Electron Transport in Semiconductors
The subject of electronic transport in semiconductors and in solids in general, is a very
old problem, which has been well studied over the past 75 years. Transport is an inherently non-equilibrium
phenomena, where the role of dissipation and the coupling to the environment play a crucial role. External
forces which drive the system out of equilibrium may be electromagnetic in origin, such as the electric fields
associated with an applied DC bias,
or the excitations of electrons from their ground to excited states due to high frequency optical excitation.
Alternately, electrochemical potentials and thermal gradients may also provide the drive for electronic transport
and its external manifestation in terms of macroscopic currents and voltages.
Electronic transport at its most fundamental level requires a full many body quantum mechanical
description. Clearly, a full many particle description of transport including the real number of particles in
both the device, its contact to the external environment, and the external environment itself, is beyond the
ability of any computational platform in the foreseeable future. Hence, successive levels of approximation,
that sacrifice information about the system, and the exact nature of transport, are necessary in any sort of
realistic description of transport. the figure below illustrates the hierarchy of transport approaches used
in describing electronic transport in semiconductors, metals, and molecular systems. At the bottom is the exact
solution of the N-body quantum mechanical problem which is computationally intractable except for small numbers
of particles (less than 100).
To treat the many-body problem, some sort of mean-field approximation is necessary
which transforms the problem into an effective one-electron problem. Non-equilibrium
Green function
methods are currently popular at the next level of approximation as they contain retain important correlations in
space and time, which are believed to be important at the nanoscale. Above this are quantum kinetic approaches in
terms of the Liouville-von Neumann equation of motion for the density matrix, or Wigner distribution approaches that
contain quantum correlations but retain the form of semi-classical approaches in terms of the distribution function.
In going from the quantum to the classical description of charge transport, information concerning the phase of
the electron and its non-local behavior are lost, and electronic transport is treated in terms of a purely particle
framework. This is the level of the Bolzmann transport equation (BTE), which represents a kinetic equation describing
the time evolution of the distribution function
of the position and momentum of the particle, and has been the primary framework for describing transport in
semiconductors and semiconductor devices with micro-scale and above dimensions. There are then approximations
to the BTE, given by moment expansions of the BTE which lead to the hydrodynamic, the drift-diffusion, and
relaxation time approximation approaches to transport. Finally, at an empirical level are non-linear circuit
models for device behavior suitable for circuit simulation in so called compact models.
One interesting aspect of transport in nanostructure systems, is that the characteristic
length scales span the transition from classical to quantum transport. Hence a single description in the
hierarchy may not be sufficient, or may be overly cumbersome for providing the correct physics of device
operation. Depending on certain critical length scales, transport may be semi-classical or purely quantum, or
even more difficult, a mixture of the two in which the effects of decoherence and dissipation play important
roles, while at the same time, quantum effects still dominant. To illustrate this notion, let us consider a
prototypical nanodevice illustrated below:
The ‘device’ is coupled to two contacts, left and right, which serve as a source
and sink (drain) for electrons. Here the contacts are drawn as metallic-like reservoirs, characterized by
chemical potentials μs and μD, and are separated by an external bias,
qVA=μs - μD. The current flowing through the device is then a
property of the chemical potential difference and the transmission properties of the active region itself.
A separate gate electrode serves to change the transmission properties of the active region, and hence modulates
the current. This separation of a nanodevice into ideal injecting and extracting contacts, and an active region
which limits the transport of charge, is a common way of visualizing the transport properties of nanoscale systems.
However, it clearly has limitations, the contacts themselves are really part of the active system, and are driven
out of equilibrium due to current flow, as well as coupling strongly to the active region through the long range
Coulomb interaction of charge carriers.
The nature of transport in a nanodevice such as that illustrated above, depends on the
characteristic length scales of the active region of the device, L. The figure below illustrates the
active region of this nanodevice in terms of a conductor of length L, and width W. The mean
free path between collisions is designated l, while the length scale over which quantum coherence is
preserved (the phase breaking length), is designated lφ. The latter is often associated
with the inelastic mean free path, or the distance between dissipative scattering events where the inelastic
coupling to the environment is associated with quantum mechanical phase breaking. Figure a) corresponds
to the case in which both L and W are much larger than both the elastic and inelastic mean free
paths. Here transport is purely diffusive, and the system behaves essentially as a semi-classical metal or
semiconductor governed by the BTE. In Figure b), the width, W, is smaller than the characteristic
mean free path, while the length, L, is still much longer. This regime corresponds to the case of a
quantum confined system, in which the motion of carriers is quantized in one dimension, but essentially behaves
as a diffusive conductor in the other directions. Quasi-two-dimensional and quasi-one-dimensional systems
correspond to this case. Finally, when both L and W are shorter than the elastic and inelastic
mean free paths, the system is purely ballistic, and the motion of charge is governed by the wave-like behavior
of the particle and its reflection and transmission properties through the structure.
Acknowledgement
More about electron transport
Semiconductor Physics Tutorial
Carrier Mobility in Semiconductors
Electron Transport in semiconductor materials
Tutorial on Electronic Transport
The Non-equilibrium Greens Function Approach to Nano-Device Simulation
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