The Drude model in optics

Dans le document The nature of Fermi-liquids, an optical perspective (Page 24-0)

1.2 Optical models and functions

1.2.1 Drude model

1.2.1.1 The Drude model in optics

In optics, the dynamical equation for the momentum has the form of an harmonic oscillator dpd

dt =−pd

τeE. (1.35)

Using an harmonic field, such as Equation 1.7, its solution gives the equivalent energy dependent complex conductivity

σ(ω) =σ1(ω) +iσ2(ω) = ne2 m

1

1/τ−iω = σdc

1−iωτ. (1.36)

In particular the dc-conductivity is recovered at zero energy. The integration along the energy axis is known as the Thomas-Reich-Kuhn or f-sum rule. It states that apart from constants, the area below the real part of the conductivity is proportional to the number of electrons. The sum rule is written

Z

0

σ1(ω)dω= ne2τ m

Z

0

1

1+ω2τ2dω= ω2p

8 (1.37)

with the plasma frequency defined as

ωp=

vt4πne2

m . (1.38)

Simple metals can be fully described –at least at low energy– in terms of their plasma frequency ωpand scattering rateγ

σ(ω) = i 4π

ω2p

i/τ+ω. (1.39)

The real and imaginary parts are given by

σ1(ω) = ω2pτ

1

1+ω2τ2 and σ2(ω) = ω2pτ

ωτ

1+ω2τ2. (1.40) Optical spectroscopy is able to directly measure n/mwhereas dc-transport needs to use the Hall effect. Figure 1.3 shows the principal parameter of the Drude model.

It is also possible to define a partial sum rule where the upper boundary of the integral is set at a certain cutoff. Unlike the f-sum rule which is independent of temperature by definition, the partial integration may be temperature dependent spectral weight transfer and thus may

0

ε ωp ε

(a)

ε1 ε2

ω

σ

γ

(c)

σ1 σ2

log(ε)

γ (b)

ε1

ε1

log(ω)

log(σ)

γ

(d)

Figure 1.3: (a)and(b)Frequency dependent dielectric constant of the Drude model in linear and logarithmic scale. The dielectric constant is effectively independent of energy up to the scattering rate after which it increase and becomes positive at the plasma frequency. The imaginary part is always positive and only has a change of slopes atγ. (c)and (d)Frequency dependent optical conductivity in linear and logarithmic scale. The real part is nearly constant well below the scattering rate and correspond to the dc-conductivity. Above it falls asω2. Whereas the imaginary part has a maximum atγand then decrease asω−1. Arrangement from[4].

also vary with temperature

Neff(ω) = 2mV πe2

Z ω 0

σ1() dΩ (1.41)

By integrating over the interband energies, the number of valence electron is recovered; at higher energy, usually above 100 eV, the core electron start to be integrated and the total number of electrons is recovered at infinite energy.

For a superconductor below Tc, the superfluid condensate can be described as a zero energy and zero scatteringδ-peak which is experimentally impossible to measure. However the FGT sum rule[5]encounter the problem by using the conservation of spectral weight below and above Tc. The superconducting weight is recovered by subtracting the sum rule above and below Tc

ωp,s=8 Z

0+

σ1(ω,Tc)−σ1(ω,T) dω (1.42) For a single isolated band around the Fermi level that can be approximated by a tight binding dispersion"k=−tcos(ka)where tis the hopping parameters and athe lattice constant. Using a low energy cutoff the sum rule also correspond to the kinetic energy of the charge carrier

Z ω 0

σ1(Ω,T) dΩ=−πe2a2

h2V Ekin(T) (1.43) 1.2.1.2 Sheet conductance

For quasi-2D copper oxides such as high-Tcsuperconductors, the sheet conductance is sometimes used in place of the optical conductivity in order to compare materials with different interlayer spacingdc=c/NL(cthe out-of-plane lattice spacing andNLthe number of CuO2layers per unit cell). This quantity is given in unit of the quantum conductanceG0=2e2/h

G(ω) = πK

1/τ−iωτG0 (1.44)

The spectral weight factorK is defined in electronvolt as ħ

2p =4πe2

dc K (1.45)

The dielectric constant yields

ε(ω) =1− ω2p

ω2−iγω. (1.46)

The Drude model is a good representation of simple metals. The estimation of the densityn

is done by multiplying the density of atoms by their valence. This is also called the free electron model and usually predicts plasma frequencies in the UV spectral range. As an example, the reflectivity of three typical metals is shown in Figure 1.4. On the negative side, the match with the experimental data in the UV spectral range is poor due to the onset of interband transitions appearing already in the NIR spectral range as depicted by the sharp rise of the DOS at the same energies. The sharp decrease of reflectivity of the material is usually wrongly attributed to the plasma frequency. As an example, the drop of reflectivity appearing for gold at 2.3 eV or silver at 4 eV is below their plasma frequency and is accounted for interband transition from the 3d to the 4s bands. An exception can be made for aluminum that has a nearly flat DOS which plasma frequency match very well, except a small dip at 1.5 eV occurring through two parallel bands. Above 5 eV most of the common metals have a poor reflection except aluminum[6]. This favors the use of the latter as mirrors for experiments working in the UV. In this thesis, gold is used as a reference below 1.2 eV, by virtue of a flat and the highest reflectivity. Above this energy and up to 2 eV silver is used instead aluminum principally due to the fact that the former is easily removable from the sample surface using adhesive tape.

1.2.1.3 Hagen-Rubens regime: (ω1/τ)

In optics, the Hagen-Rubens equation are extensively used to extrapolate the metallic behavior of samples when the low energy data is lacking (usuallyħ≤15 meV). At very low energy, when the photon energy is much smaller than the scattering rate (ω 1/τ), the optical conductivity becomes

σ1(ω)≈σdc σ2(ω)≈σdcωτ (1.47)

ε1(ω)≈1−ω2pτ2 ε2(ω)≈πσdc

2ω . (1.48)

Although the imaginary part is much smaller and scales linearly down to zero, the real part of the optical conductivity is nearly constant. On the other hand, the real part of dielectric constant is large and negative, while its imaginary part diverges to large and positive values.

The reflectivity can be thus rewritten

R(ω) = 1−p ε 1+pε ≈1−

vt 2ω

πσdc. (1.49)

1.2.1.4 Relaxation regime: (ω=1/τ)

The crossing between real part and imaginary part of both optical conductivity and dielectric constant (σ1=σ2 or|ε1|=ε2) defines the onset of the relaxation regime. In fact it happens

0 5 10 15 20 Photon energy (eV)

0 0.2 0.4 0.6 0.8 1

Reflectivity

0 0.2 0.4 0.6 0.8 (c) 1 GoldSilver Aluminum

0 5 10

EfE

DOS

states eV×Cell

(a)

0.2 0.4 0.6 0.8 1 Photon energy (eV)

0.96 0.98 1 (b)

Reflectivity

Figure 1.4: (a)Density of states of the three metals. The x-axis matches with panel (c) indicating that the onset of the rise of the DOS correspond to the sharp decrease of reflectivity. The gold DOS is from[7]and for silver[8]. The aluminum DOS was obtained by LDA calculations[9]. The DOS amplitude have been approximately scaled to match LDA calculations. (b)Low energy experimental reflectivity (full line). The aluminum data is reproduced from[10–12], gold and silver[13–15]. The Drude free electron model (dashed lines) was computed from dc-transport and an estimation of the electron density. The model parameters are[13]: gold (ħp=8.99 eV,ħ=27 meV), silver (ħp=9.02 eV,ħ=19 meV) and aluminum (ħp=14.75 eV,ħ=87 meV). (c)Experimental reflectivity (straight line) and model (dashed lines).

γ

ω

Reflectivity

Figure 1.5: Hagen-Rubens fit to the low energy reflectivity. This regime works well for energies below the scattering rate, when the real part of the conductivity is nearly constant.

when ω =1/τ shown in Figure 1.3. For energies ω 1/τ but still below ωp the optical function are well described by

σ1(ω)≈ σdc

(ωτ)2 σ2(ω)≈ σdc

ωτ (1.50)

ε1(ω)≈1−ω2p

ω2 ε2(ω)≈ ω2p

ω3τ. (1.51)

At higher energy or above ω = γ, the system moves from a dissipative to an inductive regime where σ1(ω) σ2(ω). The zero-crossing of ε1(ω) in Figure 1.3 appears at ωp =

”ω2p−(1/τ)2—1/2

. In the case of good metal 1/τ ωp the crossing appears at the plasma frequency.

1.2.1.5 Transparent regime (ω > ωp)

As shown in Figure 1.3, the reflectivity drops above the plasma frequency (ω > ωp) and the material becomes transparent. Both the real part and imaginary part of the optical conductivity decreases although with different power law behavior : σ1(ω)∝ω2 andσ2(ω)∝ω1. In addition

ε1(ω)≈1−ωp ω

2

, (1.52)

in the limitωτ1 whereε1→1. Note that this single component approach is for sure not describing real material at high energy, where the interband transitions are always taking place.

1.2.2 Drude Lorentz model

The Drude formalism can be generalized to describe finite energy modes such as phonons, magnons and other collectives modes. An example of the different collective modes is shown in Figure 2.1. It is possible to start from a quantum model taking into account electronic transitions[2] but for the sake of the simplicity an equivalent classical model is explained.

The idea is to describe collective modes as a classical harmonic oscillators. Starting from Equation 1.35 with an additional restoring forces constantK gives

d2r dt2 +1

τ dr

dt +ω20r=−e

mE(t). (1.53)

ω0 = (K/m)1/2 is the mode energy. Similarly considering oscillating fields Equation 1.6 the solution leads to the dipole momentp(ω)in the energy domain

p(ω) =−erω= e2E/m

(ω20ω2)−iγω. (1.54)

0 ε1

ω+0

ω+p ε ε+ω

2p

ω20

(a)

ω ε2

0

γ

ω0 (b)

σ1

0 (c)

ω

0 2σ

(d)

Figure 1.6: Drude-Lorentz model. (a)and(b)Real and imaginary part of the dielectric constant.

The limits are shown on the graph with dashed lines. The indices+/are used to indicate that the variable is either approaching the constant from the right (+) or from the left (-). (c)and(d)Real and imaginary par of the optical conductivity.

Identifying the susceptibility and thus the dielectric constant in the total polarization P = Np〉=χeEone reads

ε(ω) =1+ ω2p

(ω20ω2)−iγω (1.55)

σ(ω) = N e2 m

ω

i(ω20ω2) +γω. (1.56)

Plotting these function in Figure 1.6 allows to easily recognize the mode energyω0, its broad-eningγ=1/τand its strengthωp. Depending on the values used in the model, the energy whereε1 crosses zero is approaching the mode energy. The indices used in the figure indicates if the value is respectively approached from lower (-) or higher energies (+). The zero energy modeω0=0 is indeed the Drude model.

1.2.2.1 Intraband and interband

In real materials, several Drude-Lorentz oscillators are usually needed for a complete description of the optical functions. When the high energy data is not accessible by experiment, it can be described through theε constant. The total dielectric function is a sum of Drude-Lorentz

oscillatorsε(ω):

ε(ω) =ε+ Xn

j=1

ω2p,j

(ω20,jω2)−iγjω. (1.57) This model is displayed in Figure 3.13. This description has the advantage of considering all the modes separately. Intraband excitations (mobile charge carriers) of the solid can be easily decoupled from interband (bound charge carrier) provided that they are well separated and a careful fitting is done.

ε(ω) =εcond+εb+ε (1.58)

The data was fitted using the Drude-Lorentz model using RefFIT software [16]. The total intraband plasma frequency including the Drude peak (ω0 =0) is given by

ω2p

nintra

X

j=0

ω2p,j. (1.59)

where nis the number of intraband oscillators.

The other advantage of having two well separated response is that the high-energy bound-charge dielectric constant can be considered as constant when looking at the low energy conduction response of the solid

εbound(ω) =ε+ Xn j=nintra

ω2p,j

ω2jω(ω+j) (1.60)

ε+ Xn j=nintra

ω2p,j

ω2j (1.61)

This assumption is displayed in Figure 3.14.

1.2.3 Extended Drude model

It is already remarkable how far the Drude model is able to summarize most of the low energy physics of simple metals with only two constant parameters (1/τ,ωp). However in the presence of strong disorder or strong interactions this model usually breaks down. The typical example is when the typical energy scale describing the strength of interaction is comparable to the kinetic energy of the electrons[2]. The highly correlated systems usually have a more complicated conductivity which can markedly deviate from the Drude model. This is the result of complex electrons or electrons-bosons correlations, which cannot be described by an energy independent scattering rate. The simplest example is a material with an optical conductivity made of a coherent and an incoherent (broader) peak; later one intraband will refer to the process involving the free carrier (coherent+incoherent) response only

The Drude formalism is extended by taking an energy dependent and complex scattering rate [17]which can be derived through specific microscopic models such as the Fermi-liquid model.

The Drude optical conductivity is extended by replacing the constant scattering rate by an optical self-energy following Ref.[17], also-called the memory function M(ω) =M1(ω) +iM2(ω). By introducing it in place of the constant scattering rate, the optical conductivity becomes

σ(ω) = i

Comparing with Equation 1.39 gives

M1(ω) =ω

By causality, the energy-dependent scattering rate is linked to the dynamical mass renormal-ization factorm/m. Different notations are used in the literature[17–19]. Equation 1.62 is rewritten

σ(ω) = i 4π

ωp2

i/τ(ω) +ω (1.64)

The optical scattering rate 1/τ is renormalized by the effective mass and is replaced by 1/τ(ω)≡ mm(ω)1/τ(ω). The optical resistivityρ(ω) =σ−11 (ω)is also widely used and a quick comparison gives1

M2(ω)≡1/τopt(ω)≡1/τ(ω) = ω2p

ρ(ω) = m

m1/τ(ω) (1.65) Even if the extended Drude model is able to describe the complex variation of the low energy scattering rate it remain a single component model which make it unable to describe interband transitions. In such a case, the high energy modes are subtracted from the optical conductivity prior to the conversion to the memory function. Hopefully for both cuprates and the single layer ruthenate intraband transitions are well separated from the interband response[20]; the extension of the Lorentz modes is assumed to be negligible at low energy and are thus replaced

1Since the resistivity is usually expressed incm it is necessary to use conversion factor to get the scattering rate expressed in eV :ħh/τ=ρp)

2

4πħh =ρp)2134.51715 withρincm andħpin eV.ħh=6.5822×1016eV s and 1cm=1.112 650 06×10−12s.

by constant dielectric functions. At least part of the lowest energy interband transitions are accessible by experiment and can be fitted by assuming that they follow the Drude-Lorentz model. Additionally, the higher energy modes that cannot be measured in the experiment have a simple constant impact on the low-energy data so that they summarized into the constant ε. Adding all the transition together leads to

εb=ε+ Xn

i=1

ω2p,i

ω20,i. (1.66)

which can later be removed from the main component.

M1(ω) = ω2p

ω

εbε1(ω)

(εbε1(ω))2+ε22(ω)−ω (1.67a) M2(ω) = ω2p

ω

ε2(ω)

(εbε1(ω))2+ε22(ω). (1.67b) 1.2.3.1 Clausius Mossotti relation

The polarizability at low energy from interband transitions can be approximately by retrieved using the Clausius-Mossotti relation[21], which supposes a simple homogeneous cubic structure

ε−1 ε+2= 4π

3 nα. (1.68)

The dielectric constant is

ε=1+ 4πnα

1−4πnα/3. (1.69)

whereαis the ionic polarizability,n=N/V the density of atoms per unit cell. In this thesis, both the oxygen of cuprates and ruthenate have the highest polarizability compared to other atoms in the structures[22]. It is therefore possible to count only for the contribution ofO2− where α=3.88×10−24cm3 [23]For HgBa2CuO4+x the cell parameters area=b=3.85 Å,c=9.5 Å and contain 4 oxygen atoms per unit cell which leads to εb = 3.57. For Sr2RuO4 a = b= 3.8730(3)Å,c =12.7323(9)Å [24] and contain also 4 oxygen atoms which givesεb =2.55.

Taking into account all the contribution gives 2.82 which indicate the importance of the oxygen cation in these material[25].

1.2.4 Kramers-Kronig relations

The Kramers-Kronig relations relates the real and imaginary part of any causal response function by the principle that the stimulus always comes before any response. From the linear response

Reω Imω

C

c1 c2

ω0

Figure 1.7: Contour integral over the real and imaginary energy axis.

theory, the differential equation representing the response to an external perturbation j(r,t)is

(r,t) = j(r,t), (1.70)

withD, the differential operator. The responseϕ(r,t)is given by the convolution of the response –a Green’s function– with the perturbation itself

ϕ(r,t) = ZZ

−∞

G(r,r0,t,t0)j(r0,t0)dr0dt0. (1.71)

The main argument is that the system is causal and the origin of time is not essential

G(tt0) =0 t<t0. (1.72) In addition the medium is homogeneous and isotropic so that the response reduces to

ϕ(t) = Z t

−∞

G(tt0)j(t0)dt0. (1.73) In Fourier space the convolution is simply the product

ϕ(ω) =G(ω)j(ω). (1.74)

Further assuming that the energy is complexω=ω1+2the Fourier transform ofG(tt0) is therefor

G(ω) = Z

G(tt0)e1(tt0)e−ω2(tt0)dt. (1.75) eω2(tt0)is bounded in the upper half of the complex plane for tt0>0 and in the lower half

fortt0<0. By virtue of Equation 1.72, the integral is thus limited in the upper half plane.

The contour C is represented by Figure 1.7 with a small diameter c2 →0 and a large circle c1→ ∞. All the poles happen on the real axis so that the Cauchy theorem over C gives

I

C

G(ω0)

ω0ω00=0. (1.76)

SinceG(ω0→ ∞)→0 the integration over the large radius gives zero contribution. The remain-ing contribution can be achieved through Dirac identities, the small circle can be parametrized usingω0=ω0c2e and in the limit wherec2 tends to zero and gives the imaginary part of the identity. The contribution from the remaining real axis gives its principle partP

P Z

−∞

G(ω0)

ω0ω00−iπG(ω0) =0. (1.77) By redefiningω0=ωthe Kramers-Kronig relations are obtained.

G1(ω) = 1

By virtue of the causality principle, this equation permits to recover either the real or imaginary part if a full knowledge of the energy dependence of the opposite part is known. As an example, the KKR are deduced in the following sections for the optical conductivity and reflectivity. The other relations for the dielectric constant and index of refraction can be found in[2].

1.2.4.1 Optical conductivity

Looking at Ohm’s law Equation 1.9d, one can see that the complex optical conductivityσ(ω)is the response function of the current density. Since the energy axis in optic is positive, one can use the fact thatσ1 is an even function (σ1(−ω) =σ1(ω)) andσ2 is odd (σ2(ω) =−σ2(ω)) to restrict the integration to the positive part of the energy axis.

σ1(ω) = 2

For the reflectivity, it is more complicated. In particular one has to study thoroughly the analytical structure of the final equation. For s-polarized light and p-polarized light with

an angle of incidence less than 45 with the surface normal one can once again formulate Kramers-Kronig relations for the phase and the logarithm of the reflectivity amplitude

r=1−pε 1+pε =p

Rer. (1.80)

The KKR can be written using ln(r) =lnp

R(ω) +iφr(ω)to recover the phase information from the reflected power (reflectivity)

φr(ω) =−2ω π P

Z

0

lnp R(ω)

ω02ω20+φr(0). (1.81) 1.2.4.2 Complexe reflectivity

The typical experimental apparatus used in this thesis covers a wide energy range spreading from 0.003 eV to 6 eV (xFIR to UV). Nonetheless the integration of Equation 1.81 spans energies from zero to infinity which mean that the missing information beyond the boundaries may strongly affect the calculated optical function within that range. Consequently, careful extrapolation are essential to guarantee the optical functions data at least at energies far inside the experimental spectral range. Many strategies exist and a comparison of the techniques is shown in Figure 1.8.

The low-energy window of metals-like material (0 meV to 8 meV) is usually extrapolated by the Hagen-Rubens Equation 1.49. A different approaches consists of a Drude fit, a double Drude or a sum of Drude and Lorentzian oscillators. Whatever the method used, the fit is then mapped onto energies down to zero and stitched to the experimental data. In this thesis, both approaches are used and and give almost similar results. For insulating materials, the optical parameters are simply extrapolated using the Drude-Lorentz model.

The traditional way of extrapolating the data at high energy is based on the assumptions of remaining interband transitions and also on the free electron model [26, 27]. Between ωHF=5 eV and 50 eV interband transitions (see Figure 2.1) continue to occur and contribute to the reflectivity, the data is usually extrapolated with a power-law function and the lack of knowledge of the interband processes is determined by adjusting the powers(usually between 0.5 and 3) in order to match the slope of the data

R(ω) =R(ωHF) ωHF

ω s

. (1.82)

Beyond 100 eV the free-electron asymptotic limit is used withs=4 in Equation 1.82.

The method used in this thesis is rather different and use of Drude-Lorentz fit for both the reflectivity data and the ellipsometric complex function. In this respect, the Kramers-Kronig relations are anchored and an extrapolation up to 150 eV allows the Kramers-Kramers-Kronig transformed reflectivity to closely match up to the maximum energy of the data (See Figure 1.8).

3 4 5 ħ

(eV) 3

4 5

ε1

3 4 5

ħ (eV)

0 1 2

ε2 No Extrapolation Wooten(s=1.6) DL+VDF

Experiment

Figure 1.8: In-plane dielectric function of Sr2RuO4 from the ellipsometric measurement compared to Kramers-Kronig transformation of the data with and without extrapolations. In the case of the Wooten extrapolation, a good match is obtained withs=1.6. The Drude-Lorentz (DL) and variable dielectric function (VDF)[28]extrapolation give an imperceptible difference with the experiment, here ellipsometry.

Another approach and an extensive comparison for different value of the parameters is shown by Tanneret al.[29]. This method uses x-ray atomic scattering functions presented by Henkelet al.[30] on the assumption that the solid is a linear combination of its atomic constituents.

1.3 Kubo formula

The previous models are phenomenological, which directly relate classical or semi-classical effects to optical functions. On the other hand, the Kubo formula, which is also known as the fluctuation-dissipation theorem, relates the linear response of an observable –such as the optical conductivity– to a time-dependent perturbation. The resistivity is computed from the dc current-current correlation function. This framework has the advantage of being a bridge from the microscopic model and the optical functions. It allows to separately define the correlation between the constituent of the matter and include them later on a self-energy term or spectral

The previous models are phenomenological, which directly relate classical or semi-classical effects to optical functions. On the other hand, the Kubo formula, which is also known as the fluctuation-dissipation theorem, relates the linear response of an observable –such as the optical conductivity– to a time-dependent perturbation. The resistivity is computed from the dc current-current correlation function. This framework has the advantage of being a bridge from the microscopic model and the optical functions. It allows to separately define the correlation between the constituent of the matter and include them later on a self-energy term or spectral

Dans le document The nature of Fermi-liquids, an optical perspective (Page 24-0)