Resistivity minimum in granular composites and thin metallic films

We analyze the temperature dependence of conductivity in thick granular ferromagnetic compounds $\mathrm{NiSi}{\mathrm{O}}_2$ and in thin weakly coupled films of Fe, Ni, and Py in the vicinity of the metal-to-insulator transition. Development of a resistivity minimum followed by a logarithmic variation of conductivity at lower temperatures is attributed to the granular structure of compounds and thin films fabricated by conventional deposition techniques. The resistivity minimum is identified as a transition between temperature dependent intragranular metallic conductance and thermally activated intergranular tunneling.

Figure 1

TEM images of $\mathrm{N}{\mathrm{i}}_x{\left(\mathrm{Si}{\mathrm{O}}_2\right)}_{1-x}$ films grown by e-beam codeposition with Ni concentration $x=0.5$ (a), and 0.8 (b). Dark areas are crystalline Ni; light are amorphous $\mathrm{Si}{\mathrm{O}}_2$.

Figure 2

Normalized resistivity of 200-nm-thick $\mathrm{N}{\mathrm{i}}_x{\left(\mathrm{Si}{\mathrm{O}}_2\right)}_{1-x}$ films as a function of temperature for several Ni volume concentrations $x$: 0.22, 0.24, 0.26, 0.48, and 0.72. Resistivity is normalized by its room temperature value.

Figure 3

Resistivity as a function of logarithm of temperature and resistivity minima in $\mathrm{N}{\mathrm{i}}_x{\left(\mathrm{Si}{\mathrm{O}}_2\right)}_{1-x}$ samples with Ni concentration $x$: 0.3, 0.35, and 0.55. The data are normalized by the minimum resistivity.

Figure 4

Conductivity versus logarithm of temperature, left column, and logarithm of conductivity versus $T^{-n}$ with $n=1/2$ (lower axis) and $n=1/4$ (upper axis), right column, for $\mathrm{N}{\mathrm{i}}_x{\left(\mathrm{Si}{\mathrm{O}}_2\right)}_{1-x}$ samples with $x=0.55$ (a, b), $x=0.26$ (c, d), and $x=0.22$ (e, f). $g$ is the calculated normalized conductivity.

Figure 5

$T_{\min }$ as a function of ${\rho }_{\min }$ for a series of $\mathrm{N}{\mathrm{i}}_x{\left(\mathrm{Si}{\mathrm{O}}_2\right)}_{1-x}$. Solid line is a guide for the eye.

Figure 6

TEM images of 5-nm-thick (a) and 7-nm-thick (b) Fe films grown by RF sputtering on a carbon grid. Light areas are unfilled gaps in between Fe clusters. Scale ruler is 10 nm (a) and 20 nm (b).

Figure 7

Temperature dependence of resistivity of a series of Fe films between 10 and 2.5 nm thick. The data are normalized by room temperature resistivity. Inset: normalized resistivity versus logarithm of temperature for Fe films 10, 7, 6, 5, 4, and 3 nm thick. Resistivity is normalized by ${\rho }_{\min }$.

Figure 8

Change of conductance as a function of temperature in films demonstrating conductance maximum (resistance minimum): (a) Ni films 8, 7, 5, and 4 nm thick; (b) Py film as deposited and after several postdeposition annealing treatments. $\mathrm{\Delta }\sigma =\sigma -\sigma \left(4.2\mathrm{K}\right)$.

Figure 9

Resistivity of 4-nm-thick Fe film (main panel) measured as a function of temperature at zero and 16-tesla (T) magnetic field applied normal to the film plane. Inset: resistivity of Py film (after the third annealing) measured at 0, 2, 9, and 16 T fields.

Figure 10

Normalized logarithmic slope of conductance $\frac{1}{\alpha }\frac{d\sigma }{d\ln T}$ (left axis) and the screening factor ${\tilde{F}}_{\sigma }$ (right axis) as a function of sheet resistance for Fe films 2.5–10 nm thick. Solid line is a guide for the eye.

Figure 11

Normalized logarithmic temperature coefficient $C=\frac{1}{{\sigma }_0}\frac{d\sigma }{d\ln T}$ as a function of sheet resistance $R_{\mathrm{sheet}}$ (bottom axis) and the normalized conductance $g=G/G_0$, where $G=1/R_{\mathrm{sheet}}$ (upper axis), for a series of Fe (a) and Ni (b) films of various thicknesses. Symbols indicate the experimental data, and lines are calculated for $z=6$ (solid line) and $z=4$ (dotted line).

Figure 12

$T_{\min }$ as a function of resistivity (inset) and normalized conductance $g$ (main panel) for thick granular $\mathrm{NiSi}{\mathrm{O}}_2$ samples (crosses) and thin films of Ni (solid circles).