0 ;n 2 h Γ em ξ;n;n0 ðk J ; k J q J ; q J Þ ¼ ℏ j g ξ;q J j nðβph ϵq J Þ þ 1 f ðβ e ϵk J ;n0 Þ½1 f ðβe ϵk J q J ;n Þ δðϵk J q J ;n ϵk J ;n0 þ ϵq J Þ; ð7aÞ 2π n0 ;n 2 jg Γ qJ ; kJ ; qJ Þ ¼ j nðβ ph ϵq J Þ ℏ ξ;q J ½1 f ðβe ϵk J ;n0 Þf ðβ e ϵk J q J ;n Þ δðϵk J q J ;n ϵk J ;n0 þ ϵq J Þ:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 ) L2 2Jð1 JÞ 27 20J L Jð1 JÞ Jð1 JÞq2J þ q4J 9 5 2 6
( pt;a q J 1
4 ) L2 2ð1 JÞ 27 20J L ð1 JÞ ð1 JÞq2J þ q4J 9 5 2 6 ð11bÞ "
rffiffiffiffiffiffiffiffiffiffi 1J 2 L2 qJ 1 ð27 20JÞq2J 3 120
ð7bÞ
where f(x) and n(x) are the Fermi and Bose distributions of electrons
#
ωa ct L
ð11cÞ
Plugging Eqs. (10) and (11) into (5) we get 0
j g ns;q;nJ j 2 ¼ j N s j 2 ½qt q J ðq2J p2l Þ2 Sðn0 ; n; s; pl Þ; 0
8ℏE2F cos 2 ðLqt =2Þ 9ρωs
j g na;q;nJ j 2 ¼ j N a j 2 ½pt q J ðq2J p2l Þ2 Sðn0 ; n; a; pl Þ;
ð12aÞ 8ℏE2F 2 sinh ðLpt =2Þ 9ρωa ð12bÞ
where the overlap integrals Sðn0 ; n; α; pl Þ are Z 2 L=2 n 0 ϕn0 ðzÞϕn ðzÞ coshðzpl Þ dz ; Sðn ; n; s; pl Þ ¼ L=2 d Z 2 L=2 n ϕ 0 ðzÞϕn ðzÞ sinhðzpl Þ dz : Sðn ; n; a; pl Þ ¼ L=2 d n 0
ð13aÞ
ð13bÞ
Expressing all the quantities in terms of the frequency ω we obtain from Eq. (13a) in the first relevant order Sðn1 ; n2 ; s; ωÞ 1;
if n1 n2 ¼ 0; "
ab ξ;n;n0 ðk J
ð10cÞ
ð11aÞ
ð4Þ operators, whereas a†q ξ and aq J ξ are the phonon creation and J annihilation operators. The matrix elements for the electron– phonon coupling are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z L=2 0 2 ℏ ϕn0 ðzÞϕn ðzÞ g nξ;q;n ¼ EF N α J 3 2ρωq J ;ξ L=2 d n
∂wq J ;ξ;z ðzÞ dz: ð5Þ iq J wq J ;ξ ðzÞ þ ∂z
ð10bÞ
ωs 2q J ct 1 J ¼ 2q J cl Jð1 JÞ;
J
where c†k;n and ck;n are the electron creation and annihilation
ð9Þ
ð14aÞ #2
ω4 d4 ð1 2JÞ4 1 1 ; 16π 4 c4l J 2 ð1 JÞ2 ðn1 n2 Þ2 ðn1 þ n2 Þ2 if n1 n2 ¼ 2 m;
ð14bÞ
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
"
ω4 ðL d=2Þ2 d2 ð1 2JÞ4 1 1 4π 4 c4l J 2 ð1 JÞ2 ðn1 n2 Þ2 ðn1 þ n2 Þ2
#2
if n1 n2 ¼ 2 m þ 1;
10 5
ð14cÞ
whereas from (13b) we get pffiffiffi 1 ðL dÞ2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω; if n1 n2 ¼ 0; Sðn1 ; n2 ; a; ωÞ 4 cl Jð1 JÞ L
ð15aÞ
" #2 pffiffiffi 4 9 3 d ðL dÞ2 1 1 3 ω ; if n1 n2 ¼ 2 m; 4π 4 L3 c3 ½Jð1 JÞ3=2 ðn1 n2 Þ2 ðn1 þn2 Þ2
10 4
P (0) /V el [W/m 3 ]
10 3 10 2 10 1 10 0 10 -1 11
l
ð15bÞ
pffiffiffi 4 3
π
4
59
d [n
" #2 2 d 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω ; if n1 n2 ¼ 2 m þ 1; Jð1 JÞLcl ðn1 n2 Þ2 ðn1 þn2 Þ2
0.2
0.1
10.5
m]
10
0.01
Te
[K]
Fig. 2. (Color online). P ð0Þ =V el for L ¼ 100 nm.
ð15cÞ Using these approximations we can now calculate the low temperature heat flux.
3. Electron–phonon heat flux We separate P ð0Þ and P ð1Þ into the symmetric and the antisymmetric parts, according to which type of phonons contribute to ð0Þ ð1Þ ð1Þ P ð1Þ the process: P ð0Þ P ð0Þ s þP a and P s þP a . We work in the low temperature limit, using the approximations of Section 2.4. Using (10) and (12) and changing from summations over k J and q J to integrals, for the symmetric part we have Z Z 1 Z 2π A2 XX 1 dk J k J dq J q J dϕ P ð0Þ s ¼2 3 ð2π Þ n0 n 0 0 0
2 16π E2F Lqt j N s j 2 qt q q2 p2l cos 2 9ρωq J 2 0 Sðn ; n; s; ql Þℏωq J nðβe ϵq J Þ f ½β e ðϵk J ϵq J Þ ð16Þ f ðβ e ϵk J Þ δðϵk J q J ϵk J þ ϵq J Þ We write P ð0Þ s as the sum of three terms P ð0Þ s ¼A
me E2F 2 L c 4 ℏ5 l
36π
ρ
1 ð1 JÞ2
ð0Þ ð0Þ ð0Þ ð0Þ ðS1ð0Þs þ Sð0Þ 2 s þ S3 s Þ P 1 s þ P 2 s þ P 3 s
P (1) /V el [W/m 3 ]
10 5
which correspond to the summations over n and n0 satisfying the conditions (14a), (14b) and (14c), respectively. P 1ð0Þs ðT e ; dÞ forms very sharp crests almost parallel to the Te axis, which correspond to EF ϵnF (see Section 2.1). In the regions between the crests sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nF X P ð0Þ π 2 me3=2 E2F J 1 4 1s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; pffiffiffi 5 ðkB T e Þ ð18Þ 3 3 A ℏ L ρ c 135 2 ð1 JÞ n ¼ 1 EF ϵn l whereas on the crests the heat flux may increase by more than one order of magnitude and is not proportional, in general, to a single ð0Þ power of Te (see Fig. 2). The terms P ð0Þ 2 s and P 3 s are several orders of ð0Þ magnitude smaller than P 1 s and will be neglected. For the antisymmetric modes we have Z Z 1 Z 2π A2 XX 1 P ð0Þ dk J k J dq J q J dϕ a ¼2 3 ð2π Þ n0 n 0 0 0
2 16π E2F 2 Lqt j N a j 2 qt q J q2J p2l sin 9ρωa 2 Sðn0 ; n2 ; a; ql Þℏωa nðβ e ϵq J Þ f ½β e ðϵk J ;n0 ϵph Þ ð19Þ f ðβ e ϵk J ;n0 Þ δðϵk J q J ;n ϵk;n0 þ ℏωa Þ
10 3 11 10 -1
10.5
d [nm
10
]
10 -2
Te
[K]
Fig. 3. (Color online). P ð1Þ =V el for T ph ¼ 0:2 K and L ¼ 100 nm.
We split P að0Þ in three terms, P að0Þ ¼
ð17Þ
10 4
2 A me E2F J ð0Þ ð0Þ ð0Þ ð0Þ ðSð0Þ þ Sð0Þ 2 a þ S3 a Þ P 1 a þ P 2 a þP 3 a ; 3 2 ℏ3 ð1 JÞ 1 a 2 ρ c 3π L l
ð20Þ
corresponding to the conditions (15a), (15b) and (15c), respecð0Þ ðT e ; dÞ forms also very sharp crests parallel to the Te axis, tively. P 1a which correspond to EF ϵnF . In the regions between the crests
3=2 2 P 1ð0Þa 31=4 5 7 me EF ðL dÞ2 J 3=4 ¼ pffiffiffi ζ A 8 2π 3=2 2 ℏ9=2 ρc5=2 L7=2 ð1 JÞ5=4 l
ðkB T e Þ7=2
n r n nF
X n
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; E F ϵn
ð21Þ
whereas on the crests P ð0Þ 1 a =A may increase by more than one order of magnitude and is not proportional, in general, to a single power of Te (see Fig. 2). The terms P 2ð0Þa and P 3ð0Þa are several orders of magnitude smaller than P ð0Þ 1a and we neglect them. The heat power P ð1Þ from phonon to electrons is calculated like P ð0Þ , with the exception that in the integrals over ϵph one should use a Bose distribution corresponding to the temperature Tph instead of Te. Using again Eq. (14) we write me E2F 1 ð1Þ ð1Þ ð1Þ ð1Þ ðSð1Þ þ Sð1Þ P sð1Þ ¼ A 2 s þS3 s Þ P 1 s þ P 2 s þP 3 s 36π 2 Lρc4l ℏ5 ð1 JÞ2 1 s
ð22Þ
and P ð1Þ 1 s forms very sharp crests parallel to Te axis, which
60
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
10 4 2.5 (1) P av /V el [W/m 3 ]
(0) P av /V el [W/m 3 ]
10 4 10 3 10 2 10 1 10 0
2 1.5 1
0.5 11
11 10.5
10
d [n
m]
10
10
Te[
-2
-1
10.5
d[
10
nm
K]
]
10
10 -2
Te
-1
[K]
Fig. 4. (Color online). The averages of P ð0Þ and P ð1Þ of Figs. 2 and 3: for a thickness variation of 1 Å.
correspond to EF ϵnF . In the regions between the crests P ð1Þ 1s A
3=2 nF X m E2 J 1=2 1 pffiffiffi 5e 3 F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ ðkB T ph Þ4 3=2 135 2 ℏ cl L ð1 JÞ n ¼ 1 EF n
π
2
ρ
ϵ
ð23Þ
whereas on the crests the heat power may increase by more than one order of magnitude and is not described by a simple power ð1Þ law (see Fig. 3). Again, the contribution of the terms P ð1Þ 2 s and P 3 s is negligible. For the antisymmetric modes we have P ð1Þ a ¼
2 A me E2F J ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ðSð1Þ 1 a þ S2 a þ S3 a Þ P 1 a þ P 2 a þ P 3 a : 3π 2 L3 ρc2l ℏ3 ð1 JÞ
ð24Þ
ð0Þ P ð1Þ and P ð1Þ 1 a ðT e ; dÞ forms crests, like P 1 s . In the regions between the crests
3=2 2 P ð1Þ 31=4 5 7 me EF ðL dÞ2 1a ðkB T ph Þ7=2 pffiffiffi ζ 3=2 2 ℏ9=2 ρc5=2 L7=2 A 8 2π l
J 3=4 ð1 JÞ
n r nnF
5=4
X n
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; EF ϵn
ð25Þ
whereas on the crests the heat power may increase by more than one order of magnitude and is not described by a simple power ð1Þ law. The summations P ð1Þ 2 a and P 3 a are several orders of magnitude smaller that P ð1Þ and are neglected. 1a The sharp crests of P ð0Þ and P ð1Þ may be smoothened by the imperfections of the metallic film surfaces. We estimate this effect 1 R d þ δd ðiÞ by averaging P ð0Þ and P ð1Þ , namely P ðiÞ P av ðT e ; T ph ; dÞ δd d 0 0 ðT e ; T ph ; d Þ dd . The result is plotted in Fig. 4 for δd ¼ 1 Å. If we take into account the differences between the elastic properties of the metallic film and of the supporting membrane, then according to Ref. [29] we expect that this would lead to a heat power which is different only by a factor from the results presented above, but qualitatively the temperature dependence of the heat power would remain the same.
are specific to a quantum well (QW) formalism [21,16,22,25]. We choose a range of parameters L, Te (electrons temperature), and Tph (lattice temperature) such that the phonon gas has a quasi twodimensional distribution: for L¼100 nm we took T e ; T ph r0:2 K. The heat flux assumes the non-symmetric form P ¼ P ð0Þ ðT e Þ P ð1Þ ðT e ; T ph Þ, where P ð0Þ is the heat flux from electrons to phonons and P ð1Þ is the heat flux from phonons to electrons. Due to the Fermi level crossing the bottom of electronic sub-bands, both P ð0Þ and P ð1Þ exhibit very sharp “crests” approximately parallel to the temperature axis when plotted vs ðT e ; dÞ. The crests are separated by much more flat “valleys.” In the valley regions, in the low 3:5 ð0Þ and P ð1Þ temperature limit P p T 3:5 e T ph , whereas on the crests P increase by more than one order of magnitude and P does not obey a simple power law behavior. Such sharp oscillations are a characteristic of the QW description of the electronic states in the metallic films [21,16,22,25] and could be useful for applications, like thickness variation detection or surface contamination. Moreover, by varying the thickness of the membrane one might vary dramatically the electron–phonon coupling and, through this, the thermalisation or the noise level in the system. The crests are smoothened by surface imperfections.
Acknowledgments Discussions and comments from Profs. Y.M. Galperin, T.T. Heikkilä, I.J. Maasilta, and Dr. T. Kühn are gratefully acknowledged. The suggestion of Prof. Heikkilä to calculate averages, like in Fig. 4, is appreciated. This work has been financially supported by CNCSISUEFISCDI (project IDEI 114/2011) and ANCS (project PN-09370102). Travel support from Romania-JINR Collaboration grants 4436-32015/2017, 4342-3-2014/2015, and Titeica-Markov program is gratefully acknowledged.
References 4. Conclusions We calculated the heat flux P between electrons and phonons in a system which consists of a metallic film of thickness d of the order of 10 nm deposited on an insulating SiNx membrane of thickness L d of the order of 100 nm (see Fig. 1). We described the electrons as a gas of free fermions confined in the metallic film. The principal characteristic of the electron gas is that due to the quantization of the wavevector component perpendicular to the film surfaces the electron gas forms quasi-2D sub-bands, which
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Electron–phonon heat exchange in layered nano-systems D.V. Anghel n, S. Cojocaru Institutul Naţional de Fizică şi Inginerie Nucleară Horia Hulubei, RO-077125, Măgurele, Romania
art ic l e i nf o
a b s t r a c t
Article history: Received 23 September 2015 Received in revised form 16 November 2015 Accepted 30 November 2015 by E.L. Ivchenko Available online 8 December 2015
We analyze the heat power P from electrons to phonons in thin metallic films deposited on free-standing dielectric membranes in a temperature range in which the phonon gas has a quasi two-dimensional distribution. The quantization of the electrons wavenumbers in the direction perpendicular to the film surfaces lead to the formation of quasi two-dimensional electronic sub-bands. The electron–phonon coupling is treated in the deformation potential model. If we denote by Te the electrons temperature and by Tph the phonons temperature, we find that P P ð0Þ ðT e Þ P ð1Þ ðT e ; T ph Þ. Due to the quantization of the electronic states, both P ð0Þ and P ð1Þ , plotted vs ðT e ; dÞ show very strong oscillations with d, forming sharp crests almost parallel to Te. From valley to crest, both P ð0Þ and P ð1Þ 3:5 increase by more than one order of magnitude. In the valleys between the crests, P p T 3:5 e T ph in the low temperature limit, whereas on the crests P does not have a simple power law dependence on temperature. The strong modulation of P with the thickness of the film may provide a way to control the electron-phonon heat power and the power dissipation in thin metallic films. On the other hand, the surface imperfections of the metallic films can smoothen these modulations. & 2015 Elsevier Ltd. All rights reserved.
Keywords: D. Electron-phonon interaction A. Nano-systems and devices C. Lower dimensional systems
1. Introduction In a recent paper Nquyen et al. [1] reported remarkable cooling properties of normal metal-insulator-superconductor (NIS) tunnel junctions refrigerators, by reaching electronic temperatures of 30 mK or below, from a bath temperature of 150 mK, at a cooling power of 40 pW. Such micro-refrigerators have great potential for applications, since they can be mounted directly on chips for cooling qubits or ultra-sensitive detectors, like micro-bolometers or micro-calorimeters. The principle of operation of NIS micro-refrigerators has been explained in several publications (e.g. [2–7]) and consists basically in cooling of a normal metal island by evacuating the “hot” electrons (from above the Fermi sea) into a superconductor while injecting “cold” electrons (below the Fermi sea) from another superconductor using a pair of symmetrically biased NIS tunnel junctions. If the normal metal island is deposited on a chip, then it can serve as a refrigerator by cooling the chip through electron– phonon interaction. The efficiency of the electron cooling process is controlled by the bias voltages of the NIS junctions, whereas the success of the chip refrigeration strongly depends on the electron– phonon coupling. Moreover, due to the strong temperature dependence of the electric current through the junctions at fixed n
Corresponding author. E-mail address: [email protected] (D.V. Anghel).
http://dx.doi.org/10.1016/j.ssc.2015.11.019 0038-1098/& 2015 Elsevier Ltd. All rights reserved.
bias voltage (or the strong variation of voltage with temperature at fixed current) the NIS junctions can also serve as thermometers. Because of this, if the normal metal island absorbs radiation, the device turns into a very sensitive radiation detector [5,8]. When it works as a detector, the normal metal island can be kept at the nominal working temperature either by cooling it directly through NIS junctions (eventually through the thermometer junctions), or indirectly through electron–phonon coupling to a cold substrate [9–11]. Therefore in any situation, i.e. when the NIS junctions work as coolers, thermometers, or radiation detectors, the electron–phonon coupling plays a central role in the functionality of the device. A typical experimental setup [9,5] is depicted in Fig. 1 and consists of a Cu film of thickness of the order of 10 nm deposited on a dielectric silicon-nitride (SiNx) membrane of thickness of the order of 100 nm. The Cu film is the normal metal island and is connected to superconducting Al leads through NIS tunnel junctions. When it is used as a radiation detector, to reach the sensitivity required by astronomical observations, the working temperature of the device should be in the range of hundreds of mK or below [1,10,11]. At such temperatures the phonon gas in the layered structure formed by the normal metal island and the supporting membrane undergoes a dimensionality cross-over from a three-dimensional (3D) gas (at higher temperatures) to a quasi two-dimensional (2D) gas (at lower temperatures) [12–14].
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
z
x
discretization of the components of the electrons wavenumbers perpendicular to the membrane's surfaces. We work in a temperature range T o200 mK in which the phonon gas is quasi-2D and we observe that the electron–phonon heat flux P cannot be simply described by a single power-law dependence, P p T xe T xph .
A
L/2 L/2−d O
57
y
3:5 While in some ranges of d we have P ð0Þ p T 3:5 e T ph , in general the heat flux has a very strong oscillatory behavior as a function of d. At certain regular intervals the power flux increases sharply with the film thickness by at least one order of magnitude and in the regions of increased heat flux the exponent of the temperature dependence is not well defined.
2. Electron–phonon interaction
−L/2 Fig. 1. (Color online). The schematic model of the system. On a dielectric membrane with parallel surfaces perpendicular on the z axis is deposited an uniform metallic film. The surfaces of the membrane cut the z axis at L=2 and L=2 d, whereas the upper surface of the film cuts the z axis at L=2. In the (x,y) plane the area of the device is A ð⪢L2 Þ. The whole system is treated like a homogeneous elastic continuum, with the elastic properties and the density of silicon nitride (SiNx).
In the stationary regime, one may assume that the electrons in the metallic layer have a Fermi distribution characterized by an effective temperature Te, whereas the phonons have a Bose distribution of effective temperature Tph. In 3D bulk systems the heat flux between the electrons and phonons varies as T 5e T 5ph at low temperatures (see e.g. [15]). Such a model is not justified for our devices and finite size effects [16] have to be taken into account. Phenomenologically, a number of experimental studies have been interpreted by assuming that the heat flux has a temperature dependence proportional to T xe T xph , where x o 5 [17–19]. On the other hand, a theoretical investigation of the surface effects for a thin metallic film deposited on a half-space (semi-infinite) dielectric showed that the value of x is actually larger than 5 [20]. Qualitatively, the growth of the exponent x in the lowest part of the measured temperature range has been later found in some experiments when metallic films were deposited on bulky substrates [18]. The electron scattering rate caused by interaction with 2D (Lamb) phonon modes in a semiconductor quantum well (QW) has been studied in Refs. [21,16] and in a double heterostructure QW including the piezoelectric coupling in Ref. [22]. The electron–phonon heat transfer in monolayer and bilayer graphene was studied in Ref. [23] and a temperature dependence of the form T 4e T 4ph was found in the low temperature regime. For a quasi one-dimensional geometry (metallic nanowire) the electron–phonon power flux was studied theoretically in [24] and a T3 dependence was obtained. It is argued that a general temperature dependence of the form T se þ 2 T sphþ 2 should be valid, where s is the smaller of dimensions of the electron and phonon system [23]. In this context it is interesting to analyze theoretically the temperature dependence of the heat transferred between electrons and phonons in the typical experimental setup of Fig. 1. We assume that the metallic layer is made of Cu and the supporting membrane is silicon nitride (SiNx). We first observe that the dimensionality crossover of the phonon gas in a 100 nm thick SiNx membrane occurs around a scaling temperature T C cl ℏ=ð2kB LÞ 237 mK [14]. Below this temperature one may expect a quasi-2D behavior of the phonon system (rigorously, for T⪡T C ), whereas above it (rigorously, at T⪢T C ) a 3D model would suffice. We carry out our analysis employing a QW picture of the metallic film [21,16,22,25] by taking into account the
2.1. The electron gas The electron system is described as a gas of free fermions confined in the metallic film (Fig. 1). The electron wavevector will be denoted by k ðk J ; kz Þ, where k J and kz are the components of k perpendicular and parallel to z, respectively. The wavefunctions satisfy the Dirichlet boundary conditions on the film surfaces (at L=2 d and L=2) and periodic boundary pffiffiffi conditions in the ðxyÞ plane, ψ k J ;n ðr; tÞ ¼ ϕn ðzÞeiðk J r J ϵk J ;n t=ℏÞ = A, where n ¼ 1; 2; …, pffiffiffiffiffiffiffiffi kz ¼ π n=d, ϕn ðzÞ ¼ 2=d sin z þ d 2L kz , and A is the area of the device surface. The electron energy is 2 2
ℏ π 2 ϵk ¼ ℏ2mke ¼ ℏ2mk eJ þ 2m 2 n ϵk J ;n , where me is the electron mass, d 2 2
2
2
e
k j kj , and k J j k J j . We denote the highestelectronic sub-band pffiffiffiffiffiffiffiffiffiffi 2me EF populated at 0 K by ϵnF , where nF π ℏ d (⌊xc is the biggest integer smaller than x). 2.2. The phonon gas For simplicity, we treat the whole system – supporting membrane and metallic film (Fig. 1) – as a single elastic continuum of thickness L, area A, and volume V ph ¼ LA. The elastic modes of such pffiffiffi a system [16,14,26,27] have the form wq J ξ ðzÞeiðq J r J ωq J ξ tÞ = A, where q J and r J are the components of the wavevector and position vector, respectively, parallel to the (x,y) plane. The functions wq J ξ ðzÞ are normalized on the interval z A ½ L=2; L=2, R L=2 namely L=2 wq J ξ ðzÞ† wq J ξ0 ðzÞdz ¼ δξ;ξ0 . The elastic modes are divided into three main categories: horizontal shear (h), dilatational or symmetric (s) with respect to the mid-plane, and flexural or antisymmetric (a) with respect to the mid-plane. The quantization of the elastic modes in the z direction leads to the formation of phonon branches (or subbands), such that a phonon mode is identified by its symmetry α ¼ h; s; a, sub-band number ν ¼ 1; 2; …, and q J ; ξ represents the pair ðα; νÞ. The h modes are simple transversal modes. The free boundary conditions imposed at the upper and lower surfaces of the membrane quantize the z-component of the wavevector at the values qthn ¼ nπ =L, n ¼ 0; 1; …, whereas qlhn 0. If weqdenote by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q J j q J j , then the phonon frequency is ωq J ;h;ν ¼ ct
q2J þ q2t;h;ν ,
where by ct and cl we denote the transversal and longitudinal sound velocities, respectively. The sound velocities are determined by the Lamé coefficients λ and μ, and the density ρ of SiNx: ct ¼ μ=ρ, and cl ¼ ðλ þ 2μÞ=ρ. The s and a modes are superpositions of longitudinal and transversal modes, oscillating in a plane perpendicular to the surfaces. The quantization relations for qlα and qtα – which are the z components of the wavevectors corresponding to the
58
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
longitudinal and transversal oscillations, respectively – are [27]
4q2J qlα qt α tan ðqtα L=2Þ 7 1 ¼ ; ð1Þ tan ðqlα L=2Þ ðq2J q2tα Þ2 where þ1 and 1 exponents on the right hand side correspond to the α ¼ s and α ¼ a, respectively. The closure equation is secured by the Snell's law, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωq J ;ξ ¼ cl q2l;ξ þ q2J ¼ ct q2t;ξ þ q2J ; ð2Þ and the branch number ν is determined from Eqs. (1) and (2) in the limit q J -0. We are interested in the low temperatures limit, so q J is small (rigorously it should be Lq J ⪡1) in the lowest phonon sub-band ðν ¼ 0Þ. Under these conditions, qt;s;0 qt;s is real, whereas ql;s;0 ipl;s , qt;a;0 ¼ ipt;a , and ql;a;0 ¼ ipl;a are imaginary. The component qt;h;ν is always real as stated above. 2.3. The interaction Electron–phonon interaction in the volume of the film is described by the deformation potential Hamiltonian [28], Z 2 † 3 H def ¼ EF d r Ψ ðrÞΨ ðrÞ∇ uðrÞ; ð3Þ 3 V el ¼ Ad where EF is the Fermi energy. From Eq. (3) we notice that all the transversal components of the h, s, and a modes are not contributing to the electron–phonon interaction. In the second quantization Eq. (3) reads
0 n X n0 ;n † H def ¼ g ξ;q ck J þ q ;n0 ck J ;n aq J ξ þ g nξ;q;n c†k J q n0 ck J ;n a†q ξ : k J ;q J ;ξ;n;n0
J
J
J
J
and phonons, respectively. Using the identity f ðxÞ½1 f ðx yÞ ¼ nðyÞ ½f ðx yÞ f ðxÞ and taking into account the δ functions in Eq. (7) we can write ab Γ em ξ;n;n0 ðk J ; k J q J ; q J Þ Γ ξ;n;n0 ðk J q J ; k J ; q J Þ 2π n0 ;n 2 j g ξ;q j ½f ðβ e ϵk J q J ;n Þ f ðβ e ϵk J ;n0 Þ ½nðβ e ϵq J Þ nðβph ϵq J Þ ¼ J
ℏ
ð8Þ
and the power flux becomes P ¼ P ð0Þ ðT e Þ P ð1Þ ðT e ; T ph Þ: 2.4. Long wavelength approximation We find the relevant low temperature asymptotic expressions by expanding all the quantities in a Taylor series in q J . From Eqs. (1) and (2) we derive expressions for qts, pls, pta, and pla, which we then use to calculate all the other quantities. For the symmetric modes it is sufficient to express qt;s and pl;s to the lowest order in q J , whereas for the antisymmetric modes we have to express pt;a and pl;a up to the third order. In this way we obtain pffiffiffiffiffiffiffiffiffiffiffiffi qt;s q J 3 4J ; ð10aÞ pl;s q J ð1 2JÞ pffiffiffiffiffiffiffiffiffiffi
( pl;a q J 1
Because the h modes do not contribute to the interaction and we are interested in the low temperature limit, in the following we 0 shall take into account in the expression of g nξ;q;n (5) only the J modes with ξ ¼ ðs; 0Þ and ða; 0Þ. The energy transferred from electrons to phonons in a unit of time (heat flux) is h i X 0 0 em ab P¼2 ℏωq J ;ξ Γ ξ;n;n0 ðk J ; k J ; q J Þ Γ ξ;n;n0 ðk J ; k J ; q J Þ ; ð6Þ 0
k J ;k J ;q J ;ξ
where the factor 2 comes from the electrons spin degeneracy. We introduce the notations β e 1=ðkB T e Þ, β ph 1=ðkB T ph Þ, and ϵq J ;ξ ℏωq J ;α;0 (ξ may be omitted). From Eqs. (3) and (5), applying the Fermi's golden rule, we obtain the emission and absorption rates Γ em and Γ abs , i 2π n0 ;n 2 h Γ em ξ;n;n0 ðk J ; k J q J ; q J Þ ¼ ℏ j g ξ;q J j nðβph ϵq J Þ þ 1 f ðβ e ϵk J ;n0 Þ½1 f ðβe ϵk J q J ;n Þ δðϵk J q J ;n ϵk J ;n0 þ ϵq J Þ; ð7aÞ 2π n0 ;n 2 jg Γ qJ ; kJ ; qJ Þ ¼ j nðβ ph ϵq J Þ ℏ ξ;q J ½1 f ðβe ϵk J ;n0 Þf ðβ e ϵk J q J ;n Þ δðϵk J q J ;n ϵk J ;n0 þ ϵq J Þ:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 ) L2 2Jð1 JÞ 27 20J L Jð1 JÞ Jð1 JÞq2J þ q4J 9 5 2 6
( pt;a q J 1
4 ) L2 2ð1 JÞ 27 20J L ð1 JÞ ð1 JÞq2J þ q4J 9 5 2 6 ð11bÞ "
rffiffiffiffiffiffiffiffiffiffi 1J 2 L2 qJ 1 ð27 20JÞq2J 3 120
ð7bÞ
where f(x) and n(x) are the Fermi and Bose distributions of electrons
#
ωa ct L
ð11cÞ
Plugging Eqs. (10) and (11) into (5) we get 0
j g ns;q;nJ j 2 ¼ j N s j 2 ½qt q J ðq2J p2l Þ2 Sðn0 ; n; s; pl Þ; 0
8ℏE2F cos 2 ðLqt =2Þ 9ρωs
j g na;q;nJ j 2 ¼ j N a j 2 ½pt q J ðq2J p2l Þ2 Sðn0 ; n; a; pl Þ;
ð12aÞ 8ℏE2F 2 sinh ðLpt =2Þ 9ρωa ð12bÞ
where the overlap integrals Sðn0 ; n; α; pl Þ are Z 2 L=2 n 0 ϕn0 ðzÞϕn ðzÞ coshðzpl Þ dz ; Sðn ; n; s; pl Þ ¼ L=2 d Z 2 L=2 n ϕ 0 ðzÞϕn ðzÞ sinhðzpl Þ dz : Sðn ; n; a; pl Þ ¼ L=2 d n 0
ð13aÞ
ð13bÞ
Expressing all the quantities in terms of the frequency ω we obtain from Eq. (13a) in the first relevant order Sðn1 ; n2 ; s; ωÞ 1;
if n1 n2 ¼ 0; "
ab ξ;n;n0 ðk J
ð10cÞ
ð11aÞ
ð4Þ operators, whereas a†q ξ and aq J ξ are the phonon creation and J annihilation operators. The matrix elements for the electron– phonon coupling are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z L=2 0 2 ℏ ϕn0 ðzÞϕn ðzÞ g nξ;q;n ¼ EF N α J 3 2ρωq J ;ξ L=2 d n
∂wq J ;ξ;z ðzÞ dz: ð5Þ iq J wq J ;ξ ðzÞ þ ∂z
ð10bÞ
ωs 2q J ct 1 J ¼ 2q J cl Jð1 JÞ;
J
where c†k;n and ck;n are the electron creation and annihilation
ð9Þ
ð14aÞ #2
ω4 d4 ð1 2JÞ4 1 1 ; 16π 4 c4l J 2 ð1 JÞ2 ðn1 n2 Þ2 ðn1 þ n2 Þ2 if n1 n2 ¼ 2 m;
ð14bÞ
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
"
ω4 ðL d=2Þ2 d2 ð1 2JÞ4 1 1 4π 4 c4l J 2 ð1 JÞ2 ðn1 n2 Þ2 ðn1 þ n2 Þ2
#2
if n1 n2 ¼ 2 m þ 1;
10 5
ð14cÞ
whereas from (13b) we get pffiffiffi 1 ðL dÞ2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω; if n1 n2 ¼ 0; Sðn1 ; n2 ; a; ωÞ 4 cl Jð1 JÞ L
ð15aÞ
" #2 pffiffiffi 4 9 3 d ðL dÞ2 1 1 3 ω ; if n1 n2 ¼ 2 m; 4π 4 L3 c3 ½Jð1 JÞ3=2 ðn1 n2 Þ2 ðn1 þn2 Þ2
10 4
P (0) /V el [W/m 3 ]
10 3 10 2 10 1 10 0 10 -1 11
l
ð15bÞ
pffiffiffi 4 3
π
4
59
d [n
" #2 2 d 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω ; if n1 n2 ¼ 2 m þ 1; Jð1 JÞLcl ðn1 n2 Þ2 ðn1 þn2 Þ2
0.2
0.1
10.5
m]
10
0.01
Te
[K]
Fig. 2. (Color online). P ð0Þ =V el for L ¼ 100 nm.
ð15cÞ Using these approximations we can now calculate the low temperature heat flux.
3. Electron–phonon heat flux We separate P ð0Þ and P ð1Þ into the symmetric and the antisymmetric parts, according to which type of phonons contribute to ð0Þ ð1Þ ð1Þ P ð1Þ the process: P ð0Þ P ð0Þ s þP a and P s þP a . We work in the low temperature limit, using the approximations of Section 2.4. Using (10) and (12) and changing from summations over k J and q J to integrals, for the symmetric part we have Z Z 1 Z 2π A2 XX 1 dk J k J dq J q J dϕ P ð0Þ s ¼2 3 ð2π Þ n0 n 0 0 0
2 16π E2F Lqt j N s j 2 qt q q2 p2l cos 2 9ρωq J 2 0 Sðn ; n; s; ql Þℏωq J nðβe ϵq J Þ f ½β e ðϵk J ϵq J Þ ð16Þ f ðβ e ϵk J Þ δðϵk J q J ϵk J þ ϵq J Þ We write P ð0Þ s as the sum of three terms P ð0Þ s ¼A
me E2F 2 L c 4 ℏ5 l
36π
ρ
1 ð1 JÞ2
ð0Þ ð0Þ ð0Þ ð0Þ ðS1ð0Þs þ Sð0Þ 2 s þ S3 s Þ P 1 s þ P 2 s þ P 3 s
P (1) /V el [W/m 3 ]
10 5
which correspond to the summations over n and n0 satisfying the conditions (14a), (14b) and (14c), respectively. P 1ð0Þs ðT e ; dÞ forms very sharp crests almost parallel to the Te axis, which correspond to EF ϵnF (see Section 2.1). In the regions between the crests sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nF X P ð0Þ π 2 me3=2 E2F J 1 4 1s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; pffiffiffi 5 ðkB T e Þ ð18Þ 3 3 A ℏ L ρ c 135 2 ð1 JÞ n ¼ 1 EF ϵn l whereas on the crests the heat flux may increase by more than one order of magnitude and is not proportional, in general, to a single ð0Þ power of Te (see Fig. 2). The terms P ð0Þ 2 s and P 3 s are several orders of ð0Þ magnitude smaller than P 1 s and will be neglected. For the antisymmetric modes we have Z Z 1 Z 2π A2 XX 1 P ð0Þ dk J k J dq J q J dϕ a ¼2 3 ð2π Þ n0 n 0 0 0
2 16π E2F 2 Lqt j N a j 2 qt q J q2J p2l sin 9ρωa 2 Sðn0 ; n2 ; a; ql Þℏωa nðβ e ϵq J Þ f ½β e ðϵk J ;n0 ϵph Þ ð19Þ f ðβ e ϵk J ;n0 Þ δðϵk J q J ;n ϵk;n0 þ ℏωa Þ
10 3 11 10 -1
10.5
d [nm
10
]
10 -2
Te
[K]
Fig. 3. (Color online). P ð1Þ =V el for T ph ¼ 0:2 K and L ¼ 100 nm.
We split P að0Þ in three terms, P að0Þ ¼
ð17Þ
10 4
2 A me E2F J ð0Þ ð0Þ ð0Þ ð0Þ ðSð0Þ þ Sð0Þ 2 a þ S3 a Þ P 1 a þ P 2 a þP 3 a ; 3 2 ℏ3 ð1 JÞ 1 a 2 ρ c 3π L l
ð20Þ
corresponding to the conditions (15a), (15b) and (15c), respecð0Þ ðT e ; dÞ forms also very sharp crests parallel to the Te axis, tively. P 1a which correspond to EF ϵnF . In the regions between the crests
3=2 2 P 1ð0Þa 31=4 5 7 me EF ðL dÞ2 J 3=4 ¼ pffiffiffi ζ A 8 2π 3=2 2 ℏ9=2 ρc5=2 L7=2 ð1 JÞ5=4 l
ðkB T e Þ7=2
n r n nF
X n
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; E F ϵn
ð21Þ
whereas on the crests P ð0Þ 1 a =A may increase by more than one order of magnitude and is not proportional, in general, to a single power of Te (see Fig. 2). The terms P 2ð0Þa and P 3ð0Þa are several orders of magnitude smaller than P ð0Þ 1a and we neglect them. The heat power P ð1Þ from phonon to electrons is calculated like P ð0Þ , with the exception that in the integrals over ϵph one should use a Bose distribution corresponding to the temperature Tph instead of Te. Using again Eq. (14) we write me E2F 1 ð1Þ ð1Þ ð1Þ ð1Þ ðSð1Þ þ Sð1Þ P sð1Þ ¼ A 2 s þS3 s Þ P 1 s þ P 2 s þP 3 s 36π 2 Lρc4l ℏ5 ð1 JÞ2 1 s
ð22Þ
and P ð1Þ 1 s forms very sharp crests parallel to Te axis, which
60
D.V. Anghel, S. Cojocaru / Solid State Communications 227 (2016) 56–61
10 4 2.5 (1) P av /V el [W/m 3 ]
(0) P av /V el [W/m 3 ]
10 4 10 3 10 2 10 1 10 0
2 1.5 1
0.5 11
11 10.5
10
d [n
m]
10
10
Te[
-2
-1
10.5
d[
10
nm
K]
]
10
10 -2
Te
-1
[K]
Fig. 4. (Color online). The averages of P ð0Þ and P ð1Þ of Figs. 2 and 3: for a thickness variation of 1 Å.
correspond to EF ϵnF . In the regions between the crests P ð1Þ 1s A
3=2 nF X m E2 J 1=2 1 pffiffiffi 5e 3 F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ ðkB T ph Þ4 3=2 135 2 ℏ cl L ð1 JÞ n ¼ 1 EF n
π
2
ρ
ϵ
ð23Þ
whereas on the crests the heat power may increase by more than one order of magnitude and is not described by a simple power ð1Þ law (see Fig. 3). Again, the contribution of the terms P ð1Þ 2 s and P 3 s is negligible. For the antisymmetric modes we have P ð1Þ a ¼
2 A me E2F J ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ðSð1Þ 1 a þ S2 a þ S3 a Þ P 1 a þ P 2 a þ P 3 a : 3π 2 L3 ρc2l ℏ3 ð1 JÞ
ð24Þ
ð0Þ P ð1Þ and P ð1Þ 1 a ðT e ; dÞ forms crests, like P 1 s . In the regions between the crests
3=2 2 P ð1Þ 31=4 5 7 me EF ðL dÞ2 1a ðkB T ph Þ7=2 pffiffiffi ζ 3=2 2 ℏ9=2 ρc5=2 L7=2 A 8 2π l
J 3=4 ð1 JÞ
n r nnF
5=4
X n
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; EF ϵn
ð25Þ
whereas on the crests the heat power may increase by more than one order of magnitude and is not described by a simple power ð1Þ law. The summations P ð1Þ 2 a and P 3 a are several orders of magnitude smaller that P ð1Þ and are neglected. 1a The sharp crests of P ð0Þ and P ð1Þ may be smoothened by the imperfections of the metallic film surfaces. We estimate this effect 1 R d þ δd ðiÞ by averaging P ð0Þ and P ð1Þ , namely P ðiÞ P av ðT e ; T ph ; dÞ δd d 0 0 ðT e ; T ph ; d Þ dd . The result is plotted in Fig. 4 for δd ¼ 1 Å. If we take into account the differences between the elastic properties of the metallic film and of the supporting membrane, then according to Ref. [29] we expect that this would lead to a heat power which is different only by a factor from the results presented above, but qualitatively the temperature dependence of the heat power would remain the same.
are specific to a quantum well (QW) formalism [21,16,22,25]. We choose a range of parameters L, Te (electrons temperature), and Tph (lattice temperature) such that the phonon gas has a quasi twodimensional distribution: for L¼100 nm we took T e ; T ph r0:2 K. The heat flux assumes the non-symmetric form P ¼ P ð0Þ ðT e Þ P ð1Þ ðT e ; T ph Þ, where P ð0Þ is the heat flux from electrons to phonons and P ð1Þ is the heat flux from phonons to electrons. Due to the Fermi level crossing the bottom of electronic sub-bands, both P ð0Þ and P ð1Þ exhibit very sharp “crests” approximately parallel to the temperature axis when plotted vs ðT e ; dÞ. The crests are separated by much more flat “valleys.” In the valley regions, in the low 3:5 ð0Þ and P ð1Þ temperature limit P p T 3:5 e T ph , whereas on the crests P increase by more than one order of magnitude and P does not obey a simple power law behavior. Such sharp oscillations are a characteristic of the QW description of the electronic states in the metallic films [21,16,22,25] and could be useful for applications, like thickness variation detection or surface contamination. Moreover, by varying the thickness of the membrane one might vary dramatically the electron–phonon coupling and, through this, the thermalisation or the noise level in the system. The crests are smoothened by surface imperfections.
Acknowledgments Discussions and comments from Profs. Y.M. Galperin, T.T. Heikkilä, I.J. Maasilta, and Dr. T. Kühn are gratefully acknowledged. The suggestion of Prof. Heikkilä to calculate averages, like in Fig. 4, is appreciated. This work has been financially supported by CNCSISUEFISCDI (project IDEI 114/2011) and ANCS (project PN-09370102). Travel support from Romania-JINR Collaboration grants 4436-32015/2017, 4342-3-2014/2015, and Titeica-Markov program is gratefully acknowledged.
References 4. Conclusions We calculated the heat flux P between electrons and phonons in a system which consists of a metallic film of thickness d of the order of 10 nm deposited on an insulating SiNx membrane of thickness L d of the order of 100 nm (see Fig. 1). We described the electrons as a gas of free fermions confined in the metallic film. The principal characteristic of the electron gas is that due to the quantization of the wavevector component perpendicular to the film surfaces the electron gas forms quasi-2D sub-bands, which
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