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Dynamic thermoelectricity in uniform bipolar semiconductor
Author’s Accepted Manuscript Dynamic thermoelectricity in uniform bipolar semiconductor I.N. Volovichev
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S0921-4526(16)30130-2 http://dx.doi.org/10.1016/j.physb.2016.04.010 PHYSB309432
To appear in: Physica B: Physics of Condensed Matter Received date: 29 January 2016 Revised date: 30 March 2016 Accepted date: 9 April 2016 Cite this article as: I.N. Volovichev, Dynamic thermoelectricity in uniform bipolar semiconductor, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.04.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Dynamic thermoelectricity in uniform bipolar semiconductor I.N. Volovichev A. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, 12 Ac. Proscura St., Kharkov 61085, Ukraine
Abstract The theory of the dynamic thermoelectric effect has been developed. The effect lies in an electric current flowing in a closed circuit that consists of a uniform bipolar semiconductor, in which a non-uniform temperature distribution in the form of the traveling wave is created. The calculations are performed for the one-dimensional model in the quasi-neutrality approximation. It was shown that the direct thermoelectric current prevails, despite the periodicity of the thermal excitation, the circuit homogeneity and the lack of rectifier properties of the semiconductor system. Several physical reasons underlining the dynamic thermoelectric effect are found. One of them is similar to the Dember photoelectric effect, its contribution to the current flowing is determined by the difference in the electron and hole mobilities, and is completely independent of the carrier Seebeck coefficients. The dependence of the thermoelectric short circuit current magnitude on the semiconductor parameters, as well as on the temperature wave amplitude, length and velocity is studied. It is shown that the magnitude of the thermoelectric current is proportional to the square of the temperature wave amplitude. The dependence of the thermoelectric short circuit current on the temperature wave length and velocity is the nonmonotonic function. The optimum values for the temperature wave length and velocity, at which the dynamic thermoelectric effect is the greatest, have been deduced. It is found that ∗ Tel.:
+380 57 7203331 Email address: [email protected] (I.N. Volovichev)
Preprint submitted to Physica B
April 9, 2016
the thermoelectric short circuit current changes its direction with decreasing the temperature wave length under certain conditions. The prospects for the possible applications of the dynamic thermoelectric effect are also discussed. Keywords: dynamic thermoelectric effect, thermoelectricity, thermopower, Seebeck effect, nonequilibrium charge carriers PACS: 72.20.Pa, 73.50.Lw
1. Introduction Research of the thermoelectric phenomena and development of efficient thermoelectric converters represent a major challenge to the modern applied solid state physics. Currently, investigations is mainly focused on the search and creation of new efficient thermoelectric materials with a high thermoelectric figure of merit ZT [1–3]. However, despite a long history of thermoelectric research, because of its diversity the physics of transport phenomena under non-equilibrium energy conditions (a particular case of which is a non-uniform temperature distribution [4]) is as yet imperfectly understood. This is especially true in the case of simultaneous action of several perturbations in the semiconductor systems, as well as in the case of time-dependent temperature fields. Meanwhile, there are a number of papers demonstrating that the energy conversion in semiconductor systems under non-steady state conditions may differ significantly from the static case by its physical background. In particular, a significant increase in the thermocouple efficiency [5] (although this work does not deal with a semiconductor system), as well as an increase in the thermoelectric power at the p-n junction [6] are proven in the transient regime. Note that the difference between dynamic mechanisms of the electromotive force (emf) formation and their static counterparts is not an exclusive feature of thermoelectric phenomena. It is known that, for example, a photo-emf in dynamic mode can significantly exceed the static values (for example, the Dember emf [7, 8]), as well as it can appear in situations where no static emf can arise [9]. Moreover, an effective experimental technique for semiconductor pa2
rameter measurements has been developed based on the effect of the non-steady state photo-emf [10]. The main factors underlying the most significant distinctions of thermoelectric effects in the dynamics might be as follows. First, a time-dependent perturbation gives rise to an uncompensated space charge in the semiconductor. In turn, the latter affects the conventional mechanism of the thermopower generation. This way is of especial importance in multi-layer structures and polycrystalline films [6]. Second, time derivatives of the carrier densities explicitly appear in the continuity equation [11, 12], and it might directly affect the magnitude of the thermo-emf. For the non-steady state photovoltaic phenomena it is the mechanism that determines the existence of the Dember emf even when the production of the static photoelectricity is basically impossible [9]. Finally, the time dependence of the temperature can influence the transport processes through changing the thermal generation rate. As shown in Ref. [4], in the static case the thermal generation can contribute to the emf only when there is the difference in the carrier temperatures (e.g., the electron temperature differs from the temperature of holes and phonons). If there is a single temperature of all quasiparticles in the semiconductor, the thermal generation does not affect the value of the thermopower at any static temperature distribution in the semiconductor structure. The time dependence of the thermal generation rate can change the picture, leading to at first glance paradoxical results. Thus, it is widely assumed that two contacts at different temperatures between two media with different Seebeck coefficients are mandatory for the thermopower appearance. Or that in a uniform semiconductor sample having the same temperature of its ends, no thermoelectricity is generated at any temperature distribution inside the semiconductor. However, in the dynamic case the above is no longer true. Thus, in Ref. [13] it has been presented a physical mechanism that is responsible for the thermoelectric current generation by the temperature wave propagating in a uniform unipolar semiconductor. Below, we show that a similar phenomenon takes place in a uniform bipolar semiconductor. In the latter, as in a more complex physical 3
system (in which there are two types of the charge carriers: electrons and holes), the nature of the dynamic thermoelectric effect is also more complicated, and the magnitude of the thermopower generated can be substantially higher than the corresponding value in the unipolar semiconductor. The presented theory also shows that the thermoelectric short circuit current is a nonmonotonic function of the temperature wave parameters. This allows us to offer using the considered dynamic thermoelectric effect as a research technique for investigations of the bipolar semiconductor properties. 2. Mathematical model of dynamic thermoelectric effect Consider a slab of uniform bipolar semiconductor of the length L, in which a stationary (but not static) temperature distribution in the form of a traveling wave is maintained by an external heating: T = T0 + δT0 cos Φ,
(1)
where Φ = kx + ωt, k = 2π/Λ, Λ is the temperature wave length, ω = kv, v is the temperature wave velocity. To simplify the calculations, we assume that the slab length is divisible by the temperature wave length, i.e. let kL/(2π) be an integer. We will not dwell on methods for creating the temperature distribution of the form in Eq. (1) in the specimen. For discussion of this matter see the Appendix A. We will restrict our consideration to the one-dimensional model, supposing all physical values in directions normal to the temperature wave propagation to be uniform. Then, under the drift-diffusion model one can write the following continuity equations for the electron (jn ) and hole (jp ) current densities [11, 12, 14]: ∂n 1 ∂jn = − Rn , ∂t e ∂x 1 ∂jn ∂p =− − Rp , ∂t e ∂x 4
(2) (3)
with ∂n ∂T − eμn αn n , ∂x ∂x ∂p ∂T − eμp αp p , jp = eμp pE − T μp ∂x ∂x jn = eμn nE + T μn
(4) (5)
where n and p are the electron and hole densities, respectively; e is the elementary charge, Rn,p are the recombination rates of electrons and holes, μn,p are the electron and hole mobilities, E is the electric field in the semiconductor, αn,p are the Seebeck coefficients for the electron and hole subsystems. Brief mention should be made of the following. In modern literature, Eqs. (4)(5) are written frequently in the following form [15–17]: ∂ψn,p ∗ ∂T − αn,p jn,p = −σn,p , ∂x ∂x
(6)
where ψn,p = ϕ ∓ ζn,p /e are the electrochemical potentials of the carriers (here, sign ”-” refers to the electrons, and ”+” to holes), ϕ is the electric potential, ζn,p are the chemical potentials of the carriers, σn = eμn n and σp = eμp p are the conductivity of the electron and hole subsystems, respectively. Introduced in that way the Seebeck coefficients α∗n,p differ from the phenomenologically introduced Seebeck constants in Eqs. (4)-(5) and are related to them as follows: α∗n,p = αn,p ∓
1 ∂ζn,p . e ∂T
(7)
Below we will use the Seebeck coefficients αn,p , and the resulting expression can be rewritten in terms of α∗n,p in an obvious way, using Eq. (7). Here we restrict ourselves to the quasi-neutrality approximation by assuming the Debye screening length to be the smallest spatial parameter and the Maxwell relaxation time to be the smallest temporal parameter of the problem [18, 19]. Under these conditions there is no space charge in the semiconductor: ρ = e(p − nt − n) = 0, where nt is the charged impurities density (to be specific, the density of negatively charged impurity levels). Suppose that the dominant mechanism of recombination is the interband recombination (if so, the calculations are less bulky). Then [20, 21]: Rn = Rp = R = γ (np − n0 p0 ) , 5
(8)
where γ is the capture coefficient, n0 and p0 are the electron and hole densities in the thermodynamic equilibrium (i.e. at T0 ), respectively. To simplify the calculations we assume that the charged impurity density remains constant. This assumption is valid at sufficiently high temperatures, when in the thermodynamic equilibrium the majority of dopant atoms are already ionized. Thus, the local change in temperature has no effect on them. The appearance of nonequilibrium electrons and holes does not affect the equilibrium of the impurity subsystem due to the interband transitions dominate. In these conditions, the lack of space charge imposes the following relation between the nonequilibrium electron density δn = n − n0 and the nonequilibrium hole density δp = p − p0 : δn = δp [19]. In the quasi-neutrality from Eqs. (2)-(3) follows ∂j/∂x = 0, where j = jn +jp is the total electric current density. Adding up Eqs. (4) and (5), one gets: j(t) = σE + T (μn − μp )
∂δn ∂T − (αn σn + αp σp ) , ∂x ∂x
(9)
where σ = σn + σp is the total conductivity of the semiconductor. The system of differential equations (2)-(3) must be supplemented by boundary conditions (BCs). To avoid issues related to influence of contacting media and physical processes at the interfaces we choose the cyclic BCs. From a physical point of view, such a choice is equivalent to connecting the slab ends, i.e. to creating a cylinder or ring-shaped semiconductor circuit (sample). Thus one obtains a homogeneous (not containing other media than the semiconductor) closed circuit. Then: n(0, t) = n(L, t),
∂n(0, t) ∂n(L, t) = , ∂x ∂x
(10)
E(0, t) = E(L, t),
ϕ(0, t) = ϕ(L, t),
(11)
where ϕ is the electric field potential (E = −∂ϕ/∂x). Note also that for the given temperature distribution the condition T (0, t) = T (L, t) is fulfilled identically. The last-mentioned BC in Eq. (11) can be rewritten as: L E dx = 0. 0
6
(12)
Then, expressing the electric field E from Eq. (9) and substituting it into Eq. (12), we obtain:
j(t) = (μn − μp )
L
0
T ∂δn dx− σ ∂x −
L
0
L αn σn + αp σp ∂T dx / σ −1 dx. (13) σ ∂x 0
From the form of Eq. (13), which is an exact one (no assumption about the smallness of the nonequilibrium carrier densities has been yet made), it follows that: • The first non-zero term of the series expansion of j(t) in powers of a small parameter δT /T0 (δT = T − T0 ) is quadratic. • To determine j(t) with an accuracy of (δT /T0 )2 it is necessary and sufficient to find δn and E with an accuracy of δT /T0 . In other words it is sufficient to solve the linearized system of equations (2)-(3). With this in mind, the current in the circuit may be written as follows: j(t) =
μn − μp L
0
L
δT
∂δn dx− ∂x −
e2 μn μp (αn − αp )(p0 − n0 ) σ0 L
0
L
δn
∂T dx, (14) ∂x
where σ0 is the semiconductor conductivity at T0 . When deriving Eq. (14) all terms representing a total differential in the integrands, as well as small quantities of higher orders have been omitted. The linearized system of equations (2)-(5) taking into account the quasineutrality conditions δp = δn (δn n0 , δn p0 ) becomes: 1 ∂jn ∂δn = − R, ∂t e ∂x 1 ∂jp ∂δn =− − R, ∂t e ∂x
(15) (16)
∂δn ∂δT − eμn αn n0 , ∂x ∂x ∂δn ∂δT − eμp αp p0 , jp = eμp p0 E − T0 μp ∂x ∂x jn = eμn n0 E + T0 μn
7
(17) (18)
and the expression for the interband recombination (8) can be written as: 1 n0 p0 δT R= , (19) δn − τ n0 + p 0 T 0 where τ −1 = γ(n0 + p0 ); the parameter , describing the temperature dependence of the thermal generation rate, for the nondegenerate electron gas is equal to = 3 + εg /(kB T0 ); εg is the semiconductor bandgap value, kB is the Boltzmann constant [22]. Using the first harmonic approximation method (FHA), we seek the solution of equations (15)-(17) in the form of: δn = δn0 + δns sin(kx + ωt) + δnc cos(kx + ωt),
(20)
with δn0 , δns , δnc n0 , p0 . It is easily seen that this choice satisfies the BCs in Eq. (10) identically. Substitute Eq. (20) in Eq. (15) and collect terms that do not depend on Φ. As is easy to see, the latter appear in Eq. (15) solely due to the term R, defined by Eq. (19) (since the other terms in Eq. (15) are derivatives with respect to time or coordinate). This immediately implies that δn0 = 0. Thus, the temperature wave leaves the averaged carrier density unchanged. This is physically clear: the thermal generation is the only source of the excess carriers, and as the mean temperature deviation in the temperature wave is zero, it does not affect the mean carrier density. Now let us substitute Eq. (20) in Eq. (14), taking into account δn0 = 0. It is easy to see that after integration the non-zero result is owing to the terms with sin2 Φ or cos2 Φ. All terms with product sin Φ cos Φ, being integrated over the semiconductor length, vanish. Finally, one obtains the following expression for the short-circuit thermoelectric current: k e2 μn μp (αn − αp )(p0 − n0 ) δns δT0 . j0 = μn − μp + 2 σ0
(21)
It should be noted that within the framework of the FHA method the electric current in the closed circuit contains only the direct current (dc) component, while the main harmonic of the thermoelectric current is absent. The second 8
harmonic, whose amplitude is proportional to (δT0 /T0 )2 , is beyond the linear theory. In order to find the amplitude of the alternating current (ac), the linear approximation is insufficient; one must take into account the terms of higher order of smallness. In addition, the ac component is effectively suppressed by the quasi-neutrality. Therefore, below we concentrated our attention on the dc component of the thermoelectric short circuit current. 3. Results and discussion Substituting Eq. (20) in Eqs. (15)-(19), we obtain a system of algebraic equations. Solving the latter, after simple but rather cumbersome transformations, the following expressions for the non-equilibrium carrier densities can be worked out:
¯ δT0 n0 p0 e(αn − αp )k¯ 2 + ω δns = ,
2 n0 + p 0 T0 ω ¯ 2 + k¯2 + 1
n0 p0 e(αn − αp )k¯ 2 + k¯ 2 + 1 δT0 , δnc =
2 n0 + p 0 T0 ω ¯ 2 + k¯2 + 1
(22) (23)
where we have introduced the normalized wave vector and the wave frequency of √ ¯ = ωτ ; LD = Da τ is the diffusion length, the temperature wave: k¯ = kLD , ω Da = T0 (n0 + p0 )μn μp /σ0 is the ambipolar diffusion coefficient. With reference to Eq. (21) and Eq. (22), it can be seen that the short-circuit thermoelectric current (and the corresponding thermopower) contains contributions from two physical mechanisms. The first mechanism is similar by its nature to the Dember photo-emf and is due to the difference in mobility of electrons and holes. It is remarkable that this contribution is independent of the Seebeck coefficients αn,p . We emphasize that the presence of this term is not typical for the known mechanisms of the thermoelectric emf. To the contrary, the second component of the dynamic thermo-emf depends on the Seebeck coefficients of the semiconductor. However, here also there are some interesting differences from the static thermopower. Thus, this term is proportional to the difference between Seebeck coefficients of electrons and holes, rather than to the effective
9
Seebeck coefficient of the semiconductor (αn σn + αp σp )/σ0 , as it is usually the case in bipolar semiconductors. Furthermore, according to Eq. (21) the term that depends on the difference in the Seebeck coefficients, is also proportional to the difference in the equilibrium densities of electrons and holes. In turn, this means that the contribution to the thermopower for the n-type (electron) semiconductor and the p-type (hole) semiconductor has opposite sign, while entirely vanishing in intrinsic or compensated semiconductors. However, in this case the dependence on αn − αp still influence the magnitude of the nonequilibrium carrier densities, see Eq. (22). Also note that the current in the circuit does not depend on the semiconductor length. This fact has a simple explanation. Each spatial period of temperature wave makes its additive contribution to the thermo-emf magnitude. Increasing the semiconductor length, the number of periods of the temperature wave that fit in the sample also increases, so increasing the total thermopower. On the other hand, with increasing the length of the semiconductor its resistance also rises proportionally. As a result, the thermoelectric short circuit current is independent of the semiconductor length. Let us analyze the dependence of the nonequilibrium carrier densities and the thermoelectric short circuit current on the temperature wave parameters. Recall that in the framework of the method used, δnc denotes the amplitude of the inphase (with respect to the temperature wave) nonequilibrium electron wave, and δns denotes the amplitude of the phase shifted (quarter-phase) density wave. As it can be seen from Eq. (21), the short circuit current is determined by the quarter-phase nonequilibrium carrier density component only. It is easy to see from Eqs. (22)-(23) that at too high temperature wave velocities (ω → ∞) the nonequilibrium carrier densities decrease to zero. At the same time, for a slow movement of the temperature waves (in the limit, for the static temperature distribution) the quarter-phase component also vanishes, whereas the inphase component remains finite. In turn, this means that the considered thermoelectric effect disappears both at too high temperature wave velocities and at a fixed temperature distribution. The latter implies that the thermoelec10
tric effect is a strictly dynamic phenomenon. This deduction is consistent with the well-known fact that in a uniform semiconductor whose ends are kept the same temperature, there is no thermopower whatever the static temperature distribution has been created inside the semiconductor [23, 24]. The thermoelectric current vanishing as ω −1 at extremely high temperature wave velocities ω → ∞ is also quite understandable. In this case, the system is unable to follow the change of temperature, and effectively we have similarity to the static situation. Thus, there is some optimal velocity of the temperature wave, for which the magnitude of the dynamic thermoelectric effect is the largest. It is not difficult to determine the value of the optimal temperature wave velocity. Finding the extremum of δns in Eq. (22) as a function of ω, one gets: vmax =
1 + k 2 L2D . kτ
(24)
Pay attention to the simple form of the functional dependence in Eq. (24). This fact allows us to offer using the considered effect for the experimental determination of the diffusion length and the lifetime of nonequilibrium carriers. Indeed, measuring the temperature wave velocity, at which the current in the circuit reaches the maximum, for several different spatial periods1 of the temperature pattern (i.e. for different lengths of the temperature wave), it is easy to calculate the diffusion length LD and the lifetime τ by means of Eq. (24). Let us analyze the dependence of the thermoelectric short circuit current on the temperature wave length. Naturally, with k → 0 the thermoelectric current decreases linearly and vanishes at k = 0, because it is the case of a uniform heating of the semiconductor. In the opposite limiting case k → ∞ the thermoelectric current vanishes as k −1 : at an extremely small temperature wave length the heating can be regarded as effectively uniform. Accordingly, 1 Remember
that when using the experimental laser technique of the moving interference
pattern its spatial period and its velocity can be handled independently [10]. The period is controlled by the tilt angle of the sample, while the velocity is controlled by the frequency shift between two laser beams. For details, see Ref. [10]
11
there is an optimal value for the temperature wave length at which the dynamic thermoelectric effect is the most pronounced. Unfortunately, in general case it almost impossible to get a useful analytical expression for the optimal temperature wave length. However, for efficient thermoelectric materials (in which the condition [3 + εg /(kB T0 )]/[e(αn − αp )] k 2 L2D is met) the optimum value of the temperature wave length is determined by the following expression: 1 kmax = 1 + 4 + 3ω 2 τ 2 . LD
(25)
In the opposite limiting case [3 + εg /(kB T0 )]/[e(αn − αp )] k 2 L2D (e.g., in wide-gap semiconductors at low temperatures), the optimal value for the temperature wave length is somewhat different: √ 3 kmax = 4 + 3ω 2 τ 2 − 1. 3LD
(26)
Note that the above analysis of the thermoelectric short-circuit current dependence on the temperature wave length is made under the assumption of a constant temperature wave frequency ω = kv (instead of a constant temperature wave velocity v). This situation is typical for the interference methods of creating the nonuniform heating, when the temperature wave velocity decreases with decreasing wave length, i.e. with growth of k (v = ΛΔf , where Δf is the frequency shift between two laser beams, see Ref. [10]). At a fixed temperature wave velocity the situation is somewhat different. In particular, for k → ∞ the dynamic thermoelectric effect does not disappear. So, for some combination of semiconductor parameters the monotonous functional dependence of the short circuit thermoelectric current on the temperature wave length becomes possible. However, if the condition [3 + εg /(kB T0 )]/[e(αn − αp )] k 2 L2D holds true, the extremum still has a place, and the optimal value for the temperature wave length is given by kmax = 1/LD . The considered dynamic thermoelectric effect has another interesting property. As seen from Eq. (22), if αp > αn the thermoelectric short circuit current changes its sign with increasing k at
3 + εg /(kB T0 ) 1 . k0 = LD e(αp − αn ) 12
(27)
The physical meaning of the sign change is as follows. In the presence of the temperature wave in the initially uniform semiconductor two simultaneous competing processes take place: the thermal generation of the nonequilibrium carriers (its rate is determined both by the parameter and by the amplitude of the temperature wave), and the flow of nonequilibrium carriers into colder areas of the semiconductor due to the temperature gradient (for clarity, here we put aside the diffusion of carriers). The rate of the second process is determined by the Seebeck coefficient, as well as by the temperature gradient, i.e. both by the amplitude and by the wavelength of the temperature wave. One can choose the temperature wave length (k = k0 ), so that these two processes cancel each other (of course, this is only true in the linear approximation and for the harmonic temperature distribution). At this, the nonequilibrium carriers do not appear at all (the factor e(αn − αp )k¯ 2 + in Eqs. (22)-(23) is zero). On opposite sides of this value k0 there dominates either the generation of carriers or their flow away. Thus, at the point of the temperature maximum either the maximum of the carrier densities or their minimum takes place. The latter corresponds to a phase shift of π/2, i.e. it means different sign of the thermoelectric power. Vanishing of the thermoelectric current at a certain value of the temperature wave length makes it possible by means of Eq. (27) to determine the semiconductor bandgap (if the Seebeck coefficients are known) or the difference in the electron and hole Seebeck coefficients (with a known bandgap) in a single experiment along with the diffusion length and the carrier lifetime. Finally, we estimate the magnitude of the thermoelectric short circuit current. By choosing the optimum velocity of the temperature wave according to Eq. (24) for the wavelength of the order of 1 μm, assuming μn ∼ 1500 cm2/(V s), μp ∼ 500 cm2/(V s), n0 ∼ p0 ∼ 1011 cm−3 , αn,p ∼ 1 mV/K, the ambipolar diffusion length of the order of 1 μm, the semiconductor bandgap above 1 eV, at room temperature, and δT0 ∼ 1 K one gets the density of the dc thermoelectric short-circuit current of the order of 10−7 A/cm2 . This pretty low current density is due to the choice of a very low carrier density, being close to the value of the intrinsic carrier density in silicon. This choice is due to the fact 13
that the factor n0 p0 /(n0 + p0 ) in Eq. (22) decreases rapidly with increasing unipolarity of the semiconductor (since n0 p0 = const [12, 14]). In compensated semiconductors, where high density both for electrons and holes can be achieved simultaneously, one would expect a significant increase in the value of the thermoelectric power. However, in this case the assumption of the interband recombination dominance is open to question, the recombination via impurity centers (e.g., the Shockley-Read-Hall recombination mechanism [20, 25]) is more likely. So, this case requires a separate study. Also narrow-gap semiconductors with high intrinsic carrier density look promising for pronounced manifestation of the considered effect. Note that the above estimation of the thermoelectric current is much higher than the typical currents in the experiments dealing with the non-steady-state photo-emf [10]. This rough estimation suggests that the considered effect more likely should be regarded as a possible way to the experimental study of semiconductors, rather than the basis for the thermocouple development. On the other hand, the thermoelectric current caused by the classical thermo-emf in a semiconductor with the above parameters at the temperature difference about 1 K in the sample of 1 cm in length is dozens of times less. Noteworthy that in the considered problem the heat flow from the semiconductor to the thermostat is orthogonal to the direction of the temperature wave propagation, and thus, both to the thermoelectric current direction and to the temperature gradient creating this current. This configuration allows surmising that the efficiency of the thermal converter based on the dynamic thermoelectric effect may be larger than of the conventional thermocouple. Unfortunately, within the above-stated one-dimensional model it is impossible to obtain correct estimations for the efficiency of the dynamic thermoelectric converter. 4. Conclusions The above investigation shows that in a uniform bipolar semiconductor the temperature distribution in the form of the travelling wave is accompanied by
14
an electric current flowing in the direction of the temperature wave propagation. As this takes place, the dc component of the short circuit thermoelectric current prevails, while the fundamental harmonic of the ac component is absent. This result is very unusual: at every point in the semiconductor the temperature distribution is an oscillating function with the spatially constant mean value, there is no external bias, the system is uniform and has no rectifying properties in its static current-voltage characteristic, but the direct current appears in the circuit. The physical basis for this effect is the presence of a phase shift between the nonequilibrium carrier density wave and the temperature wave in the semiconductor. This phase shift is formed by the combined influence of a number of mechanisms: the lag effect in the generation-recombination processes, finite response times of the diffusion and drift in a nonequilibrium thermoelectric field. This is responsible for some unusual properties of the considered dynamic thermoelectric effect. Namely, the value of the thermoelectric short circuit current contains a contribution that generally is independent of the Seebeck coefficient of the semiconductor. The magnitude of the thermoelectric short circuit current is proportional to the square of the temperature wave amplitude and does not depend on the semiconductor length. However, the current significantly and nonmonotonically depends on the length of the temperature wave and its propagation velocity. The thermoelectric short circuit current vanishes for any static temperature distribution. Also the current is suppressed at extremely high velocities of the temperature wave. Thus, there is an optimal value for the temperature wave velocity at which the dynamical thermoelectric effect manifests itself to the utmost. The proposed theory provides a way to determine the optimum velocity. It proved to be completely determined by the ambipolar diffusion length of the semiconductor, as well as by the nonequilibrium carrier lifetimes. In turn, this fact allows offering using the dynamic thermoelectric effect to measure these parameters of the semiconductor. The thermoelectric current also vanishes in the case of an extremely large 15
temperature wave length (Λ → ∞), i.e. in the case of the almost uniform temperature perturbation. In the opposite limiting case of a very small temperature wave length (Λ → 0), in context of the interference technique for creating the temperature profile in the form of a traveling wave, the effect is also suppressed. This is due to the fact that in these experimental conditions ω = const, and the temperature wave velocity decreases with decreasing Λ. Thus, there is also an optimum value for the temperature wave length at which the current in the circuit reaches its maximum. In experimental conditions of the fixed temperature wave velocity the thermoelectric short circuit current remains nonzero when the temperature wave length decreases to the zero limit. At the same time, the functional dependence of the thermoelectric short circuit current on the temperature wave length may be monotonous. The theory developed also shows that if the Seebeck coefficients of electrons and holes fulfill the condition αp > αn , the magnitude of the thermoelectric short circuit current changes its sign with decreasing the temperature wave length. The value of the temperature wave length, at which the sign reversal occurs, is independent of the temperature wave velocity. This fact makes it possible to measure the band gap or the Seebeck coefficient simultaneously in a single experiment with the measurement of the diffusion length and the carrier lifetime. An essential feature of the effect studied is its manifestation even in uniform semiconductors, not requiring a contact between two media with different Seebeck coefficients. Moreover, the magnitude of the effect may not depend on the Seebeck coefficient of the semiconductor at all. Hence, the dynamic thermopower can appear in conditions that are completely atypical for manifestation of conventional thermoelectric phenomena. In turn, this means that the dynamic thermoelectric effect may manifest itself in totally unexpected situations. In particular, it may serve as a source of noise and may cause malfunction of semiconductor electronic devices. Indeed, the operation of modern microprocessors is accompanied by propagation of temperature perturbations over the chip crystal that is favorable for the dynamic thermoelectric effect manifesta16
tion. The thermoelectric field, following the temperature wave propagation, may appear in a completely unexpected place and affect the operation of the integrated circuit. Thus, the account of the considered effect may be important in the design of new high-performance computing devices. Acknowledgments The author thanks Dr. Yury Gurevich (CINVESTAV-I.P.N., Mexico) for useful discussions. Appendix A. Temperature wave creation Let us show that in the semiconductor slab, externally heated from one side and having a thermal contact with the thermostat at the opposite side, under certain conditions the temperature distribution can be represented in the form of a traveling wave. Here the heating method is of no concern: either direct contact with a moving heater or nonuniform heating by infrared radiation, etc. One of the most convenient implementation of the heating can be the experimental technique similar to the moving photocarrier grating technique [10] (that now is widely used for simultaneous measurements of carrier mobility and lifetime), but with the laser wavelength making the photocarriers generation impossible. In this case, at the upper surface of the semiconductor slab two laser beams (with some frequency shift) form interference fringes moving in the direction of the axis x and heating the semiconductor. The temperature distribution in the semiconductor slab is described by the thermal conductivity equation [26, 27]: 2 ∂T ∂ T ∂2T ∂2T 2 =a + + , ∂t ∂x2 ∂y 2 ∂z 2
(A.1)
where a2 is the thermal diffusivity. As mentioned above, in the present paper the temperature along the y-axis is considered uniform. For the considered problem the BCs for Eq. (A.1) are as follows. If the upper surface of the sample (z = 0) is illuminated with light of intensity 17
I = I0 (1 + cos Φ) (where Φ ≡ kx + ωt), completely absorbed in an infinitely thin near-surface layer of the sample without photocarrier generation, the appropriate BCs are as follows: ∂T g0 I0 (1 + cos Φ) , =− ∂z z=0 κ
(A.2)
where κ is the thermal conductivity of the semiconductor, g0 is the heating efficiency of the semiconductor by the light. In the absence of reflection for the single-mode laser beam the heating efficiency can be considered as g0 = ω0 (here ω0 is the laser radiation frequency). On the opposite side of the sample (z = h), which is the interface with the substrate serving as the thermostat, the BC is written in the form [26, 27]: ∂T χ = − (T − T0 ), (A.3) ∂z z=h κ where χ is the surface thermal conductivity. Along the x axis we use the cyclic BC: ∂T ∂T = . ∂x x=0 ∂x x=L
(A.4)
The solution to Eq. (A.1) will be sought in the form of: T (x, z, t) = A1 (z) sin Φ + A2 (z) cos Φ + A3 (z).
(A.5)
Substituting Eq. (A.5) in Eq. (A.1) one obtains the system of equations for the temperature wave amplitudes A1−3 (z): ω A2 (z) = 0, a2 ω A2 (z) − k 2 A2 (z) − 2 A1 (z) = 0, a
A1 (z) − k 2 A1 (z) +
A3 (z) = 0,
(A.6) (A.7) (A.8)
with BC A1 (0) = 0,
A2 (0) = A3 (0) = −
χ A1,2 (h) = − A1,2 (h), κ χ A3 (h) = − (A3 (h) − T0 ). κ 18
g0 I0 , κ
(A.9) (A.10) (A.11)
The general solution to the system of Eqs. (A.6)-(A.8) has the following form: 1 k1 z 1 k2 z + (C1 − iC3 ) cosh + A1 (z) = (C1 + iC3 ) cosh 2 a 2 a iag0 I0 k1 z k2 z + − k1 sinh k2 sinh , 2k1 k2 a a i i k2 z k1 z − (C1 + iC3 ) cosh + A2 (z) = (C1 − iC3 ) cosh 2 a 2 a ag0 I0 k1 z k2 z − k1 sinh + k2 sinh , 2k1 k2 a a A3 (z) =C5 z + C6 ,
(A.12)
(A.13) (A.14)
where k12 = a2 k 2 + iω, k22 = a2 k 2 − iω, C1−6 are the integration constants. For a thermally thin sample |k1,2 |h/a 1 Eqs. (A.12)-(A.13) take the simple form: A1 (z) = C1 + C2 z,
A2 (z) = C3 + C4 z.
(A.15)
Once the integration constants have been calculated, one obtains the following temperature distribution: T = T0 −
g0 I0 κ
κ + χh z− (1 + cos Φ) . χ
(A.16)
As can be seen from Eq. (A.16), if the semiconductor is weakly coupled to the substrate (χh κ), then the temperature distribution in the sample becomes: T = T0 +
g0 I0 (1 + cos Φ) , χ
(A.17)
that formally corresponds to Eq. (1) with T0 being substituted by T0 + g0 I0 /χ and δT0 = g0 I0 /χ. References [1] K. Biswas, J. He, I. D. Blum, C. Wu, T. P. Hogan, D. N. Seidman, V. P. Dravid, M. G. Kanatzidis, Nature 489 (2012) 414–418. [2] A. J. Minnich, M. S. Dresselhaus, Z. F. Ren, G. Chen, Energy Environ. Sci. 2 (2009) 466–479. 19
[3] G. Ding, G. Gao, K. Yao, Sci. Rep. 5 (2015) 9567. [4] I. N. Volovichev, Yu. G. Gurevich, Curr. Appl. Phys. 16 (2016) 191–196. [5] J. T. Abrahamson, B. Sempere, M. P. Walsh, J. M. Forman, F. Sen, S. Sen, S. G. Mahajan, G. L. C. Paulus, Q. H. Wang, W. Choi, M. S. Strano, ACS Nano 7 (2013) 6533–6544. [6] V. N. Agarev, Semiconductors 31 (1997) 784–785. [7] S. Kono, P. Gu, M. Tani, K. Sakai, Appl. Phys. B 71 (2000) 901–904. [8] A. Reklaitis, J. Appl. Phys. 108 (2010) 053102. [9] M. P. Petrov, I. A. Sokolov, S. I. Stepanov, G. S. Trofimov, J. Appl. Phys. 68 (1990) 2216–2225. [10] U. Haken, M. Hundhausen, L. Ley, Phys. Rev. B 51 (1995) 10579–10590. [11] A. J¨ ungel, Transport Equations for Semiconductors, Springer, 2009. [12] V. L. Bonc-Bruevic, S. G. Kalasnikov, Physics of the Semiconductors, Berlin: VEB Deutscher Verlag der Wissenschaften, 1982 (in German). [13] I. Volovichev, J. Appl. Phys. 119 (2016) 095712. [14] D. Neamen, Semiconductor physics and devices: basic principles, McGrawHill Series in Electrical and Computer Engineering, McGraw-Hill, 2003. [15] H. J. Goldsmid, Introduction to Thermoelectricity, Springer-Verlag Berlin Heidelberg, 2009. [16] V. Zlatic, R. Monnier, Modern Theory of Thermoelectricity, Oxford University Press, 2014. [17] L. Villegas-Lelovsky, G. Gonzalez de la Cruz, I. N. Volovichev, Phys. Stat. Sol. (b) 242 (2005) 971–982. [18] S. R. in’t Hout, J. Appl. Phys. 79 (11) (1996) 8435–8444. 20
[19] Yu. G. Gurevich, J. E. Vel´azquez-P´erez, G. Espejo-L´opez, I. N. Volovichev, O. Yu. Titov, J. Appl. Phys. 101 (2) (2007) 023705. [20] P. T. Landsberg, Recombination in Semiconductors, CUP, 2003. [21] I. N. Volovichev, Yu. G. Gurevich, Semiconductors 35 (3) (2001) 306–315. [22] I. N. Volovichev, G. N. Logvinov, O. Yu. Titov, Yu. G. Gurevich, J. Appl. Phys. 95 (8) (2004) 4494–4496. [23] D. M. Rowe, Thermoelectrics Handbook: Macro to Nano, CRC Press, 2005. [24] J. Tauc, Photo and thermoelectric effects in semiconductors, Pergamon Press, 1962. [25] W. Shockley, W. T. Read, Phys. Rev. 87 (1952) 835–842. [26] J. Welty, C. Wicks, and R. Wilson, Fundamentals of momentum, heat, and mass transfer, Wiley, 1976. [27] J. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, CUP, 1984.
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S0921-4526(16)30130-2 http://dx.doi.org/10.1016/j.physb.2016.04.010 PHYSB309432
To appear in: Physica B: Physics of Condensed Matter Received date: 29 January 2016 Revised date: 30 March 2016 Accepted date: 9 April 2016 Cite this article as: I.N. Volovichev, Dynamic thermoelectricity in uniform bipolar semiconductor, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.04.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Dynamic thermoelectricity in uniform bipolar semiconductor I.N. Volovichev A. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, 12 Ac. Proscura St., Kharkov 61085, Ukraine
Abstract The theory of the dynamic thermoelectric effect has been developed. The effect lies in an electric current flowing in a closed circuit that consists of a uniform bipolar semiconductor, in which a non-uniform temperature distribution in the form of the traveling wave is created. The calculations are performed for the one-dimensional model in the quasi-neutrality approximation. It was shown that the direct thermoelectric current prevails, despite the periodicity of the thermal excitation, the circuit homogeneity and the lack of rectifier properties of the semiconductor system. Several physical reasons underlining the dynamic thermoelectric effect are found. One of them is similar to the Dember photoelectric effect, its contribution to the current flowing is determined by the difference in the electron and hole mobilities, and is completely independent of the carrier Seebeck coefficients. The dependence of the thermoelectric short circuit current magnitude on the semiconductor parameters, as well as on the temperature wave amplitude, length and velocity is studied. It is shown that the magnitude of the thermoelectric current is proportional to the square of the temperature wave amplitude. The dependence of the thermoelectric short circuit current on the temperature wave length and velocity is the nonmonotonic function. The optimum values for the temperature wave length and velocity, at which the dynamic thermoelectric effect is the greatest, have been deduced. It is found that ∗ Tel.:
+380 57 7203331 Email address: [email protected] (I.N. Volovichev)
Preprint submitted to Physica B
April 9, 2016
the thermoelectric short circuit current changes its direction with decreasing the temperature wave length under certain conditions. The prospects for the possible applications of the dynamic thermoelectric effect are also discussed. Keywords: dynamic thermoelectric effect, thermoelectricity, thermopower, Seebeck effect, nonequilibrium charge carriers PACS: 72.20.Pa, 73.50.Lw
1. Introduction Research of the thermoelectric phenomena and development of efficient thermoelectric converters represent a major challenge to the modern applied solid state physics. Currently, investigations is mainly focused on the search and creation of new efficient thermoelectric materials with a high thermoelectric figure of merit ZT [1–3]. However, despite a long history of thermoelectric research, because of its diversity the physics of transport phenomena under non-equilibrium energy conditions (a particular case of which is a non-uniform temperature distribution [4]) is as yet imperfectly understood. This is especially true in the case of simultaneous action of several perturbations in the semiconductor systems, as well as in the case of time-dependent temperature fields. Meanwhile, there are a number of papers demonstrating that the energy conversion in semiconductor systems under non-steady state conditions may differ significantly from the static case by its physical background. In particular, a significant increase in the thermocouple efficiency [5] (although this work does not deal with a semiconductor system), as well as an increase in the thermoelectric power at the p-n junction [6] are proven in the transient regime. Note that the difference between dynamic mechanisms of the electromotive force (emf) formation and their static counterparts is not an exclusive feature of thermoelectric phenomena. It is known that, for example, a photo-emf in dynamic mode can significantly exceed the static values (for example, the Dember emf [7, 8]), as well as it can appear in situations where no static emf can arise [9]. Moreover, an effective experimental technique for semiconductor pa2
rameter measurements has been developed based on the effect of the non-steady state photo-emf [10]. The main factors underlying the most significant distinctions of thermoelectric effects in the dynamics might be as follows. First, a time-dependent perturbation gives rise to an uncompensated space charge in the semiconductor. In turn, the latter affects the conventional mechanism of the thermopower generation. This way is of especial importance in multi-layer structures and polycrystalline films [6]. Second, time derivatives of the carrier densities explicitly appear in the continuity equation [11, 12], and it might directly affect the magnitude of the thermo-emf. For the non-steady state photovoltaic phenomena it is the mechanism that determines the existence of the Dember emf even when the production of the static photoelectricity is basically impossible [9]. Finally, the time dependence of the temperature can influence the transport processes through changing the thermal generation rate. As shown in Ref. [4], in the static case the thermal generation can contribute to the emf only when there is the difference in the carrier temperatures (e.g., the electron temperature differs from the temperature of holes and phonons). If there is a single temperature of all quasiparticles in the semiconductor, the thermal generation does not affect the value of the thermopower at any static temperature distribution in the semiconductor structure. The time dependence of the thermal generation rate can change the picture, leading to at first glance paradoxical results. Thus, it is widely assumed that two contacts at different temperatures between two media with different Seebeck coefficients are mandatory for the thermopower appearance. Or that in a uniform semiconductor sample having the same temperature of its ends, no thermoelectricity is generated at any temperature distribution inside the semiconductor. However, in the dynamic case the above is no longer true. Thus, in Ref. [13] it has been presented a physical mechanism that is responsible for the thermoelectric current generation by the temperature wave propagating in a uniform unipolar semiconductor. Below, we show that a similar phenomenon takes place in a uniform bipolar semiconductor. In the latter, as in a more complex physical 3
system (in which there are two types of the charge carriers: electrons and holes), the nature of the dynamic thermoelectric effect is also more complicated, and the magnitude of the thermopower generated can be substantially higher than the corresponding value in the unipolar semiconductor. The presented theory also shows that the thermoelectric short circuit current is a nonmonotonic function of the temperature wave parameters. This allows us to offer using the considered dynamic thermoelectric effect as a research technique for investigations of the bipolar semiconductor properties. 2. Mathematical model of dynamic thermoelectric effect Consider a slab of uniform bipolar semiconductor of the length L, in which a stationary (but not static) temperature distribution in the form of a traveling wave is maintained by an external heating: T = T0 + δT0 cos Φ,
(1)
where Φ = kx + ωt, k = 2π/Λ, Λ is the temperature wave length, ω = kv, v is the temperature wave velocity. To simplify the calculations, we assume that the slab length is divisible by the temperature wave length, i.e. let kL/(2π) be an integer. We will not dwell on methods for creating the temperature distribution of the form in Eq. (1) in the specimen. For discussion of this matter see the Appendix A. We will restrict our consideration to the one-dimensional model, supposing all physical values in directions normal to the temperature wave propagation to be uniform. Then, under the drift-diffusion model one can write the following continuity equations for the electron (jn ) and hole (jp ) current densities [11, 12, 14]: ∂n 1 ∂jn = − Rn , ∂t e ∂x 1 ∂jn ∂p =− − Rp , ∂t e ∂x 4
(2) (3)
with ∂n ∂T − eμn αn n , ∂x ∂x ∂p ∂T − eμp αp p , jp = eμp pE − T μp ∂x ∂x jn = eμn nE + T μn
(4) (5)
where n and p are the electron and hole densities, respectively; e is the elementary charge, Rn,p are the recombination rates of electrons and holes, μn,p are the electron and hole mobilities, E is the electric field in the semiconductor, αn,p are the Seebeck coefficients for the electron and hole subsystems. Brief mention should be made of the following. In modern literature, Eqs. (4)(5) are written frequently in the following form [15–17]: ∂ψn,p ∗ ∂T − αn,p jn,p = −σn,p , ∂x ∂x
(6)
where ψn,p = ϕ ∓ ζn,p /e are the electrochemical potentials of the carriers (here, sign ”-” refers to the electrons, and ”+” to holes), ϕ is the electric potential, ζn,p are the chemical potentials of the carriers, σn = eμn n and σp = eμp p are the conductivity of the electron and hole subsystems, respectively. Introduced in that way the Seebeck coefficients α∗n,p differ from the phenomenologically introduced Seebeck constants in Eqs. (4)-(5) and are related to them as follows: α∗n,p = αn,p ∓
1 ∂ζn,p . e ∂T
(7)
Below we will use the Seebeck coefficients αn,p , and the resulting expression can be rewritten in terms of α∗n,p in an obvious way, using Eq. (7). Here we restrict ourselves to the quasi-neutrality approximation by assuming the Debye screening length to be the smallest spatial parameter and the Maxwell relaxation time to be the smallest temporal parameter of the problem [18, 19]. Under these conditions there is no space charge in the semiconductor: ρ = e(p − nt − n) = 0, where nt is the charged impurities density (to be specific, the density of negatively charged impurity levels). Suppose that the dominant mechanism of recombination is the interband recombination (if so, the calculations are less bulky). Then [20, 21]: Rn = Rp = R = γ (np − n0 p0 ) , 5
(8)
where γ is the capture coefficient, n0 and p0 are the electron and hole densities in the thermodynamic equilibrium (i.e. at T0 ), respectively. To simplify the calculations we assume that the charged impurity density remains constant. This assumption is valid at sufficiently high temperatures, when in the thermodynamic equilibrium the majority of dopant atoms are already ionized. Thus, the local change in temperature has no effect on them. The appearance of nonequilibrium electrons and holes does not affect the equilibrium of the impurity subsystem due to the interband transitions dominate. In these conditions, the lack of space charge imposes the following relation between the nonequilibrium electron density δn = n − n0 and the nonequilibrium hole density δp = p − p0 : δn = δp [19]. In the quasi-neutrality from Eqs. (2)-(3) follows ∂j/∂x = 0, where j = jn +jp is the total electric current density. Adding up Eqs. (4) and (5), one gets: j(t) = σE + T (μn − μp )
∂δn ∂T − (αn σn + αp σp ) , ∂x ∂x
(9)
where σ = σn + σp is the total conductivity of the semiconductor. The system of differential equations (2)-(3) must be supplemented by boundary conditions (BCs). To avoid issues related to influence of contacting media and physical processes at the interfaces we choose the cyclic BCs. From a physical point of view, such a choice is equivalent to connecting the slab ends, i.e. to creating a cylinder or ring-shaped semiconductor circuit (sample). Thus one obtains a homogeneous (not containing other media than the semiconductor) closed circuit. Then: n(0, t) = n(L, t),
∂n(0, t) ∂n(L, t) = , ∂x ∂x
(10)
E(0, t) = E(L, t),
ϕ(0, t) = ϕ(L, t),
(11)
where ϕ is the electric field potential (E = −∂ϕ/∂x). Note also that for the given temperature distribution the condition T (0, t) = T (L, t) is fulfilled identically. The last-mentioned BC in Eq. (11) can be rewritten as: L E dx = 0. 0
6
(12)
Then, expressing the electric field E from Eq. (9) and substituting it into Eq. (12), we obtain:
j(t) = (μn − μp )
L
0
T ∂δn dx− σ ∂x −
L
0
L αn σn + αp σp ∂T dx / σ −1 dx. (13) σ ∂x 0
From the form of Eq. (13), which is an exact one (no assumption about the smallness of the nonequilibrium carrier densities has been yet made), it follows that: • The first non-zero term of the series expansion of j(t) in powers of a small parameter δT /T0 (δT = T − T0 ) is quadratic. • To determine j(t) with an accuracy of (δT /T0 )2 it is necessary and sufficient to find δn and E with an accuracy of δT /T0 . In other words it is sufficient to solve the linearized system of equations (2)-(3). With this in mind, the current in the circuit may be written as follows: j(t) =
μn − μp L
0
L
δT
∂δn dx− ∂x −
e2 μn μp (αn − αp )(p0 − n0 ) σ0 L
0
L
δn
∂T dx, (14) ∂x
where σ0 is the semiconductor conductivity at T0 . When deriving Eq. (14) all terms representing a total differential in the integrands, as well as small quantities of higher orders have been omitted. The linearized system of equations (2)-(5) taking into account the quasineutrality conditions δp = δn (δn n0 , δn p0 ) becomes: 1 ∂jn ∂δn = − R, ∂t e ∂x 1 ∂jp ∂δn =− − R, ∂t e ∂x
(15) (16)
∂δn ∂δT − eμn αn n0 , ∂x ∂x ∂δn ∂δT − eμp αp p0 , jp = eμp p0 E − T0 μp ∂x ∂x jn = eμn n0 E + T0 μn
7
(17) (18)
and the expression for the interband recombination (8) can be written as: 1 n0 p0 δT R= , (19) δn − τ n0 + p 0 T 0 where τ −1 = γ(n0 + p0 ); the parameter , describing the temperature dependence of the thermal generation rate, for the nondegenerate electron gas is equal to = 3 + εg /(kB T0 ); εg is the semiconductor bandgap value, kB is the Boltzmann constant [22]. Using the first harmonic approximation method (FHA), we seek the solution of equations (15)-(17) in the form of: δn = δn0 + δns sin(kx + ωt) + δnc cos(kx + ωt),
(20)
with δn0 , δns , δnc n0 , p0 . It is easily seen that this choice satisfies the BCs in Eq. (10) identically. Substitute Eq. (20) in Eq. (15) and collect terms that do not depend on Φ. As is easy to see, the latter appear in Eq. (15) solely due to the term R, defined by Eq. (19) (since the other terms in Eq. (15) are derivatives with respect to time or coordinate). This immediately implies that δn0 = 0. Thus, the temperature wave leaves the averaged carrier density unchanged. This is physically clear: the thermal generation is the only source of the excess carriers, and as the mean temperature deviation in the temperature wave is zero, it does not affect the mean carrier density. Now let us substitute Eq. (20) in Eq. (14), taking into account δn0 = 0. It is easy to see that after integration the non-zero result is owing to the terms with sin2 Φ or cos2 Φ. All terms with product sin Φ cos Φ, being integrated over the semiconductor length, vanish. Finally, one obtains the following expression for the short-circuit thermoelectric current: k e2 μn μp (αn − αp )(p0 − n0 ) δns δT0 . j0 = μn − μp + 2 σ0
(21)
It should be noted that within the framework of the FHA method the electric current in the closed circuit contains only the direct current (dc) component, while the main harmonic of the thermoelectric current is absent. The second 8
harmonic, whose amplitude is proportional to (δT0 /T0 )2 , is beyond the linear theory. In order to find the amplitude of the alternating current (ac), the linear approximation is insufficient; one must take into account the terms of higher order of smallness. In addition, the ac component is effectively suppressed by the quasi-neutrality. Therefore, below we concentrated our attention on the dc component of the thermoelectric short circuit current. 3. Results and discussion Substituting Eq. (20) in Eqs. (15)-(19), we obtain a system of algebraic equations. Solving the latter, after simple but rather cumbersome transformations, the following expressions for the non-equilibrium carrier densities can be worked out:
¯ δT0 n0 p0 e(αn − αp )k¯ 2 + ω δns = ,
2 n0 + p 0 T0 ω ¯ 2 + k¯2 + 1
n0 p0 e(αn − αp )k¯ 2 + k¯ 2 + 1 δT0 , δnc =
2 n0 + p 0 T0 ω ¯ 2 + k¯2 + 1
(22) (23)
where we have introduced the normalized wave vector and the wave frequency of √ ¯ = ωτ ; LD = Da τ is the diffusion length, the temperature wave: k¯ = kLD , ω Da = T0 (n0 + p0 )μn μp /σ0 is the ambipolar diffusion coefficient. With reference to Eq. (21) and Eq. (22), it can be seen that the short-circuit thermoelectric current (and the corresponding thermopower) contains contributions from two physical mechanisms. The first mechanism is similar by its nature to the Dember photo-emf and is due to the difference in mobility of electrons and holes. It is remarkable that this contribution is independent of the Seebeck coefficients αn,p . We emphasize that the presence of this term is not typical for the known mechanisms of the thermoelectric emf. To the contrary, the second component of the dynamic thermo-emf depends on the Seebeck coefficients of the semiconductor. However, here also there are some interesting differences from the static thermopower. Thus, this term is proportional to the difference between Seebeck coefficients of electrons and holes, rather than to the effective
9
Seebeck coefficient of the semiconductor (αn σn + αp σp )/σ0 , as it is usually the case in bipolar semiconductors. Furthermore, according to Eq. (21) the term that depends on the difference in the Seebeck coefficients, is also proportional to the difference in the equilibrium densities of electrons and holes. In turn, this means that the contribution to the thermopower for the n-type (electron) semiconductor and the p-type (hole) semiconductor has opposite sign, while entirely vanishing in intrinsic or compensated semiconductors. However, in this case the dependence on αn − αp still influence the magnitude of the nonequilibrium carrier densities, see Eq. (22). Also note that the current in the circuit does not depend on the semiconductor length. This fact has a simple explanation. Each spatial period of temperature wave makes its additive contribution to the thermo-emf magnitude. Increasing the semiconductor length, the number of periods of the temperature wave that fit in the sample also increases, so increasing the total thermopower. On the other hand, with increasing the length of the semiconductor its resistance also rises proportionally. As a result, the thermoelectric short circuit current is independent of the semiconductor length. Let us analyze the dependence of the nonequilibrium carrier densities and the thermoelectric short circuit current on the temperature wave parameters. Recall that in the framework of the method used, δnc denotes the amplitude of the inphase (with respect to the temperature wave) nonequilibrium electron wave, and δns denotes the amplitude of the phase shifted (quarter-phase) density wave. As it can be seen from Eq. (21), the short circuit current is determined by the quarter-phase nonequilibrium carrier density component only. It is easy to see from Eqs. (22)-(23) that at too high temperature wave velocities (ω → ∞) the nonequilibrium carrier densities decrease to zero. At the same time, for a slow movement of the temperature waves (in the limit, for the static temperature distribution) the quarter-phase component also vanishes, whereas the inphase component remains finite. In turn, this means that the considered thermoelectric effect disappears both at too high temperature wave velocities and at a fixed temperature distribution. The latter implies that the thermoelec10
tric effect is a strictly dynamic phenomenon. This deduction is consistent with the well-known fact that in a uniform semiconductor whose ends are kept the same temperature, there is no thermopower whatever the static temperature distribution has been created inside the semiconductor [23, 24]. The thermoelectric current vanishing as ω −1 at extremely high temperature wave velocities ω → ∞ is also quite understandable. In this case, the system is unable to follow the change of temperature, and effectively we have similarity to the static situation. Thus, there is some optimal velocity of the temperature wave, for which the magnitude of the dynamic thermoelectric effect is the largest. It is not difficult to determine the value of the optimal temperature wave velocity. Finding the extremum of δns in Eq. (22) as a function of ω, one gets: vmax =
1 + k 2 L2D . kτ
(24)
Pay attention to the simple form of the functional dependence in Eq. (24). This fact allows us to offer using the considered effect for the experimental determination of the diffusion length and the lifetime of nonequilibrium carriers. Indeed, measuring the temperature wave velocity, at which the current in the circuit reaches the maximum, for several different spatial periods1 of the temperature pattern (i.e. for different lengths of the temperature wave), it is easy to calculate the diffusion length LD and the lifetime τ by means of Eq. (24). Let us analyze the dependence of the thermoelectric short circuit current on the temperature wave length. Naturally, with k → 0 the thermoelectric current decreases linearly and vanishes at k = 0, because it is the case of a uniform heating of the semiconductor. In the opposite limiting case k → ∞ the thermoelectric current vanishes as k −1 : at an extremely small temperature wave length the heating can be regarded as effectively uniform. Accordingly, 1 Remember
that when using the experimental laser technique of the moving interference
pattern its spatial period and its velocity can be handled independently [10]. The period is controlled by the tilt angle of the sample, while the velocity is controlled by the frequency shift between two laser beams. For details, see Ref. [10]
11
there is an optimal value for the temperature wave length at which the dynamic thermoelectric effect is the most pronounced. Unfortunately, in general case it almost impossible to get a useful analytical expression for the optimal temperature wave length. However, for efficient thermoelectric materials (in which the condition [3 + εg /(kB T0 )]/[e(αn − αp )] k 2 L2D is met) the optimum value of the temperature wave length is determined by the following expression: 1 kmax = 1 + 4 + 3ω 2 τ 2 . LD
(25)
In the opposite limiting case [3 + εg /(kB T0 )]/[e(αn − αp )] k 2 L2D (e.g., in wide-gap semiconductors at low temperatures), the optimal value for the temperature wave length is somewhat different: √ 3 kmax = 4 + 3ω 2 τ 2 − 1. 3LD
(26)
Note that the above analysis of the thermoelectric short-circuit current dependence on the temperature wave length is made under the assumption of a constant temperature wave frequency ω = kv (instead of a constant temperature wave velocity v). This situation is typical for the interference methods of creating the nonuniform heating, when the temperature wave velocity decreases with decreasing wave length, i.e. with growth of k (v = ΛΔf , where Δf is the frequency shift between two laser beams, see Ref. [10]). At a fixed temperature wave velocity the situation is somewhat different. In particular, for k → ∞ the dynamic thermoelectric effect does not disappear. So, for some combination of semiconductor parameters the monotonous functional dependence of the short circuit thermoelectric current on the temperature wave length becomes possible. However, if the condition [3 + εg /(kB T0 )]/[e(αn − αp )] k 2 L2D holds true, the extremum still has a place, and the optimal value for the temperature wave length is given by kmax = 1/LD . The considered dynamic thermoelectric effect has another interesting property. As seen from Eq. (22), if αp > αn the thermoelectric short circuit current changes its sign with increasing k at
3 + εg /(kB T0 ) 1 . k0 = LD e(αp − αn ) 12
(27)
The physical meaning of the sign change is as follows. In the presence of the temperature wave in the initially uniform semiconductor two simultaneous competing processes take place: the thermal generation of the nonequilibrium carriers (its rate is determined both by the parameter and by the amplitude of the temperature wave), and the flow of nonequilibrium carriers into colder areas of the semiconductor due to the temperature gradient (for clarity, here we put aside the diffusion of carriers). The rate of the second process is determined by the Seebeck coefficient, as well as by the temperature gradient, i.e. both by the amplitude and by the wavelength of the temperature wave. One can choose the temperature wave length (k = k0 ), so that these two processes cancel each other (of course, this is only true in the linear approximation and for the harmonic temperature distribution). At this, the nonequilibrium carriers do not appear at all (the factor e(αn − αp )k¯ 2 + in Eqs. (22)-(23) is zero). On opposite sides of this value k0 there dominates either the generation of carriers or their flow away. Thus, at the point of the temperature maximum either the maximum of the carrier densities or their minimum takes place. The latter corresponds to a phase shift of π/2, i.e. it means different sign of the thermoelectric power. Vanishing of the thermoelectric current at a certain value of the temperature wave length makes it possible by means of Eq. (27) to determine the semiconductor bandgap (if the Seebeck coefficients are known) or the difference in the electron and hole Seebeck coefficients (with a known bandgap) in a single experiment along with the diffusion length and the carrier lifetime. Finally, we estimate the magnitude of the thermoelectric short circuit current. By choosing the optimum velocity of the temperature wave according to Eq. (24) for the wavelength of the order of 1 μm, assuming μn ∼ 1500 cm2/(V s), μp ∼ 500 cm2/(V s), n0 ∼ p0 ∼ 1011 cm−3 , αn,p ∼ 1 mV/K, the ambipolar diffusion length of the order of 1 μm, the semiconductor bandgap above 1 eV, at room temperature, and δT0 ∼ 1 K one gets the density of the dc thermoelectric short-circuit current of the order of 10−7 A/cm2 . This pretty low current density is due to the choice of a very low carrier density, being close to the value of the intrinsic carrier density in silicon. This choice is due to the fact 13
that the factor n0 p0 /(n0 + p0 ) in Eq. (22) decreases rapidly with increasing unipolarity of the semiconductor (since n0 p0 = const [12, 14]). In compensated semiconductors, where high density both for electrons and holes can be achieved simultaneously, one would expect a significant increase in the value of the thermoelectric power. However, in this case the assumption of the interband recombination dominance is open to question, the recombination via impurity centers (e.g., the Shockley-Read-Hall recombination mechanism [20, 25]) is more likely. So, this case requires a separate study. Also narrow-gap semiconductors with high intrinsic carrier density look promising for pronounced manifestation of the considered effect. Note that the above estimation of the thermoelectric current is much higher than the typical currents in the experiments dealing with the non-steady-state photo-emf [10]. This rough estimation suggests that the considered effect more likely should be regarded as a possible way to the experimental study of semiconductors, rather than the basis for the thermocouple development. On the other hand, the thermoelectric current caused by the classical thermo-emf in a semiconductor with the above parameters at the temperature difference about 1 K in the sample of 1 cm in length is dozens of times less. Noteworthy that in the considered problem the heat flow from the semiconductor to the thermostat is orthogonal to the direction of the temperature wave propagation, and thus, both to the thermoelectric current direction and to the temperature gradient creating this current. This configuration allows surmising that the efficiency of the thermal converter based on the dynamic thermoelectric effect may be larger than of the conventional thermocouple. Unfortunately, within the above-stated one-dimensional model it is impossible to obtain correct estimations for the efficiency of the dynamic thermoelectric converter. 4. Conclusions The above investigation shows that in a uniform bipolar semiconductor the temperature distribution in the form of the travelling wave is accompanied by
14
an electric current flowing in the direction of the temperature wave propagation. As this takes place, the dc component of the short circuit thermoelectric current prevails, while the fundamental harmonic of the ac component is absent. This result is very unusual: at every point in the semiconductor the temperature distribution is an oscillating function with the spatially constant mean value, there is no external bias, the system is uniform and has no rectifying properties in its static current-voltage characteristic, but the direct current appears in the circuit. The physical basis for this effect is the presence of a phase shift between the nonequilibrium carrier density wave and the temperature wave in the semiconductor. This phase shift is formed by the combined influence of a number of mechanisms: the lag effect in the generation-recombination processes, finite response times of the diffusion and drift in a nonequilibrium thermoelectric field. This is responsible for some unusual properties of the considered dynamic thermoelectric effect. Namely, the value of the thermoelectric short circuit current contains a contribution that generally is independent of the Seebeck coefficient of the semiconductor. The magnitude of the thermoelectric short circuit current is proportional to the square of the temperature wave amplitude and does not depend on the semiconductor length. However, the current significantly and nonmonotonically depends on the length of the temperature wave and its propagation velocity. The thermoelectric short circuit current vanishes for any static temperature distribution. Also the current is suppressed at extremely high velocities of the temperature wave. Thus, there is an optimal value for the temperature wave velocity at which the dynamical thermoelectric effect manifests itself to the utmost. The proposed theory provides a way to determine the optimum velocity. It proved to be completely determined by the ambipolar diffusion length of the semiconductor, as well as by the nonequilibrium carrier lifetimes. In turn, this fact allows offering using the dynamic thermoelectric effect to measure these parameters of the semiconductor. The thermoelectric current also vanishes in the case of an extremely large 15
temperature wave length (Λ → ∞), i.e. in the case of the almost uniform temperature perturbation. In the opposite limiting case of a very small temperature wave length (Λ → 0), in context of the interference technique for creating the temperature profile in the form of a traveling wave, the effect is also suppressed. This is due to the fact that in these experimental conditions ω = const, and the temperature wave velocity decreases with decreasing Λ. Thus, there is also an optimum value for the temperature wave length at which the current in the circuit reaches its maximum. In experimental conditions of the fixed temperature wave velocity the thermoelectric short circuit current remains nonzero when the temperature wave length decreases to the zero limit. At the same time, the functional dependence of the thermoelectric short circuit current on the temperature wave length may be monotonous. The theory developed also shows that if the Seebeck coefficients of electrons and holes fulfill the condition αp > αn , the magnitude of the thermoelectric short circuit current changes its sign with decreasing the temperature wave length. The value of the temperature wave length, at which the sign reversal occurs, is independent of the temperature wave velocity. This fact makes it possible to measure the band gap or the Seebeck coefficient simultaneously in a single experiment with the measurement of the diffusion length and the carrier lifetime. An essential feature of the effect studied is its manifestation even in uniform semiconductors, not requiring a contact between two media with different Seebeck coefficients. Moreover, the magnitude of the effect may not depend on the Seebeck coefficient of the semiconductor at all. Hence, the dynamic thermopower can appear in conditions that are completely atypical for manifestation of conventional thermoelectric phenomena. In turn, this means that the dynamic thermoelectric effect may manifest itself in totally unexpected situations. In particular, it may serve as a source of noise and may cause malfunction of semiconductor electronic devices. Indeed, the operation of modern microprocessors is accompanied by propagation of temperature perturbations over the chip crystal that is favorable for the dynamic thermoelectric effect manifesta16
tion. The thermoelectric field, following the temperature wave propagation, may appear in a completely unexpected place and affect the operation of the integrated circuit. Thus, the account of the considered effect may be important in the design of new high-performance computing devices. Acknowledgments The author thanks Dr. Yury Gurevich (CINVESTAV-I.P.N., Mexico) for useful discussions. Appendix A. Temperature wave creation Let us show that in the semiconductor slab, externally heated from one side and having a thermal contact with the thermostat at the opposite side, under certain conditions the temperature distribution can be represented in the form of a traveling wave. Here the heating method is of no concern: either direct contact with a moving heater or nonuniform heating by infrared radiation, etc. One of the most convenient implementation of the heating can be the experimental technique similar to the moving photocarrier grating technique [10] (that now is widely used for simultaneous measurements of carrier mobility and lifetime), but with the laser wavelength making the photocarriers generation impossible. In this case, at the upper surface of the semiconductor slab two laser beams (with some frequency shift) form interference fringes moving in the direction of the axis x and heating the semiconductor. The temperature distribution in the semiconductor slab is described by the thermal conductivity equation [26, 27]: 2 ∂T ∂ T ∂2T ∂2T 2 =a + + , ∂t ∂x2 ∂y 2 ∂z 2
(A.1)
where a2 is the thermal diffusivity. As mentioned above, in the present paper the temperature along the y-axis is considered uniform. For the considered problem the BCs for Eq. (A.1) are as follows. If the upper surface of the sample (z = 0) is illuminated with light of intensity 17
I = I0 (1 + cos Φ) (where Φ ≡ kx + ωt), completely absorbed in an infinitely thin near-surface layer of the sample without photocarrier generation, the appropriate BCs are as follows: ∂T g0 I0 (1 + cos Φ) , =− ∂z z=0 κ
(A.2)
where κ is the thermal conductivity of the semiconductor, g0 is the heating efficiency of the semiconductor by the light. In the absence of reflection for the single-mode laser beam the heating efficiency can be considered as g0 = ω0 (here ω0 is the laser radiation frequency). On the opposite side of the sample (z = h), which is the interface with the substrate serving as the thermostat, the BC is written in the form [26, 27]: ∂T χ = − (T − T0 ), (A.3) ∂z z=h κ where χ is the surface thermal conductivity. Along the x axis we use the cyclic BC: ∂T ∂T = . ∂x x=0 ∂x x=L
(A.4)
The solution to Eq. (A.1) will be sought in the form of: T (x, z, t) = A1 (z) sin Φ + A2 (z) cos Φ + A3 (z).
(A.5)
Substituting Eq. (A.5) in Eq. (A.1) one obtains the system of equations for the temperature wave amplitudes A1−3 (z): ω A2 (z) = 0, a2 ω A2 (z) − k 2 A2 (z) − 2 A1 (z) = 0, a
A1 (z) − k 2 A1 (z) +
A3 (z) = 0,
(A.6) (A.7) (A.8)
with BC A1 (0) = 0,
A2 (0) = A3 (0) = −
χ A1,2 (h) = − A1,2 (h), κ χ A3 (h) = − (A3 (h) − T0 ). κ 18
g0 I0 , κ
(A.9) (A.10) (A.11)
The general solution to the system of Eqs. (A.6)-(A.8) has the following form: 1 k1 z 1 k2 z + (C1 − iC3 ) cosh + A1 (z) = (C1 + iC3 ) cosh 2 a 2 a iag0 I0 k1 z k2 z + − k1 sinh k2 sinh , 2k1 k2 a a i i k2 z k1 z − (C1 + iC3 ) cosh + A2 (z) = (C1 − iC3 ) cosh 2 a 2 a ag0 I0 k1 z k2 z − k1 sinh + k2 sinh , 2k1 k2 a a A3 (z) =C5 z + C6 ,
(A.12)
(A.13) (A.14)
where k12 = a2 k 2 + iω, k22 = a2 k 2 − iω, C1−6 are the integration constants. For a thermally thin sample |k1,2 |h/a 1 Eqs. (A.12)-(A.13) take the simple form: A1 (z) = C1 + C2 z,
A2 (z) = C3 + C4 z.
(A.15)
Once the integration constants have been calculated, one obtains the following temperature distribution: T = T0 −
g0 I0 κ
κ + χh z− (1 + cos Φ) . χ
(A.16)
As can be seen from Eq. (A.16), if the semiconductor is weakly coupled to the substrate (χh κ), then the temperature distribution in the sample becomes: T = T0 +
g0 I0 (1 + cos Φ) , χ
(A.17)
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