Temperature-dependent infrared ellipsometry of Mo-doped VO2 thin films across the insulator to metal transition

We present a spectroscopic ellipsometry study of Mo-doped VO2 thin films deposited on silicon substrates for the mid-infrared range. The dielectric functions and conductivity were extracted from analytical fittings of Ψ and Δ ellipsometric angles showing a strong dependence on the dopant concentration and the temperature. Insulator-to-metal transition (IMT) temperature is found to decrease linearly with increasing doping level. A correction to the classical Drude model (termed Drude-Smith) has been shown to provide excellent fits to the experimental measurements of dielectric constants of doped/undoped films and the extracted parameters offer an adequate explanation for the IMT based on the carriers backscattering across the percolation transition. The smoother IMT observed in the hysteresis loops as the doping concentration is increased, is explained by charge density accumulation, which we quantify through the integral of optical conductivity. In addition, we describe the physics behind a localized Fano resonance that has not yet been demonstrated and explained in the literature for doped/undoped VO2 films.


Additional optical measurements 11
Imaginary part of pseudo-refractive index k . Figure S1 shows the imaginary part of the pseudo refractive index for different 12 temperatures across the percolation transition of VO2:Mo/Si-p++ thin films. 13 Effective optical responses are calculated from an expression which is valid only for bulk samples, where the inversion of 14 ellipsometric measures quantities really yields the dielectric function (refractive index) of the optical medium under consideration. 15 In the case of layered structures, this inversion is a convolution of optical responses of the different individual layers and their 16 corresponding thicknesses. The inversion relation is given by (1) 17 ˜ = sin 2 θ0 1 + tan 2 θ0 1 − ρ 1 + ρ , [1] 18 where θ0 is the angle of incidence, and ρ is as defined in Eq. (1) of the main part of the text. Further, the pseudo refractive 19 index n + i k = ˜ . Pseudo-dielectric function ˜ . Figure S2 shows the real (left panels) and imaginary (right panels) parts of the pseudo-dielectric 21 function for different temperatures across the percolation transition of VO2:Mo/Si-p++ thin films. Pseudo-refractive index of silicon substrate. Figure S3 shows the real and imaginary parts of the pseudo-refractive index of 23 the employed silicon substrate as a function of the frequency. The substrate owns a native silicon oxide layer of ∼ 8 nm (as 24 obtained from ellipsometric measurements). The plot displays a set of spectra from 18 to 115 • C, which are observed to slightly 25 change as a function of temperature (right insets), however, they do it with an appreciably lower shift as compared to those 26 with VO2 layers. Left insets show a close up to the Fano resonance associated to the highly doped Si substrate.

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It is observed that the integrals of n better represent the trends of integral conductivity, whereas the integrals of 1 33 always fail to the left. The reason for using real parts of both representations is that when dealing with overlayers there is 34 an apparent interchange of real and imaginary parts (2, 3) that tends to be really marked when the overlayer has strong 35 optical response as compared to the substrate (4). Actually, integrals for 2 were also made, but led to no fair comparison, as 36 expected from the aforementioned interchange: they also form hysteresis but failed too much to the right. The reason for 37 this lies in the fact, as our anonymous referees rightly pointed out the refractive index might peak at different position as the 38 dielectric function when transforming one to the other, a situation that is clearly inherited to pseudo-transforms.

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A further explanation for choosing n as a good representation of the observed physical phenomena should be given. In

Spectral deconvolution example 49
The deconvolution of a selected experimental spectrum for pseudo-refractive index and pseudo-dielectric function by means of 50 Drude-Smith, Fano and Drude-Smith+Fano models is shown in Figure S6.  where 1(ω) and 2(ω) are the real and imaginary parts of complex function˜ (ω), respectively.˜ DS (ω) and˜ F (ω) correspond 63 to the complex equations of Drude-Smith and Fano, respectively, and ∞ is the so called high-frequency permittivity limit.

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The Drude-Smith model is described by and ωp, c, and τ , are the plasmon frequency, the persistency of velocity, and the collision-modified lifetime, respectively.

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The Fano model is described by where AF , q, Γ and En are the amplitude, the phase, the width of the resonant energy and the resonant energy, respectively.

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Optical conductivity. The real part of the optical conductivity in the Drude-Smith approach for a single scattering event, is (5) 76 σ1(ω) = N e 2 τ /m * 1 + ω 2 τ 2 1 + where the second term in the RHS, weighed by the Drude denominator, is a function with the line shape of a single simple 78 oscillation with equal weights above and below the ω axis.

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Applying the f -sum rule: The first term within the brackets yields the well-known f -sum rule result ω 2 p /8 independent of measurement system, provided 82 ω 2 p = 4πN e 2 /m * (cgs) and N e 2 /(m * 0) (SI). The second term contributes 0 to the integral.  where the equivalences at the right are the expressions used in Eq. (1) of the main text, for reasons of simplicity.

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Temperature-dependent q-Fano parameter 103 Figure S7 shows the dependence of q-Fano parameter as a function of the temperature for VO2 samples with the indicated

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Mo-doping percentage. It is noted that q, also called the shape factor of the Fano resonance, tends to zero (from below) when