6 Mesoscopic Physics

Prof. Subhasis Ghosh

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2.3.1 Mesoscopic  Transport in 1D

2.3.2    Mesoscopic Thermal Transport

 

2.3  MESOSCOPIC PHYSICS

 

The mesoscopic physics that deals with the intermediate region between macroscopic and either microscopic or nanoscopic world, is mostly applicable to the objects between nm to µm length scales. In case of transport of charge carriers, it deals where dimension of the object is more than mean free path but less than phase coherence length.

 

Macroscopic Conductor:  ≤           ≤             ≪

Mesoscopic conductor:   ≪         ≪          ≤

 

Where, , , are phase coherence length, mean free path between elastic scattering length and distance travelled by electron before losing energy kT. Let us consider the case of current carrying conductor. According to Ohm’s law, the resistance R is related to the cross-sectional area A and the length L by the relation

 

Fig. 2.9 Schematics of current conduction through conductor in different dimensions. In 3D, the length of the conductor is more than the inelastic man free path (lφ), hence the conduction is diffusive. In 2D and 1D, the length of the conductor is less than lφ hence the conduction is ballistic.

 

The reciprocal of the resistance is the conductance and given by

G= (2.41)

 

Fig. 2.9 shows that if we reduce the length, L of the channel by half, the resistance should go down by half. But what if we reduce the length to very small dimensions let us say zero? Will the resistance tend to zero and become superconductor? Conventionally when an electron passes through a solid, it follows a random path travelling in one direction for short interval of time and then scattered in other direction. This is called diffusive transport. What will happen if the inelastic mean free path of electron is more than the dimension of wire, the electron will pass through the wire without suffering any scattering just like a bullet. This is called ballistic transport. Will the Ohm’s law still follow? Or will there be resistance in case of few atoms thick nanowires. On the other hand, it has been shown experimentally the minimum conductance is finite, G = Gc (=80μS) and not infinity as L tends to 0 [Ref.10].

 

2.3.1 Mesoscopic Transport in 1D

 

Whenever, the dimension of the systems is less than inelastic mean free path lφ the current conduction mechanism in a conductor has to be treated quantum mechanically. In a quantum conductor, the resistance through a wire connected between two electrodes can be given by Landauer formula

 

=h/22 (2.42)

Where T is transmission probability through a scatter in conductor. Eq. 2.42 can be rewritten as

=h/22+h/221-

 

The first part which does not contain T is called contact resistance however the second part is the actual contribution from the conductor.

Fig. 2.10 Schematics of current carrying conductor as a barrier with transmission probability, T. The total resistance is sum of contact and actual resistance of wire.

 

From eq. 2.43, it is clear that even in the case of 100% transmittance (no barrier, T=1) there is always a finite resistance known as contact resistance which leads to the finite conductance in case of nanoscale.

 

Let us solve the Schrodinger equation for a conductor in 2D (Fig.2.11). The time independent Schrodinger equation of current carrying wire is given as

The energy eigenvalues of Eq. 2.44 are  given by

In this case, energy is quantized along z-direction as conductor is reduced from 3D to 2D.

Fig. 2.11 Schematics of Current carrying conductor in 2D (a) without gate bias (b) with gate bias.

 

In presence of gate bias, Eq. 2.44 will take following  form

WhereV ( y) 1/2 m2 y2 (gate bias can be modeled as simple harmonic potential along y-direction). The eigenvalues of the Eq. 2.46 are given by

In this case, energy is quantized along z- and y-directions. The energy is quantized along y-direction due to gate bias. The second term is simply the contribution of free electron along x-direction,

The third term in Eq.2.48 decides the no. of conduction channels  along x-direction. Let us consider the electron has energy E, the channels with energy greater than E are not allowed however the electron will traverse through the channels below the energy, E. Since the barrier height increases with the channel index, there are always be the finite number of channels which are open. The number of open channels in the mesoscopic conductor can be given by

Hence, the total current carried by the channels is given by

∞

= 2(        ) ∑ ∫ 2     ( ) ( )

−∞

 

where, f(E) is filling factor i.e. the fraction of filled states and is given by  Fermi-Dirac statistics

 

Where µ is the chemical potential or the Fermi energy at zero temperature.

 

where R,L are the chemical potentials of right and left contacts respectively.

(2.51)

 

 

It is clear that more be the gate voltage the less be number of channels and hence the conductance. As M can take only integral values conductance Gc is quantized. It is to be noted that Gc does not depend on any materials parameters. When there is only one conduction channel available for current i.e. M=1, Gc = 80µS or Rc(=1/ Gc )=12.6kΩ. Experimental verification of the quantization of Gc is presented in Fig.2.12.

Fig. 2.12 Point-contact conductance as a function of gate voltage (Ref.10). The conductance shows plateaus at multiples of 2 2⁄ℎ. It is clear that more be the gate voltage the less be number of channels and hence the conductance.

 

 

2.3.2 Mesoscopic Thermal Transport

 

As discussed before that generally mesoscopic phenomena in transport processes occur when the coherence length of the carriers becomes larger than the sample dimensions. When two reservoirs are connected by quasi-one-dimensional mesoscopic conductor, electrical conduction becomes quantized and the conductance of the conductor is determined by the number of participating quantum `channels’ and the conductance is given by Landauer formula (Eq.2.42). So, the similar behavior should be applicable for heat transport in mesoscopic phonon system. The maximum heat current for a single channel can be derived by Landauer formula for elastic transport between two electron reservoirs, L and R,

 

 ℎ = ∫∞ ( )⌈ ( ) − ( )⌉

 

where fα(ε) is the Fermi function of reservoir α=L, R and T(ε) is the transmission function. This formula can be extended to bosonic system and can be given by

 

 

where nα (ω) are the reservoir Bose functions. When there is full transmission, i.e. T (ω)= 1, the thermal conductance Gth is given in both cases integration by

 

which is quantum thermal conductance, g0=(9.456×10-13W/K2)T. This is maximum possible value of energy transported per phonon mode through mesoscopic thermal conductor. This is also amazingly results that quantum thermal conductance does not depend on statistics. Indeed it has been shown [Ref.12] that thermal conductance is quantized and the limiting value for the thermal conductance, Gth, in suspended insulating nanostructures, as shown in Fig.2.13 at very low temperatures. This result is very general and can be applied for heat carried by excitations of any statistics and it has also been experimentally demonstrated for electrons, phonons, and photons [11-14].

 

Fig. 2.13 Suspended mesoscopic device for thermal conductance measurement. A series of progressive magnications are shown. (a) Overall view of the showing 12 wirebond pads that converge via thin niobium leads into the centre of the device. The central region is a 60-nm-thick silicon nitride membrane, which appears dark in the electron micrograph. (b) View of the suspended device, which consists of a 4×4 µm `phonon cavity’ (centre) patterned from the membrane. In this view, the bright “c” shaped objects on the device are thin gold transducers, whereas in the dark regions the membrane has been completely removed. The transducers are connected to thin niobium leads that run atop the `phonon waveguides’; these leads ultimately terminate at the wirebond pads. (c) Close-up of one of the catenoidal waveguides, displaying the narrowest region which necks down to <200-nm width.(Taken from Ref.12) Experimental proof of the quantization of thermal conductance is given in Fi.2.14.

Fig. 2.14 Quantization of thermal conductance. For temperatures above Tco < 0:8 K, a cubic power-law behaviour consistent with a mean free path of ,0.9mm is observed. For temperatures below Tco, saturation in Gth at a value near the expected quantum of thermal conductance (taken from Ref.12)

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References

 

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