Linear THz conductivity of nanocrystals

T. Ostatnický

doi:10.1364/OE.27.006083

1. Introduction

Electron confinement in nanostructures, in particular semiconductor nanocrystals, manifests itself by characteristic features in the terahertz conductivity spectra [1–7], namely slowly increasing real part and linearly decreasing imaginary part at the low frequency. Despite the expectation of the same behavior due to the localized plasmon resonance [8,9], the decay of conductivity at the low frequency is the characteristic property of the microscopic conductivity itself, without the effect of the depolarizing fields.

The characteristic shape of the spectra was initially interpreted in terms of the electron back-scattering within the Drude-Smith (DS) model [10]. The drawback of the DS model is, however, that there is an unclear microscopic physical interpretation of its parameters. Subsequent development of the Monte-Carlo (MC) methods [4] confirmed the origin of the low-frequency conductivity depletion due to the electron confinement. Since then, the MC methods have been accepted as the most reliable classical model of the electron linear response and it was shown [4] that the DS model significantly overestimates the scattering time in the most interesting region of the nanocrystal sizes in the interval 1–100 times the mean free path of electrons. It has been later argued independently by Cocker et al. [11] and by us [12] that the low-frequency cut of the conductivity is caused by the diffusion restoring current.

While the both two aforementioned models [11,12] resolve the dominant role of the diffusion at low frequencies, they differ in the level of description of the physical system. Cocker et al. consider the classical electron gas and, on the contrary to the DS model, take explicitly into account the microscopic characteristics of the system which enter the formula for the conductivity:

σ (ω) = \frac{n e^{2} τ^{'} / m}{1 - i ω τ^{'}} (1 - \frac{1}{1 - i ω / a}),

where n, m and e are the electron density, effective mass and charge, respectively, τ′ is the effective scattering time and a is the diffusion parameter. The formula (1) separates the Drude peak with the renormalized scattering time due to the electron reflection on the crystal surface (first term) and the diffusive counter-current (second term in parentheses).

Our approach is, on the other hand, fully quantum-mechanical (QM) and has therefore a larger variability in setting the microscopic parameters of the system. We have also shown [12] that the QM model gives the same predictions as the MC simulation in the classical limit (large crystals, room temperature) and consequently, we may regard it as a better theoretical standard for the linear conductivity spectra, applicable beyond the classical limit of the MC simulations. The clear disadvantage of our QM formula for the electron conductivity is, nevertheless, that it requires a long evaluation time to sum up all the contributing terms. The QM model is not therefore convenient for everyday use to fit experimental curves as well as the MC simulations due to the demands on the computation time despite the high accuracy.

In this paper, we aim at the simplification of the QM model to reach a convenient form of an expression for the electron conductivity in the classical limit which can be evaluated directly. To this end, we derive the semi-classical (SC) model in Section 2, we state which parameters limit its precision and we show that it almost coincides with the QM model within the given range of validity. We show in Section 3 that under an additional restriction on the system parameters, the SC model may be approximated by the formula of the modified DS model. Finally, we numerically evaluate the accuracy of the SC and the modified DS model in Section 4 by calculating the correlation of their spectra with the spectra of the QM model. We conclude that the SC model gives significantly better predictions than the modified DS model.

2. Derivation of the SC model

We start our derivation with the formula (11) of [12], which stands for the conductivity of an electron confined in a three-dimensional rectangular potential well (the QM model):

σ (ω) = \frac{e^{2} n}{ℏ N} \sum_{k, l} \frac{f_{k} - f_{l}}{ω - ω_{k l} + i γ} [\frac{2 D γ L x_{k l}}{D π^{2} {(k_{x} - l_{x})}^{2} - i ω L} - \frac{2 D γ L x_{k l}}{D π^{2} {(k_{x} + l_{x})}^{2} - i ω L} - i ω_{k l} x_{k l}^{2}],

where we sum over the multi-indices k = (k_x, k_y, k_z) and l = (l_x, l_y, l_z) which are positive integers and denote the quantum state of the electron in the directions x, y, z. Energy of the particular state is ħω_k and we denote ħω_kl = ħ(ω_k − ω_l). We consider a non-degenerate electron gas with the Boltzmann distribution function f_k = exp[−ħω_k/k_BT] where k_B is the Boltzmann constant and T is the thermodynamical temperature. The symbol L stands for the crystal size, D is the diffusion constant, γ = 1/τ is the scattering rate (τ is the scattering time), x_kl is the dipole matrix element and N stands for the number of electrons in the nanocrystal:

N = \sum_{k} f_{k}

.

With the particular choice of the Boltzmann distribution function, it can be shown that the sums over indices k_y, k_z, l_y, l_z cancel out with the appropriate sums in the electron number N in the denominator and the formula reduces to a double-sum over the indices k_x and l_x. It is then favorable to rearrange the double sum so that we define a new index p = l_x − k_x and we rename k = k_x. At this point, we start calculating the classical limit where the energy spacing of the electron levels is small, compared to the dephasing and the thermal energy and therefore the density of states may be considered as quasi-continuous as expressed by the criteria: 𝒞₁ = γ/ω_k,k+p ≫ 1 and 𝒞₂ = k_BT/ħω_k,k+p ≫ 1. Under these assumptions, the maximum contribution which stems from the distribution function f_k − f_k+p arises for large values of the index k ≫ 1 and therefore we may neglect all contributions to the Eq. (2) which contain the term k_x + l_x = 2k + p in the denominator. Equation (2) then may be rewritten:

σ (ω) = \frac{e^{2} n}{ℏ \sum_{k = 0}^{\infty} f_{k}} \sum_{p = 1, 3, \dots} \sum_{k = - \infty}^{+ \infty} \frac{f_{k} - f_{k + p}}{ω - ω_{k, k + p} + i γ} [\frac{2 D γ L x_{p}}{D π^{2} p^{2} - i ω L} - i ω_{k, k + p} x_{p}^{2}],

where the dipole matrix element x_p = (−2L/π²p²). The limits of the double sum were rearranged in order to sum only over positive and odd index p. This rearrangement leads to an unintentional addition of terms which are not present in the sums in Eq. (2), for example the terms with k = 0. The effect of these terms is not supposed to be significant in the semi-classical limit. We perform, however, a partial correction by adding the k = 0 state into the sum in the denominator. Within the approximations outlined above, we may also use the approximate expressions f_k − f_k+p ≈ pk f_kħ²π²/mL²k_BT and ω_k,k+p ≈ −2pkħπ²/mL². Since the significant contributions span over a large interval of values of k and the summand is a smooth function of k, we may substitute the sum by the integral:

\sum_{k} \to \int d k

which is then evaluated analytically, giving the final formula of the SC model:

σ (ω) = - \frac{8 \sqrt{2} i e^{2} L n}{π^{3} m v_{th}} \sum_{p} \frac{ξ_{p}}{p^{3}} [1 - \frac{1}{1 - i ω / γ} \frac{1}{1 - i ω / p^{2} Γ_{D}}] [1 + i \sqrt{π} ξ_{p} erfcx (- i ξ_{p})] .

Here we defined the thermal velocity

v_{th} = \sqrt{k_{B} T / m}

, the diffusion rate Γ_D = Dπ²/L² and the dimensionless parameter ξ_p = (ω + iγ)t_th/p is defined with the help of a characteristic time constant:

t_{th} = L / π \sqrt{2} v_{th}

. The complex error function is defined in terms of the standard error function as erfcx z = exp[z²](1 − erf z). The magnitude of the summands rapidly decreases with the increasing index p and therefore it is sufficient in the most cases that we consider only the terms with p = 1, 3, 5. The accuracy of the SC model is shown in Figs. 1(a)–1(b) where we compare the QM model with the results of Eq. (4) and we also show the partial contributions for several values of the index p. The particular crystal sizes in the plots in Figs. 1(a)–1(b) are chosen to give the same geometrical factor mL² which appears in the constants 𝒞_j while keeping 𝒞_j ≫ 1. (We plot the mobility in all graphs to remove the dependency on the electron density.)

Fig. 1 (a), (b) Calculated mobility spectra for typical values of (a) GaAs and (b) Si at T = 300 K and mL²/m₀ = 7 × 10⁻⁴ μm² where m₀ is the free electron mass. Dephasing rates 1/γ = 270 fs for GaAs and 30 fs for Si. (c) Mobility spectra at 300 K by Eqs. (2) and (4) with mL²/m₀ = 7 × 10⁻⁴ μm², 1/γ = 1.16 ps (𝒞₁ = 1.0, bottom curves) and mL²/m₀ = 5.93 × 10⁻⁵ μm², 1/γ = 30 fs (𝒞₂ = 0.4, topmost curves).

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3. Derivation of the modified DS model

We show in this section that the modified DS model [11] is an approximation of our SC model Eq. (4). We approximate the error function by considering the first terms of its series expansion:

1 - erf z \approx \frac{exp [- z^{2}]}{\sqrt{π} z} (1 - \frac{1}{2} \frac{1}{z^{2} + 1}) .

The last square brackets of the Eq. (4) can be then approximated by the function

1 / 2 (1 - ξ_{p}^{2})

. We plot in Fig. 2(a) the comparison of the term

[1 + i \sqrt{π} ξ erfcx (- i ξ)]

and its approximate form for different values of the imaginary part of ξ. Clearly the curves coincide if Im ξ ≫ 1 and we may therefore approximate the error function if 𝒞₃ = γL/v_th ≫ 1: thermal electrons undergo at least one scattering between two subsequent reflections on the crystal boundaries. Within this limit, |ξ_p| ≫ 1 and we may therefore approximate

ξ_{p} / (1 - ξ_{p}^{2}) \approx - ξ_{p}^{- 1}

. The formula (4) then reads:

σ (ω) = \frac{8 e^{2} n}{π^{2} m γ^{'}} \frac{1}{1 - i ω / γ^{'}} \sum_{p} \frac{1}{p^{2}} [1 - \frac{1}{1 - i ω / γ} \frac{1}{1 - i ω / p^{2} Γ_{D}}] .

Here we intentionally write the renormalized damping coefficient γ′ instead of the bare damping γ to correct our approximations for the high-frequency region. The correction is calculated at the end of this section. Our assumption 𝒞₃ ≫ 1 implies Γ_D ≫ γ and therefore the first fraction in the square brackets can be omitted. Then we analytically sum the series

\sum_{p} (1 / p^{2}) = π^{2} / 8

and we use the identity

\sum_{p} {(p^{2} z - 1)}^{- 1} = (π / 4 \sqrt{z}) tan (π / \sqrt{4 z})

:

\sum_{p} \frac{1}{p^{2}} \frac{1}{1 - i ω / p^{2} Γ_{D}} \approx \frac{π^{2}}{8} (1 + \frac{i π^{2} ω}{12 Γ_{D}}) \approx \frac{π^{2}}{8} {(1 - \frac{i π^{2} ω}{12 Γ_{D}})}^{- 1} .

We denote Γ′_D = 12Γ_D/π² = 12D/L² and write finally:

σ (ω) = \frac{e^{2} n}{m γ} \frac{1}{1 - i ω / γ^{'}} [1 - \frac{1}{1 - i ω / Γ_{D}^{'}}] .

Following the derivation of Eq. (6), the effective damping γ′ should be equal to the bare damping γ. The classical derivation of Cocker et al. [11] reveals, however, a different coefficient which may possibly correct some omissions which we have made in our derivation. We observe that, putting D = 0, the real part of the Eq. (8) is a Lorentzian curve with the width γ′ and we compare it directly with the prediction of Eq. (2) where we a priori assume t_th → ∞ and ω/γ → 0 in one case and ω/γ → ∞ in the other. From Eq. (8), we see that [Re σ(∞)/Re σ(0)]_D=0 = γ′²/ω² thus we get the effective damping by multiplying the result by ω². Direct evaluation of the real part of Eq. (3) then gives:

γ^{' 2} = ω^{2} {[\frac{Re σ (\infty)}{Re σ (0)}]}_{D = 0} = {[2 t_{th}^{2} (1 - \sqrt{π} γ t_{th} erfcx (γ t_{th}))]}^{- 1 / 2} \approx γ \sqrt{1 + {(γ t_{th})}^{- 2}} .

Fig. 2 (a) Approximation of the error function. (b) Calculated mobility for GaAs D = 0 T = 300 K, mL²/m₀ = 7 × 10⁻⁴ μm², 1/γ = 270 fs. (c), (d) Comparison of mobility spectra for Si and GaAs calculated by various models with the parameters of Figs. 1(a)–1(b). (e), (f) Comparison of the effective dephasing rates calculated from Eq. (9) and according to Cocker et al. [11].

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4. Discussion of the results

We have shown in the Section 2 that the QM model Eq. (2) can be simplified to a more convenient SC model Eq. (4) assuming mathematically 𝒞₁, 𝒞₂ ≫ 1. The physical interpretation is that the energy levels are sufficiently broadened by dephasing so that they form a quasi-continuum in the density of states (C₁ ≫ 1) and simultaneously the distribution function is smooth enough to be regarded as a classical Botzmann’s distribution function (C₂ ≫ 1). The role of the assumptions is illustrated in Fig. 1(c) where we compare the real parts of Eqs. (2) and (4) when one of the assumptions is not obeyed. Too small ħγ leads to discretization of the particular quantum resonances in the spectrum but still the envelope is well reproduced by Eq. (4). When k_BT is too small, on the other hand, we observe a sharp onset of the conductivity at a low frequency which cannot be reproduced semi-classically.

Further simplification of Eq. (4) under the assumption 𝒞₃ ≫ 1 leads to the Eq. (8) which is the same as the modified DS model of [11]. Our SC model has therefore a wider range of applicability (the criterion 𝒞₃ ≫ 1 does not apply) and has a higher accuracy which is illustrated in Figs. 2(c)–2(d). We compare mobilities of silicon and GaAs nanocrystals with the same value of the scaling parameter mL²/m₀ = 7 × 10⁻⁴μm². The number 𝒞₃ = 13 for Si while it is 1.44 for GaAs. Clearly the modified DS model fails in the case of GaAs due to the small dephasing (large electron scattering time) while the SC model Eq. (4) still gives an accurate prediction despite the small ratio γL/v_th. To uncover the difference between the predictions of Eqs. (4) and (8), we plot in Fig. 2(b) the predictions of Eqs. (2), (4) and (8) with artificially set D = 0: while the QM model and the SC model predict the shift of the conductivity peak to a non-zero frequency due to the resonances in the electron motion, the Lorentzian Drude-like conductivity of the modified DS model remains peaked at ω = 0. The modified DS models therefore cannot reproduce the behavior of some systems even though they fit well the limit 𝒞₁, 𝒞₂ ≫ 1.

Our formula (8) for the modified DS model is exactly the same as Eq. (44) of [11]. Also both the models are limited to the case 𝒞₃ ≫ 1 but the difference is in the definition of the renormalized damping coefficient γ′ and the renormalized diffusion rate Γ′_D. Provided that 𝒞₃ ≫ 1, the definitions practically coincide what is a valuable conclusion since both expressions have been derived in very different ways: each of the routes has its mathematical or physical justification for doing particular, in general dissimilar approximations. The resulting good coincidence of the final parameters which rise from the two very different approaches therefore demonstrates the robustness of the calculations. The differences between the effective damping coefficients are illustrated in Fig. 2(e). The condition γL ≫ v_th means that the dephasing must be on the right hand side from the dashed vertical line and therefore above the intersection point of the curves which appears for γ ≈ 4v_th/L. The curves of our model Eq. (9) deviate up to 10% from that of [11] in this range as seen in Fig. 2(f). We may then conclude that the definitions of the coefficients of our modified DS model and that of [11] coincide.

In Fig. 3, we compare the three models discussed above directly by plotting their deviations from the 100% correlation with the QM model, following the dependencies on mL², 1/γ and T. The vertical lines and arrows denote the area in which apply the respective conditions 𝒞_j > 1. We see that when 𝒞_j ≫ 1, modified DS models give at least 90% correlation with the QM model while our SC model fits with better than 99% fidelity. When comparing the two modified DS models, that of [11] gives better predictions for large crystals and long scattering times while our model Eq. (8) seems to work better for small crystal sizes and short scattering times. The differences are not, however, as large as the difference from Eq. (4) which then turns out to be more accurate. The only exception appears for extremely short scattering times as seen in Fig. 3(b) but the correlation is still above 99% what indicates a good fit.

Fig. 3 Correlation of the models in Eqs. (4), (8) and the modified DS model of [11] with the full model Eq. (2). We plot the dependency on (a) parameter mL², (b) scattering time, and (c) temperature. The vertical lines and the arrows show the ranges 𝒞_j > 1. Parameters are T = 300 K, 1/γ = 25 fs, mL²/m₀ = 10⁻⁴ μm².

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5. Conclusions

We derived in this paper an analytic formula (4) (SC model) for the electron conductivity of semiconductor nanocrystals in the semi-classical limit. Its correlation with the QM model of [12] is shown to be better than 99% within the full range of its validity. We also unify our SC model [12] with the modified Drude-Smith model of Cocker et al. by showing that the Eq. (4) can be simplified into the Eq. (8) which is equal to the result of the modified DS model. Although we end up with a slightly different definitions of the two parameters of the modified DS model as compared to [11], we show that the differences are not large and the microscopic interpretation of the terms is the same. Our analysis reveals, however, that the system must fulfill additional conditions, namely 𝒞₁, 𝒞₂ ≫ 1, which are not discussed by Cocker et al. and which limit the applicability of the modified DS model. Numerical analysis shows that the accuracy of our SC model Eq. (4) notably exceeds both variants of the modified DS model (in some cases by orders of magnitude). We therefore suggest that the SC model is a better alternative for fitting experimental data than the Drude-Smith [10] or the modified Drude-Smith [11] model.

Funding

Czech Science Foundation (GAČR) (17-03662S).

References

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Linear THz conductivity of nanocrystals

Abstract

1. Introduction

2. Derivation of the SC model

3. Derivation of the modified DS model

4. Discussion of the results

5. Conclusions

Funding

References

Cited By

Figures (3)

Equations (9)

Optics Express