Conductivity of Metallic Nano Wires

CONDUCTIVITY OF METALLIC NANO WIRES

Consider a circular cross-section wire of radius 'r' and length L. Let us assume that L is very large compared to mean free path and Fermi wavelength.

When radius 'r' is macroscopic, the conductivity for a bulk material is given by

…………(1)

In the above case, the resistance is given by

………(2)

Where,

P → Resistivity of the material

A → Area of cross section

Illustration.

Let us compute the value of 'L' for various 'r" values.

(i) Lets us consider a copper wire having

Radius r = 10 mm

Resistance R = 5.395×10^-5 ohms/m and

Conductivity σ = 5.9×10⁷ Ω^-1m^-1.

We know, 

………..(3)

Substituting the above values, we get

L=5.9×10⁷ x5.395×10^-5 x3.14x (10x10^-3)²

For resistance 5.39×10 /m,

L= 0.9994

For resistance 1 Ω = 0.9994m /5.395×10^-5 we get L=18536m

Thus, one would need approximately 18,536m of wire to amount to a resistance of one ohm, which is why we can often ignore wire resistance in electrical circuits.

(ii) If r = 10 μm, R = 53.05 ohms/m, equation (3) becomes

L= Rσπг² = 53.95×5.9×10⁷ x 3.14×100×10^-12

L = 0.9994

53.95 Ω = 0.9994m

1Ω = 0.9994/53.95 = 0.0185 m

Thus, in this case one would need 0.0185m [1.85 cm] of wire only to amount to a resistance of 1 ohm.

(iii) If r = 10 nm, the resistance is huge R = 5.395 10⁷ Ω/m

However, it is vital to note that for wires having radius values on the order of mean free path or less, the conductivity value is changed for the case of a bulk material.

Thus, the values of resistance for the 10 mm and 10 μm radius are reasonable, although the value for the 10nm radius wire is not large enough.

Explanation for various nano meter range

For example, copper has a mean free path of approximately 40nm, and in this range, radius dependant effects are usually obvious.

In fact, one may consider that radius-dependant effects may occur even for values 80-100nm.

In the 1-20nm radius range, the conductivity of the wire certainly will differ from the bulk value and normally the conductivity significally decreases as r is reduced.

This decrease is due to the scattering from the wire's surface, grain boundaries, defects etc.,

As a rule of thumb, one can use the bulk value of conductivity for many conductors when the radius value r is 80-100nm. Below this point, down to radius value of 5-10nm [above quantum wire dimensions], one may expect to need to use a size-dependent value of conductivity, based on measurement.

A relatively simple approximate formula for the resistivity [ρ]

…….(3)

Where …….(4)

ρ_o.→ bulk resistivity

w → wire width

AR → aspect ratio [wire height divided by wire width]

d → average grain size [for relatively narrow wires this can be taken as the wire width]

L_m → mean free path

P → specularity parameter [relating to reflection from the wire surface]

R_c → egrain boundary reflectivity coefficient

C → constant [1.2 for this model]

The first term is related to grain boundary scattering and the second term to wire-surface scattering.

Both the P and R_c can take values between 0 and 1.

Experimentally P = 0.3 - 0.5 and R_c = 0.2 - 0.3 will be taken.

For example P = 0.50 and R_c = 0.27 we have σ = 1.22×10⁷ Ω^-1m for a 10×10 nm² copperwire.

The equation (1) may work for cross-sectional dimensions of wires which has a order of 5-10 nm.

Factors affecting the conductivity of nano wire

Apart from surface and grain boundary scattering, other factors also determine the conductivity of a nanowire.

I-V Charactertics of a Cu nano wire

For example, the I-V characteristic of a 30nm radius, 2.4 μm long single -crystalline copper nanowire is shown in Fig. 5.26 (a)

In Fig. 5.26(a), the room temperature characteristics are shown along with an SEM image of the wire contacting two Au electrodes. The resistance is approximately 10 times the value expected when we put the value of o bulk copper in eqn (2). The difference is due to electron scattering, large contact resistance between the electrodes and the wire or surface oxidation.

Fig. 5.26(a) Room temperature I-V characteristics of a long single crystalline copper nanowire. A SEM image of the wire contacting two Au electrodes is shown above plot (b) Current-voltage characteristics of a copper nanowire for various time over a 12-hour period. The increasing oxidation of the copper resulted in the wire connecting like a p-type semiconductor.

In Fig. 5.26(b) the I-V characteristics of a 25nm radius copper nanowire are shown for various times over a 12-hour period. The increasing oxidation of the copper resulted in the material becoming more like Cu₂O and after complete oxidation, the wire acts like a P-type semiconductor.

Fig. 5.27 The Cu wire from Fig. 5.26(a) after complete oxidation. The Cu₂O wire acts factor, creating Schottky contacts with each electrode, and so the I-V curve back-to-back diode) behavior.

Fig 5.27 shows the wire in Fig. 5.26(a) after complete oxidation, at both room temperature and 4.2k.

The Cu₂O wire creates a schottky contact with each electrode, and so the I-V curve has a double-diode behavior. Because of large surface-to-volume ratio of nanostructures, these kinds of effects can be extremely harmful to performance.