Drude Model of metals

Fundamentals of Optical Science – OSE 5312 Fall 2003Tuesday, September 9, 2003Some comments on the applicability of the Lorentz model to real materials: (a) InsulatorsThe Lorentz model works surprisingly well, provided we remember that real materials correspond to a collection of Lorentz oscillators with different frequencies. The outer, or valence, electrons predominantly determine the characteristics of the optical properties a solid. In an ionically – bonded material, e.g. alkali-halides such as KCl, the valence electrons are quite strongly localized at the negative ion (for KCl, this would be the Cl atom), and hence the optical spectrum contains some atomic-like features, with many resonances. As the valence electrons are tightly bound, the resonance frequency is high so that these materials may have a transparency range that extends far into the uv. This can be seen in the reflectance spectrum for KCl shown below (taken from Wooten, Ch. 3.) For these types of materials, the external field and the local field can be quite different and it is not trivial to calculate the local field. For this reason, the Lorentz model does not give quantitatively accurate results for ionic materials.(ii) Doped InsulatorsDoped insulators, for example ions in glass, behave somewhat like the ions would in a gas, except that the locally strong electric fields of the host materials may distort the spectrum slightly. The figure below shows the absorption of Nd 3+ ions in a glass host material.Usually, the absorption of the dopant material is in a region of transparency of the host so that we can approximate the polarization as a superposition of polarizations due to the host and dopant material. For the case of a single resonant absorption line, we may write:E i P P P phost dopanthost tot ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧Γ--+=+=ωωωωχε22020 where χhost is assumed to be real and constant. Hence;ωωωωχωεΓ--++=i phost r 22021)(Often, we label 1 + χhost as the “high frequency dielectric constant”, ε∞ , so that:ωωωωεωεΓ--+=∞i pr 2202)(.The static dielectric constant, defined as εst = ε(ω=0) is therefore given by setting ω = 0 in the above expression, so that:202ωωεεpst +=∞.Hence, the static dielectric function of a material is affected by dopants, even though the resonant frequency for the dopant is far away from ω = 0.(iii) Semiconductors:Semiconductors are covalently bonded materials where the electrons are evenly shared between neighboring atoms. (Some insulators are covalently bonded, too.) This means that the electrons are smeared out into broader bands and that their resonance frequencies are lower than for ionically bonded materials. Usually these materials can be described by a single energy gap and single broad absorption bandabove the energy gap. The example of Silicon is shown below:Estimation of εr (ω=0) for Si:Noting that the reflectance of Si rises sharply at about 3 eV, we may take this as an estimate for ω0. Hence ω0 ≈(3 x 1.6 x10-19)/ = 4.53 x 1015 rad/s.Now, ωωωωωεΓ--+=i pr 22021)(, so that 221)0(ωωεp r +=, so that if we candetermine ωp , we can estimate ε(0). Now m Ne p 02/∈=ω, and since each Si atom has 4 valence electrons, N = 4N Si ≈ 4x 2 ∙1028 m -3. This gives an estimate of ωp ≈ 1.6 x 1016 rad/s (corresponding to about 10.5 eV) and hence ε(0) ≈ 14. This is compared to a measured value for ε(0) of 12, so the approximations are reasonable. Note that Si appears as a grayish reflector throughout the visible spectrum. (1.7 ~ 3.2 eV)(iv) Metals:- Drude theory of optical properties of metals.We can extend the Lorentz model to metals, in which case, since the electrons are unbound or "free", they experience zero restoring force and hence the resonance frequency, ω02 = K/m is also zero. This is known as the “Drude” model. The equation of motion is then:,)()()(22t E e tt r m tt r m-=∂∂Γ+∂∂which has solution;()ωωωΓ+=i E m e t r 2)()(and hence χ(ω) is given by,ωωωωχΓ+-=i p22)(where once again the plasma frequency is defined by ωp 2 = Ne 2/ε0m . Hence,222222/)(",1)('Γ+Γ=Γ+-=ωωωωχωωωχppor,222222/)(",11)('Γ+Γ=Γ+-=ωωωωεωωωεpr prNow, in a metal, the damping term Γ is just the electron collision rate, which is just the inverse of the mean electron collision time, τ, i.e. Γ = τ-1. Hence,()22222221)(",11)('τωωτωωετωτωωε+=+-=p r p rThe collision rate can be quite rapid - tens of femtoseconds. But for opticalfrequencies, (e.g. for λ = 500 nm, ω = 2πc/λ = 3.8x1015 rad/s) (ωτ)2 >> 1. Under this approximation, we find:323222)(",1)('ωωτωωωεωωωεΓ=≈-≈p pr p r .This approximation may break down in the far-infrared spectral region, where damping may be significant. Note that damping is absolutely necessary to have an imaginary part of χ(ω) or εr (ω).It is useful to look at some plots of εr (ω), n(ω), α(ω) and R(ω). These are plotted on the next page for ωp = 10 and for Γ ≈ 0 or Γ = 0.5. In the limit of no damping, the n = 0 and R =1 for 0 < ω < ωp . Above ωp , κ is zero and the reflectance drops as n rises from zero to unity. Note that even for εr ” = 0, κ and hence α is not zero. Introducing some damping causes R to be < 1 and the reflectance drop at ωp is less severe. The behavior of εr , n and κ is consistent with what we now expect for a Lorentz oscillator with ω0 = 0.nω()αω() n gω()n gω()1 -5The last plots show the real and imaginary parts of the dielectric constant on a log scale. It is interesting to note that only the real part of ε indicates notable behavior around the plasma frequency. One can not see evidence of the plasma frequency by looking at the imaginary part of ε alone, yet both n(ω) and κ(ω) clearly show evidence of the plasma frequency.Optical absorption in low electron density materials – Semiconductors: Recalling that the absorption coefficient is given by α(ω) = 2k 0κ = 2ωκ(ω)/c = ωεr ”(ω)/cn(ω). Now for very high frequencies, or for low electron densities, as may be found in doped semiconductors, ωp 2 << ω2 , n(ω) ~ 1 so that,2222)(")(p p c c cλλωωωεωωαΓ=Γ≅≅,where λp is the wavelength corresponding to the plasma frequency. The λ2dependence of α is commonly seen in semiconductors, where dopant densities are typically in the range of 1016 to 1019 cm -3 as compared to ~ 1022 cm -3 in metals. This absorption is commonly referred to as free-carrier absorption.Tin-doped Indium Oxide (ITO) a transparent conducting electrode materialITO is a semiconducting material that gives quite high electricalconductivity, yet is transparent in the visible. It is particularly useful in low-current applications, such as liquid crystal displays. This is achieved by having a material with low electron density, but those electrons should be highly mobile, which means they travel through the material with relatively few collisions. By choosing the right density of tin doping, ITO can be highly effective. Below, we show the real and imaginary part of ε for ITO from a paper by Hamberg and Granqvist, Journal of Applied Physics, Volume 60, Issue 11, 1986, Pages R123-R159. The plasma frequency, dependent of the Sn density, is typically around 0.7 eV, which corresponds to λ ~1.7μm. Due to this, and the free carrier absorption described above, ITO is not as useful in the near infrared (λ > 1 μm) as it is in the visible.From Hamberg and GranqvistA more recent study by D. Tanner et. al. at the University of Florida, shows that the properties of ITO can be well modeled by the combination of Drude and Lorentz models. Using a model for εr that sums contributions from the Drude model to describe the free-carriers, and from the Lorentz model to describe bound carriers, which have a resonant absorption at ν = 40,000 cm -1 (c //1νλν==) or λ= 250 nm. i.e.ωωωωωωωεεf pfb pbr i i Γ+-Γ--+=∞222202where ε∞ is a background high-frequency dielectric constant, and subscripts, b and f, correspond to bound and free electrons (i.e. Drude and Lorentz contrtibutions, respectively. The plasma frequencies and damping times forthe bound and free electrons are of course different. The results of theirmodel is shown below. (Courtesy, David Tanner, University of Florida.)Significance of ωp:The expressions have been written for εr rather than for χ as they more clearly reveal something significant about the plasma frequency in this form. Notice that at ω = ωp, the real part of the dielectric constant becomes zero. Hence n(ωp) = 0 , which means the phase velocity - ∞. A more rational way to describe this is that the wavelength, λ = 2πc/nω→∞ as ω→ωp. This means that all the electrons are oscillating in phase throughout the propagation length of the material.metalNote that as all the electrons are moving together, there is no charge separation (polarization) and hence no restoring force or sustained oscillation after the field is removed..Plasma oscillations:The above figure shows an entirely transverse field (compared to the surfaces of the material). Should there be a component of the field perpendicular to the surface, there can be a net surface charge as a result of the applied field.The attractive (restoring) force between the surface charges can result in a free oscillation.For no net charge, (q f = 0) then D = 0 = ε0εr E. But E ≠0, so thenεr = εr ’+ i εr ” =0. Hence, εr ’=0. Now, P = charge x displacement /volume. Thus:.&00∈=∈-=-=⋅-=xNe P E xNe AL Lx NeA P δδδThe restoring force is given by: -eE:2∈=-∴xNe eE δ,which is equal and opposite to the acceleration: 0222∈=∂∂x Ne tx m δδ, which is theequation of motion:00222=∈+∂∂x m Netx δδ.Hence the resonance frequency for the plasma oscillation is given by22∈=m Ne p ω.Modifications of Drude theory to account for properties of real metals: The Drude model implies that the only the plasma frequency should dictate the appearance of metals. This works for many metals – see the example of Zinc (Fig. 3.12 in Wooten.) – But is does not explain why copper is red, gold is yellow and silver is colorless. In fact the appearance of these metals is characterized by an edge in the reflectance spectrum, similar to that predicted by the Drude model, but the problem is that all three metals have the same number of valence electrons. Also, the calculated plasma frequency for all three should lie at about 9 eV, - well outside the visible region, so the plasma frequency cannot in itself account for the colors of Cu and Au.All three have filled d-shells. Copper has the electronic configuration[Ar ].3d 10.4s 1 , Silver [Kr ].4d 10.5s 1 and Gold [Xe ].4f 14.5d 10.6s 1. (These metals are known as the “Noble Metals”.) The d -electron bands lie below the Fermi energy of the conduction band: Transitions from the d-band to the empty states above theFermi level can be occur over a fairly narrow band of energies, aroundd F E E -=0ω which can be modeled as additional Lorentz oscillator. The combined effects of the free-electrons (Drude model) and the bound d-electrons (Lorentz model) influence the reflectance properties of the metal.i energy D -bandCHence, εr = εfree + εbound . Where εf is described by the Drude model (ω0 = 0), and εb is described by the Lorentz model. (ω0= [E F – E d ]/ .)Example: Silver:The reflectance spectrum of silver shows a string drop at about 4 eV, well below the expected plasma frequency. The reflectance also rises again for frequencies just above 4 eV. (See Wooten, Fig. 3.15, shown below.)It turns out that this behavior is because Silver has a d-band resonance at ≈ω 4 eV. This can be determined from experimental data by fitting the Drude model to the low frequency data, as shown in Wooten, Fig. 3.18.(Shown below.) Thedifference between dielectric functions from Drude model and from experiment gives the dielectric function due to the d-band resonance, εbound (written as δε(b) in Wooten.) The effect is to “pull” the εr’ = 0 frequency in from 9 eV to about 3,9 eVWooten Fig. 3.18. Realpart of dielectricconstant for SilverThis shift in ωp means that there is a shift in the free plasma oscillation in silver due to the d-electrons. This can be explained by noting that the highly polarizable d-electrons will reduce the electric field that provides the restoring force involved in these oscillations, illustrated on page 6. A reduced restoring force gives a reduced oscillation frequency. See Wooten fig. 3.20 for an illustration of how the d-electrons do this.CopperThe case of copper is almost identical to that of silver, except that the d-band resonance is at about 2 eV. Now since ε’free becomes very large and negative at low frequencies, it turns out that ε’bound due to the d-electrons is not sufficient to pull the net ε through zero. Hence ε’ becomes small at about 2 eV, but there is no true plasma frequency there. However, the effect of this is sufficient to cause R to start to drop at 2 eV, but the reduction is gradual throughout the visible. This gives copper its characteristic red-orange appearance.Reflectance of Copper. From Wooten, Fig. 3.21Fig 3.22 from Wooten. Dielectric function of Copper。