Investigation of density fluctuations in graphene using the fluctuation-dissipation relations

Computational Condensed Matter xxx (2017) 1e5

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Investigation of density fluctuations in graphene using the fluctuation-dissipation relations T.P. Mehay*, R. Warmbier, A. Quandt Materials for Energy Research Group and DST-NRF Centre of Excellence in Strong Materials, School of Physics, University of the Witwatersrand, Private Bag 3, 2050 Johannesburg, South Africa

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 July 2017 Received in revised form 30 August 2017 Accepted 30 August 2017 Available online xxx

Based on the philosophy of density functional theory, the charge-density contains, in principle, all the information about a given system. Thus charge-density fluctuations determine all other fluctuation phenomena, and so are of great importance in the study of nanosystems, where quantum fluctuations strongly influence system behaviour. Despite this, little work has been done to investigate the effects of charge-density fluctuations in such systems. We present a previously uninvestigated relation connecting charge-density fluctuations and the electron energy-loss spectrum using the fluctuation-dissipation relations. Using this relation we perform an analysis of the autocorrelation function of charge-density fluctuations in graphene using linear-response time-dependant density functional theory calculations. It is found that the charge-density autocorrelation function contains information pertaining to both plasmonic resonance of a system, and noise characteristics. As the electron energy-loss spectrum can be studied both numerically and experimentally, this relation provides a useful analysis tool for the study of charge-density fluctuations. © 2017 Published by Elsevier B.V.

Keywords: Fluctuations and noise Dielectric properties Graphene

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction In the analysis of measured signals, whether obtained experimentally or numerically, the presence of noise is unavoidable. This noise is caused by a large variety of physical processes [8] as well as intrinsic limitations in the accuracy of computation [18]. Due to the inherent difficulty in removing an unknown type of noise from a signal, as well as the lack of a comprehensive theory of noise phenomena, the noise in numerical computations is often assumed

* Corresponding author. E-mail address: [email protected] (T.P. Mehay).

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to be uncorrelated white noise [18]. However, experiment has shown that nano-devices, in particular those based on graphene, exhibit a multitude of noise types, in particular 1∕f noise [1]. These ”coloured” noise types have nonzero correlations, which raises questions about the nature of correlated noise in the data produced by numerical simulations. The fluctuation-dissipation theorem [2,3,9,15] relates fluctuations to dissipative forces, and enables the study of fluctuation properties of systems, including noise. In this paper we present a previously uninvestigated method to study charge-density fluctuations. This is achieved by deriving a formal relationship between the electron energy-loss spectrum, and the charge-density autocorrelation function given by the fluctuationdissipation theorem. Based on the philosophy of density-

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Please cite this article in press as: T.P. Mehay, et al., Investigation of density fluctuations in graphene using the fluctuation-dissipation relations, Computational Condensed Matter (2017), http://dx.doi.org/10.1016/j.cocom.2017.08.008

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T.P. Mehay et al. / Computational Condensed Matter xxx (2017) 1e5

functional theory, such charge-density fluctuations determine virtually all other fluctuation processes of a given material. Further, as the properties of nanosystems are highly influenced by quantum fluctuations, an analysis of the effects of these fluctuations is important. Therefore the present approach is very far-reaching, and it may ultimately lead to a better understanding of the effects of noise in nanosystems.

symmetry of S(u, q)

Sðu; qÞ ¼ ebZu Sðu; qÞ;

which, when combined with Eq. (5) allows us to rewrite Eq. (6) in terms of the ACF of rq, y

〈rq ðtÞrq ð0Þ〉 ¼ 2. Theory Let us first point out some of the essential concepts from timeseries analysis. Consider the time series fXðtÞ : t2Tg. Assuming covariance stationarity, we define the autocorrelation function (ACF) by

〈XðtÞXðt þ tÞ〉 ¼

1

s2

E½ðXðtÞ  mÞðXðt þ tÞ  mÞ;

(1)

where m is the mean, s2 is the variance, and EðXðtÞÞ denotes the expectation value of the variable X(t). The ACF thus quantifies the self - similarity of a time series as a function of time. The Wiener Khinchin theorem [19] states that the spectral density and the ACF form a Fourier transform pair, and therefore, we may define the spectral density associated with the above time series by

1 SðuÞ ¼ 2p

Z∞

iut

dt e

〈XðtÞXðt þ tÞ〉;

(2)

∞

with the corresponding transform for the ACF given by

Z∞ 〈XðtÞXðt þ tÞ〉 ¼

du eiut SðuÞ:

(3)

∞

We now turn our attention to the special case of density fluctuations, which are related to the basic dielectric properties of a given material like graphene. Using many - body techniques, it can be shown that the imaginary component of the inverse dielectric function is given by [7,9,14]

n o 4pe2 X     2 I ε1 ðu; qÞ ¼ j n rq 0 j  ½dðu þ un0 Þ  dðu  un0 Þ; q2 n (4)

(7)

Z

du eiut

  q2 I ε1 ðu; qÞ 2e2 ebZu  1

(8)

Comparing Eq. (8) with Eq. (3) it is clear that we can interpret the integrand of Eq. (8) as the spectral density of the charge density fluctuations, which shows how knowledge of the EELS enables the investigation of charge - density fluctuations. Note that the relation given in Eq. (8) is one of many related expressions, describing the relationship between dissipative forces and the associated fluctuation processes. Collectively they are known as the fluctuation - dissipation relations. While a similar form of the fluctuation - dissipation relations has been derived using other methods [9], it is our understanding that an explicit study of charge - density fluctuations using the EELS has never been done. 3. Results and discussion In order to illustrate the behaviour of the ACF of charge - density fluctuations, EELS data for ideal graphene was used [17], which was determined using linear response time - dependant density functional theory [4,5,12]. Graphene was chosen as its dielectric properties are well understood [11,13,21], and since graphene - based electronic devices exhibit low frequency noise [1]. Fig. 1 shows the EELS of ideal graphene for different values of q, and for energies below 8eV, with the associated ACF shown in Fig. 2. It should be noted that the noise shown in Fig. 1 may be due to a combination of physical processes and numerical limitations related to k - point sampling. The initial decay of the ACF exhibits damped sinusoidal behaviour with a period of roughly 1 fs, which corresponds to a resonance at roughly 4eV. This indicates that density fluctuations are highly oscillatory, with the rapid decay indicating that charge density fluctuations are a stationary process. This behaviour is also exhibited by autoregressive (AR) models [20], suggesting that charge - density fluctuations may be modeled using such statistical methods. Fig. 2 further reveals that at long time lags, random fluctuations in the charge - density ACF become apparent, which

P where q is the momentum transfer, rq ¼ j eiq,xj denotes the charge - density fluctuations, and d is the Dirac delta function. The electron energy - loss spectrum (EELS) is defined simply as Ifε1 ðu; qÞg. We define the dynamic structure factor S(u, q) by

Sðu; qÞ

¼

P   y   2 j n rq 0 j dðu  un0 Þ n

1 ¼ 2p

Z

dt eiut 〈rq ðtÞrq ð0Þ〉; y

(5)

where the final expression follows from the properties of the delta function and the definition of rq. Comparison of Eq. (5) with Eq. (2) shows that the structure factor can be interpreted as the spectral density of the charge - density fluctuations. Using Eq. (5) we may rewrite Eq. (4) as

n o 4pe2 I ε1 ðu; qÞ ¼ 2 ½Sðu; qÞ  Sðu; qÞ: q

(6)

The principle of detailed balance, central to the fluctuation dissipation theorem, is summarized by the time translation

Fig. 1. EELS of ideal graphene computed using the linear response time dependant density functional theory package of the GPAW simulation software [4,5,12].

Please cite this article in press as: T.P. Mehay, et al., Investigation of density fluctuations in graphene using the fluctuation-dissipation relations, Computational Condensed Matter (2017), http://dx.doi.org/10.1016/j.cocom.2017.08.008

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Fig. 2. ACF of charge - density fluctuations for ideal graphene. The inset shows the long - time behaviour of the ACF at time lags 40 - 80fs.

Fig. 4. ACF of charge - density fluctuations for Lorentz oscillator model. The inset shows the long - time behaviour of the ACF at time lags 40 - 80fs.

may be connected to the fluctuations observed in the EELS. In order to further investigate these random fluctuations, we make use of a Lorentz oscillator model. In this model, the atom is approximated by a damped, driven harmonic oscillator, with the imaginary component of the dielectric function given by Ref. [10].

in Fig. 4 shows a drastic departure compared to the ACF produced for graphene. We see perfect sinusoidal behaviour enveloped by an exponential decay, as is expected given the nature of the Lorentz model. This suggests that the long - time behaviour of the charge density ACF is strongly connected to the noise present in the EELS, or equivalently, the charge - density fluctuations. For the sake of a comparison, we introduce artificial noise into the Lorentz model, computed using the AR model [6,16].

IfεðuÞg ¼ 

u2p gu

u20  u2

2

þ u2 g2

:

(9)

Fig. 3 shows a fit of the Lorentz oscillator model to the EELS for graphene, with Fig. 4 showing the corresponding ACF. A comparison of the ACF produced for graphene with that of the Lorentz model shows that although the first resonant peak in the EELS of Fig. 3 is shifted forward in energy, the initial decay of the ACF is virtually identical in both Fig. 2 and Fig. 4. This indicates that the initial decay of the ACF is largely dependant on the second resonant peak, which remains roughly the same for both EELS. This peak is associated with plasmonic resonance in graphene [17], indicating that the initial decay of the ACF of charge - density fluctuations may give information relating to plasmonic resonance. In particular, the 1fs period of the ACF of graphene is directly related to the 4eV plasmon in graphene. The long - time behaviour of the Lorentz ACF

Fig. 3. EELS computed using Lorentz oscillator model.

N X

fk Xðt  kÞ ¼ wðtÞ;

(10)

k¼0

where w(t) is a zero - mean white noise process, X(t) is the noise process time - series, which has the physical interpretation of charge - density fluctuations rq, and the coefficients fk are given by

fk ¼

1 fk1 þk1 ; 2 k

(11)

where the initial term is defined by f0 ¼ 0. Fig. 5 shows the EELS produced from a combination of the Lorentz model with 1∕f noise, with Fig. 6 showing the related ACF, while Fig. 7 shows the EELS of the Lorentz model combined with

Fig. 5. EELS of Lorentz oscillator model with added 1∕f noise.

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T.P. Mehay et al. / Computational Condensed Matter xxx (2017) 1e5

virtually identical to that of graphene, with the noise again appearing in the long-time behaviour. This strongly suggests that the noise content of the charge-density fluctuations can be analysed by studying the long-time behaviour of the ACF. Comparing Fig. 2 with Fig. 8, it is clear that the noise in the charge-density ACF of graphene is clearly non-white, suggesting it may be a combination of ”coloured” noise processes. It is unclear whether this noise arises from purely physical processes or also from numerical limitations. However, as the EELS may be measured experimentally, a more detailed, and also more powerful analysis of this noise may be done by combining experimental, numerical, and theoretical studies of the dielectric function. Also, a deeper understanding of the noise characteristics contained in the charge-density fluctuations, may be gained from sophisticated denoising techniques and statistical AR models may be used. However, due to the intricacies involved in these methods, more detailed analyses are left for future work. Fig. 6. ACF of charge - density fluctuations for Lorentz oscillator model with added 1∕f noise. The inset shows the long - time behaviour of the ACF at time lags 40 - 80fs.

4. Conclusion In conclusion, we presented a method based on the fluctuationdissipation theorem, to study charge-density fluctuations via the ACF. We have analysed the numerical ACF for ideal graphene, as well as for the Lorentz model with and without random noise. A comparison of the data produced indicates that the initial decay of the charge-density ACF is connected with a strong plasmonic resonance, while the long-time behaviour contains information pertaining to the noise contained in the charge-density fluctuations. The current approach can easily be extended to study other types of nanomaterials, and to analyze experimental EELS data, making it a useful method for the study of charge-density fluctuations and noise in nanomaterials. Acknowledgements The authors wish to acknowledge the support of the Materials for Energy Research Group, the DST-NRF Centre of Excellence in Strong Materials and the University of the Witwatersrand.

Fig. 7. EELS of Lorentz oscillator model with added white noise.

References band-limited white noise, and Fig. 8 showing the corresponding ACF. From Fig. 6 and Fig. 8, we again see an initial decay of the ACF

Fig. 8. ACF of charge - density fluctuations for Lorentz oscillator model with added white noise. The inset shows the long - time behaviour of the ACF at time lags 40 - 80fs.

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