CC BY
15
Macroscopic thermoelectric efficiency of carbon nanocomposites
E. D. Eidelman1,2, A. P. Meilakhs1 11Ioffe Physical Technical Institute, 26 Politekhnicheskaya, 194021 Saint Petersburg, Russia
2 Saint Petersburg State Chemical-Pharmaceutical Academy, 14 Professora Popova, 197376 Saint Petersburg, Russia [email protected], [email protected]
PACS 84.60.Rb, 85.80.Fi DOI 10.17586/2220-8054-2016-7-6-919-924
The subject of this study is the thermoelectric efficiency (Z) and the thermoelectric parameter (ZT) of carbon nanocomposites, namely, the structures consisting of graphite-like (gr) and diamond-like (d) regions made of sp2 and sp3 hybridized carbon atoms, respectively. The impact of heat transfer across the boundary between sp2 and sp3 areas is analyzed for the first time. It is shown that the interfacial thermal resistance (Kapitza resistance) is not lower than the thermal resistance in the macroscopic gr region. The influence of various factors on the Kapitza resistance is analyzed. The value of ZT & 3.5 at room temperature, taking into account the interfacial thermal resistance, is significantly higher than it would be in gr films (ZT & 0.75). Keywords: carbon nanostructures, thermoelectricity. Received: 28 October 2016
1. Introduction
In recent years, a lot of attention has been paid to the influence of size on transport phenomena in solids and, in particular, on thermoelectric effects [1-5]. Currently, attention is focused on the creation of new materials with the highest possible thermoelectric figure of merit (quality factor) Z, which makes it possible to use this material in refrigerators or generators [2,5]: Z = S2a/x. Here, S is the thermoelectric coefficient or the Seebeck coefficient (V/K), a is the conductivity ratio ((Ohm-m) -1), and x is the thermal conductivity (J/(m-s-K)).
The higher the value of the parameter ZT (where T is the temperature in Kelvin) is for any material, the more useful this material is for thermoelectric conversion. Therefore, ZT is used as a dimensionless quantity that characterizes thermoelectric materials.
Serious efforts were made to increase Z, both by selecting an appropriate semiconductor and by determining its optimal layer thickness [3, 4]. To the best of the authors' knowledge, the highest value ZT = 2.6 at room temperature is reported in [6] for SnSe crystals.
Part of this effort is the study of thermoelectric phenomena in carbon nanostructures [3-5]. In these nanos-tructures, hybridized carbon atoms coexist. The coexistence of regions with very different electrical and thermal properties at such small distances from each other is a unique feature of these structures. The gr areas are semimetallic with a high electrical conductivity but a relatively low thermal conductivity. The d areas are wide bandgap semiconductors, dielectrics in fact, but with a high thermal conductivity. The corresponding kinetic coefficients differ in value by many orders of magnitude.
The article deals with the Z values of the composite produced from 4 nm diamond nanoparticles generated by detonation synthesis. Such diamond nanoparticles were prepared by milling [7-9] and before long by purification techniques [10,11] from agglomerates about 100 nm in size. Then, solid composite was produced from 4 nm particles by sintering at high pressure and high temperature (HPHT). Regions with sp3 hybridization, as determined by areas of coherence, reached sizes of up to 4-12 nm. The composite consists of such particles with a d core, which can be covered by up to four layers of the sp2 phase, but may have no coating at all. The composite is completely free from the separate gr phase. The average particle size becomes 4-35 nm [12]. Then, the size of gr regions can be assumed at an average of l « 10-30 nm. The thermoelectric effect and mechanisms that contribute to the increase in the thermoelectric coefficient S, the main component of the Z, were discussed in [13,14] for carbon nanostructures.
As for the ratio of electrical and thermal conductivity coefficients a/x being part of Z, it is always considered to be proportional to the temperature according to the Wiedemann - Franz law (see, e.g., [15] §78). Of course, there may be limitations associated with the possible flow of heat in the gr areas. Let us further assume that the percolation threshold (see, eg, [16]) is reached [9], but the heat carried by electrons is small compared with the lattice vibrations heat transfer.
It was recently shown that the presence of borders between the gr and d regions greatly influences the thermal conductivity of carbon nanostructures [17,18]. In this paper, we examine the effect of borders on the thermoelectric efficiency.
2. Statement of the problem of heat transfer through the interface between phases
Let us assume that the gr area is heated or cooled on one side and contacts with a d area on the other side. In this case, the gr phonons migrate into d or d phonons migrate into gr. In accordance with Fourier's law, this heat source lowers or raises the temperature of the gr lattice. The heat of the electron gas from the gr area is not directly transferred to the d area. However, the electrons can heat or cool the crystal lattice in gr. An increase in the phonon flux component takes place, and then the heat is transferred from this component to the d area. This nonlinear process occurs due to the temperature difference between the electronic and phonon subsystems in gr. It appears as a result of presence of the boundary with d. The ideal structure would be the one with plane borders perpendicular to the direction of heat propagation. Any possibility of heat flow bypassing the border increases the thermal conductivity, and thus reduces the thermoelectric efficiency Z.
The temperature at the border undergoes a jump AT, which is proportional to the heat flux q (W/m2). Such dependence makes it possible to use the electrothermal analogy. The proportionality coefficient r (m2 •K/W) between the AT and q is called interfacial thermal resistance (Kapitza resistance). The Kapitza resistance is a characteristic of the interface; in the heat transfer theory the reciprocal value of 1/r is called the heat transfer coefficient.
The process being considered is a special case of heat transfer through the metal - dielectric boundary. The Kapitza resistance at the metal-dielectric interface has been widely studied (see, e.g., [17]). It was found that the Kapitza resistance, as calculated considering the nonlinearity of the process, does not depend on the dielectric characteristics and is fully determined by the properties of the metal. This thesis was experimentally confirmed in [19].
In fact, each subsystem in gr is characterized by its set of values: electron Te; qe: xe, and phonon Tph, qph, Xph. Each triplet of values is related according to the Fourier law.
The interaction coefficient 6 (W/(m3-K)) characterizes the efficiency of heat transfer between subsystems and relates the heat flux to the difference Te — Tph.
It should be noted that in experiments, the temperature is measured far away from the boundary. At this location, the temperature is Te = Tph = T, and the thermal conductivity x = Xe + Xph can be attributable to the general heat flow in gr: q = qe + qph. In gr, it can be assumed that xe >> Xph and heat transfer through the electronic subsystem dominates.
Only phonons can pass from gr to d, hence qph = q; qe = 0 at the boundary.
According to [19], the problem is posed so that the properties of d do not affect the Te and Tph temperature in gr.
3. Temperature and thermal resistance alignment at the boundary between gr and d
The solution of the problem is completely similar to that provided in [17]. The result is that the characteristic distance A, at which the temperatures of the electron and phonon subsystems in gr are actually aligned, is related to the system characteristics as:
A = yXXr " (Xph/6)1/2. (1)
The components of the thermal resistance are determined by the jumps of the subsystems temperatures, Te (0) and Tph(0), relative to the temperature Td in d, which is equal to Td(0) at the boundary (Fig. 1). In the course of the experiment, the temperature gradient from a region far from the boundary is usually extrapolated linearly to the boundary. This temperature Tgr is considered to be the gr temperature; accordingly, it is Tgr (0) at the border, and the temperature jump is considered equal to AT = Tgr (0) — Td(0). However, almost always Tgr (0) « Te(0).
The Kapitza thermal resistance caused by the measured jump AT can be divided into rph, arising due to the temperature jump ATph of phonons in gr and d areas, and rr, arising because the electronic and phonon subsystems in gr are heated unequally near the border. Therefore, r = rph + rr.
The thermal resistance (relative!) rr arises due to the difference in temperature jumps of the electron ATe and phonon ATph subsystems: ATr = Te — Tph = ATe — ATph. In practice, the temperature in gr is determined by the electronic subsystem temperature. So, the measured jump AT « ATe and ATr can be calculated as ATr = AT — ATph.
Fig. 1. This figure presents the temperatures in the graphite-like (gr) area Tgr (on the right) and the diamond-like (d) area Td (on the left), near the boundary located at x = 0. At the boundary, the temperatures are, respectively, Tgr(0), Td(0), Te(0) and Tph(0). Also shown are values of the corresponding temperature jumps ATph and ATe at the border to the temperature Td(0). The jump AT = Tgr(0) — Td(0) is measured experimentally. Finally, the temperature differences in gr subsystems were shown to be caused by the Kapitza resistance due to the presence of the boundary ATr
The value of the jump ATph determines the proportion of phonons reflected from the boundary back into gr. This proportion was determined in [13] for a system very similar to the carbon nanocomposites under study. There, it was found to be usually small. Therefore, ATph << ATr and accordingly rph << rr. This inequality is the major, roughest approximation that is used for the solution of the problem. The values in Fig. 1 are provided using this approximation.
Fig. 2. A possible thermoelement design, where the heat Q and the temperature difference Thot — Tcoid provide the conditions (see [13]) under which the quasi-ballistic drag of electrons by phonons prevails in the carbon nanostructure
Next, it will be assumed that the Kapitza resistance in carbon nanostructures and, in particular, carbon nanocomposites produced by HPHT, is determined by the relative temperature jump, which is approximately equal to the measured temperature jump ATr « AT, and therefore, the thermal resistance is determined approximately by the relative thermal resistance r « rr.
Then the solution obtained in [17] can be written for the simplest one-dimensional heat propagation as:
Tgr - Te = - qx + — e
1 A« ,-x/A.
X
Tgr - Tph = 1 qx - ^^e-x/\
Then the thermal resistance:
r ~ = rr =
Xq
X ~ X Xph AXe 1
(2)
(3)
xxph V^XPh
is expressed in terms of the thermal conductivity of the phonon subsystem in gr and heat transfer efficiency between the electron and phonon subsystems in the same area.
4. Thermoelectric efficiency
The thermoelectric efficiency of the system at the boundary between gr and d can be represented now as:
Zb
1
(4)
-« S2a I - +
Xef VX 1,
This formula is stated using the electrothermal analogy. The magnitude of the thermoelectric efficiency is expressed through the effective thermal conductivity Xef of the system that takes into account the thermal conductivity in the gr area with the overall size of l, and the thermal resistance of the boundary.
Fig. 3. A possible thermoelement design. In addition to the conditions corresponding to the structure shown in Fig. 2, the Wiedemann - Franz law is violated because of the introduction of diamond nanoparticles (ND). The thermal conductivity is determined by the Kapitza thermal resistance, i.e. by heat transfer at the interface of sp2/sp3 areas
Of course, even for macroscopic and microscopic samples, this formula provides the value Z mentioned in the Introduction. For small values of l << A in nanosystems, the addition generated by a decrease in scattering due to heat transfer from the electron subsystem to the phonon subsystem, can significantly increase the thermoelectric efficiency.
Let us estimate, first of all, the value of 0, the efficiency coefficient of heat transfer between the subsystems. As the primary heat flux is carried by electrons, we can assume that energy is transmitted from them to the crystal lattice. It is clear that energy can be transmitted by quanta only and the characteristic energy of the quanta is determined by the temperature of phonons: Tph when Tph < TD, or Debye temperature TD in case Tph > TD. Of course, the temperature difference between the subsystems is small compared with each subsystem temperature, it may be a few tenths of a degree or a few degrees, but the heat transfer occurs with the phonon transition. The phonons have the specified temperature Tph « TD in the sp2 region. On the other hand, the gradient of the thermal
flow of electrons is expressed in terms of the heat generated by the current, which would correspond to this heat flux. The result is:
6 =
„2 „2^,2
aTD
Here, a is the order of unity factor,
l25™-3
1.5
103 m/s is the speed at which the electrons move,
->3
(5) because
they captivate phonons, n0 « 1025m-3 is the conduction electron density, a « 103 (Om-m)-1 is the conductivity coefficient, TD « 0.5 • 103 K, as already indicated, is the sp2 region Debye temperature, and e = 1.6 • 10-19 Cl is the electron charge. The values of all variables, except, of course, the universal value of the electron charge, are given for graphite. All values are weakly dependent on the temperature and are borrowed from [20]. By substituting these values, we get 0 « 1015 W/(m3 K). Apparently, smaller values should be selected in reality. Of course, this assessment is by the order of magnitude.
In [21] and [22] the following values of graphite thermal conductivity versus temperature are given (see. Tab. 1).
Table 1. Graphite thermal conductivity versus temperature
Xgr Bt/(m-K) 114 86 61 47 40 34 25
T K 293 473 873 1173 1473 1774 2073
Data for diamond Xd = 1000 W/(m-K) at T = 273 and 42 W/(m-K) at 4323 K are also provided. It is obvious that the thermal conductivity for gr must be somewhat intermediate. In [23,24] xgr values between 200 and 300 W/(m-K) are provided. On the other hand, the phonon part of thermal conductivity accounts for a small part of the total value. For further estimates, xph «5-10 W/(m-K) will be taken (see [25]).
According to the formula (1) it turns out that A «100 nm, which confirms the above assumption on the size ratio of gr and the characteristic distance at which the temperatures of the electron and phonon subsystems can be regarded as being equal. Thus, the thermal conductivity in the carbon nanocomposites is no longer dependent on the size of the gr area, just like in the case where the interface is less than 4 nm thick (for oxides) the thermal conductivity is no longer dependent on the layer thickness [26-28].
Now, using the formulas (5) and (3) it is easy to estimate the thermal resistance. We obtain r « 10-9 m2K/W. The same value of the thermal resistance is derived with an entirely different reasoning in [24]. Finally, let us find the relative thermoelectric efficiency:
Zef
Z
1
rxy
I
(6)
As stated above (see Introduction), we can assume that if the size of the graphite-like area is on the order of l « 10 — 30 nm, it turns out that the thermoelectric efficiency of carbon nanocomposites is ten times higher than that of graphite. At room temperature, the thermoelectric parameter is 5-20 times higher than that of graphite.
5. Conclusion
As is known (see, for example, [20]), the Seebeck coefficient of graphite depends on its purity, but it is of the order of S = 2 — 3^V/K. Thus, the graphite-like layer, whose surfaces have the temperature difference 100 K, produces the voltage of 0.3 mV. This value does not depend on the layer thickness, but the layer geometry affects the conditions of heating and heat dissipation and, consequently, the thermoelectric efficiency. At the surface with the lower temperature of 300 K, subject to the Wiedemann - Franz law, such a thermoelement would have the thermoelectric parameter ZT = 3-10-4. Materials from carbon nanostructures have S =10 — 20^V/K [28] and they could produce the voltage of 1-2 mV with ZT = 3 • 10-2 with the same temperature differences. However, if such conditions are provided under which the quasi-ballistic drag of electrons by phonons prevails in the carbon nanostructure [13], it would be possible to obtain 5-10 mV and ZT = 0.75 (Fig. 2). In nanocomposites with the quasi-ballistic drag, in which the Wiedemann - Franz law is not fulfilled, namely, the thermal conductivity is determined by the Kapitza thermal resistance, it is possible to achieve the thermoelectric parameter 3.5 (Fig. 3).
Acknowledgement
The authors are grateful to Professor Dieter Gruen who pointed to the existence of the efficiency problem for the engineering of generators and refrigerators operating on the thermoelectric effect. We would like to acknowledge A.Ya. Vul' for his attention, which enabled us to complete this work. We are also thankful to F.M. Shakhov for fruitful discussions and the design of the figures. This work was supported by RSF grant No. 16-19-00075.
0
a
c
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