Semiconductor transport and LED’s; MOSFETs.
Phys 489, 11/5/15
(1) Currents in a p-n junction: This material is similar to the text’s description of the p-n
junction (ch. 17). When p- and n-type materials are joined a depletion region forms due to
exchange of carriers at the junction, forming a dipole layer with imbalance of electric charge.
The exchange of carriers allows the chemical potential to equalize, with random carrierexchange processes continuing in the equilibrium situation. These include the opposing
generation and recombination processes.
To picture the different contributions in equilibrium, recall that to good approximation we have
only electrons in the n-type material (outside of the depletion region), and only holes in the ptype material. There is an energy barrier, magnitude eVbi , preventing electrons from traversing
the depletion region and entering the p-type side. The holes on the p side see a barrier of the
same height keeping them from crossing to the n side (see the first sketch below, for which the p
side is on the left).
Generation current ( IG ) comes from electron-hole pairs that are randomly generated in the
depletion region. Following such an event, the electron will tend to be swept downhill towards
the n-type side, while the hole will tend to be swept toward the p-type side. If this occurs before
they can recombine, the generated carriers contribute to a net current: generated electrons and
holes go in opposite directions, but they contribute to a current in the same direction because of
their opposite charges. Generation of these pairs might come from random thermal processes, or
illumination by light at the wavelength that matches or exceeds the band-gap energy. IG changes
rather slowly with applied voltage; we will treat it as a constant.
The recombination current ( I R ) flows from p to n, the direction opposite to IG . This is the
“forward” direction for a forward-biased diode. I R is defined as the current of majority electrons
and holes that manage to “climb the hill” of the depletion-region potential barrier. Since the
probability of a carrier having energy ε is proportional to exp(−ε / kT ) , and since carriers require
an energy of at least eVbi to surmount the barrier, it is easily seen that I R varies according to,
[1]
I R = C exp(−Vbi / kT ) ,
where C is a constant.
For the biased junction, an external voltage ( Vext ) drops only across the depletion region to good
approximation because of its high resistivity. As a result Vext will raise or lower the energy
barrier, depending upon the applied voltage direction. Forward bias (positive voltage applied to
the p side) is illustrated in the sketch on the right on the first page; in this case the barrier is
lowered to e(Vbi − Vext ) . Thus the recombination current [1] becomes,
[2]
I R = C exp(−[Vbi − Vext ]/ kT ) = C exp(−Vbi / kT )exp(+Vext / kT ) .
From [1] we see that the last expression in [2] can be rewritten:
[3]
I R = I R (0)exp(+Vext / kT ) ,
adopting the notation I R (0) for the current with Vext = 0 . However, as argued above, equilibrium
conditions dictate equivalent zero-bias magnitudes, so I R (0) = IG . Because they are defined to
flow in opposite directions, the total current must be I = I R − IG . Combining, [3] becomes,
I = IG (exp[+Vext / kT ] −1) .
[4]
The last expression is the exponentially-changing diode I-V behavior, which can easily be
verified experimentally. The large asymmetry lends itself to rectifier applications for
semiconductor diodes, which are devices containing a single pn junction.
(2) Light emission: Light Emitting Diodes (LED’s) and Lasers normally emit with hν at
approximately the bandgap, as injected electrons and holes recombine to produce photons. If the
recombination involves itinerant electrons and holes propagating in states at the band edge,
energy conservaton yields hν = ε g . Sometime the carriers may be captured before
recombination, for example by shallow defect or exciton states, in which case the emission is at a
somewhat smaller energy. Light emission processes such as these normally require a direct-gap
material. There has been a huge recent increase in the use of LED’s for applications such as
flashlights, auto tail-lights, etc.
To vary the emission spectrum, the following methods are used:
(a) Alloying to change the band-bap. Mixed semiconductors can produce a bandgap between
those of the pure materials. Examples: AlxGa1-xAs produces the common red led color, with the
GaAs bandgap beyond the visible, in the infrared. InxGa1-xN can make blue, green and violet
light (GaN’s bandgap is in the UV).
(b) Minibands in superlattices, or related quantum-confinement effects, can adjust the gap
energies in devices with thin layers (or possibly nanocrystals). For example, quantum-cascade
lasers are used for infrared telecommunications devices, and feature a complicated GaAs-based
layered structure. (In our department, see the very active work of Prof. Alexey Belyanin and his
group.) Si and Ge nanocrystals have also been shown to emit blue light, a confinement effect.
(c) Commercial white and yellow LED’s actually use a phosphor which is driven by a UV GaN
LED. The phosphor is typically a cerium-doped yttrium aluminum garnet.
(d) Organic LED’s and displays are also commercially available now. Polymers such as poly(pphenylene vinylene) have been used, with chemical derivatives of these molecules providing the
wavelength variation.
(3) MOSFET stands for Metal-Oxide-Silicon Field Effect Transistor. The device has a
configuration similar to what is shown in the figure.
SiO2 is the natural surface oxide on
silicon, and can as an insulating
barrier, thickness ~ 20-40 Å (though
currently for advanced devices a
different “high k dielectric” oxide
layer is instead used as a barrier). The
Gate (with voltage VG) is a capacitor
plate whose voltage can raise and
lower the bands close to the gate.
This makes a triangle-shaped well,
which confines electrons (or holes) to
a thin layer near the surface, by
altering the depletion region. Electrons confined in this region make up a conducting channel;
altering VG changes the density of electrons in this channel, and the channel thickness. Thus, VG
can switch, or modulate, the conductivity, and this is the basis of the switching behavior of this
type of transistor.
Electric fields near the interface can be large: for example SiO2 thickness may be 20 Å. At 1 V
gate bias, the field is 5 × 108 V/m, divided by the dielectric constant. Dielectric strengths can
easily be exceeded, and purity of the insulating layer is of great concern in the fabrication of
integrated devices. These fields also define the nearly triangle-well shape of the bands at the
interface. A figure showing the band-bending process is given on the next page. This figure
shows band inversion, whereby with positive bias on the gate, for the case of p-type Si, electrons
accumulate at the interface. At suitable densities and sufficiently low temperatures, the electrons
may be confined to the lowest bound state in this well, giving quantum confinement.
(From http://www.mst.or.jp/prize/english/winners/semiconductor/semiconductor2012_en.html.)
The MOSFET was used for the original observation of the quantum hall effect (QHE), by von
Klitzing (Nobel prize, 1985). By shaping the gate layer, he laterally confined his 2D layer,
forming a Hall bar. The conduction in this system was found to be highly quantized at low
temperatures. The Fractional Quantum Hall Effect (FQHE; Nobel prize, 1998) relies on
electron-electron interactions to place the electrons in a complicated ground state, while the
integer QHE can be understood on the basis of single-particle states. The quantum steps in the
Hall resistance (transverse V divided by current) are equal to h/e2 (or a rational fraction for the
FQHE), while the longitudinal resistance goes to zero at the plateaus. The unit h/e2 = 25813....
Ω, and measured values agree with theory to within 10-8.
MOSFETs provide just one example of electron confinement. One feature is that the electrons
are confined at the interface, where surface roughness dominates the scattering, limiting the
mobilities. For ultimate quantum behavior, one wants τ, and hence l (the mean free path) to be
long. The highest mobilities (such as used for FQHE measurements) are typically obtained in
GaAs/AlGaAs confined layers. Such structures provide much cleaner systems for observations
of quantum effects in semiconductor devices. Delta-doping also helps to improve the mobility in
this case (as may be described in class).