CHAPTER 3
High-Field Effects
At sufficiently high fields, a number of physical phenomena can occur. The basic physics of
these phenomena is discussed in this Chapter. Chapter Outline the following phenomena:






Drift-velocity with increasing Electric-field
Hot Electrons
Negative Differential Resistance (NDR)
Transfer Electron Devices (Gunn Diodes)
Transferred Electrons
Avalanche Breakdown
Drift-velocity with Increasing Electric Field
The semiconductor drift-velocity is linear with the electric-field if the drift-velocity is small
compared to the thermal velocity., thus Ohm's law can be explained in terms of
drift velocity :
Vn = −μn E
for electrons(1)
(1)
vp = μp E
for holes
(2)
where:
vn =velocity of electrons (cm/s)
vp =velocity of holes (cm/s)
μn =electron mobility, 1350 cm2/V·s in intrinsic Si
μp =hole mobility, 500 cm2/V·s
in intrinsic Si
As the electric-field is increased, the drift-velocity departs from this linear relationship.
The drift-velocity starts to decrease and eventually saturates. For semiconductors that show
saturation. It can be approximated by:
where E is the electric-field,  is the mobility and v
sat is
the saturation velocity
Table 1. Values of parameters used in (3) for some common semiconductor
materials.
Using this data, the drift-velocity of carriers against electric field is shown below:
Figure 1. Drift-velocity against electric-field for electrons and holes in Silicon at
T=300 K.( 27°C )
 T=300 K.( 27°C),

at room temperature is about 107 cm/s ‫هى نفسها على التدريج الرأسى للرسم‬
th

Saturation occurs because as the electric field heats the carriers giving them more
energy. Higher energy carriers scatter more strongly and so the drift-velocity
reaches a constant value.
 Even without applied field, carriers are constantly
moving with a high thermal velocity  th , which is given
by the thermodynamic relation
1/ 2 
2
 3 / 2KT
th
The  th at room temperature is about 107 cm/cm
 Under an electric field, the carrier is accelerated by
the field and a net drift velocity arises, superimposed
on the random motion of vth
K = °C + 273 . 15
300-273=27°C
27°C and 300 K
•At low electric field( drift velocity <
thermal velocity) ,the drift velocity is linearly
proportional to the applied field
( Ohm’s law valid)
•As the drift velocity approaches the thermal velocity, its field dependence on the electric
field will begin to depart from the linear relationship
•
For large electric fields (>103 V/cm) current shows sublinear
dependence on field. Ohm’s law invalid
•
At high fields Ohm’s law doesn’t hold and current is lower
than expected.
• “Hot carriers” have velocities exceeding thermal velocity.
Hot carrier effect: drift velocity ≥ thermal velocity (~107 cm/s)
• Effective temperature Te due to increasing kinetic energy.
• Begins scattering with “optical” phonons, which transfers
energy to the lattice effectively
Hot Carriers Effect;. The term 'hot carriers' refers to either holes or electrons that
have gained very high kinetic energy after being accelerated by a strong electric
field in areas of high field intensities within a semiconductor
Figure 2.5 Carrier velocity versus electric field in semiconductors at 300 K.
Source: Semiconductor Devices: Physics and Technology by S. M. Sze © 1985
Reproduced with permission of John Wiley & Sons, Inc.
Hot-Electrons
Hot Carriers Effect;. The term 'hot carriers' refers to either holes or electrons that
have gained very high kinetic energy after being accelerated by a strong electric
field in areas of high field intensities within a semiconductor
Hot-electrons occur when a strong electric field is applied across a semiconductor.
The electrons obtain energy from the applied electric field, which they dissipate by
emitting phonons, exciting other electrons or emitting photons. When an electron's
energy exceeds the average thermal energy of the semiconductor, the increase in
kinetic energy of the electrons is statistically interpreted ‫ تفسر احصائيا‬as a raising of
the electrons' temperature, so they are termed 'hot electrons' .
Where E 0 is the energy with zero applied field, and  and E are the mobility and
the energy relaxation time measured at zero field. Typically, E th is about 1kV cm - .
What is relaxation of electron?
When an excited electron falls back to‫ يعود إلى‬a state of lower energy, it undergoes
‫ يتعرض‬electron relaxation ‫（ اسرتضاء‬deexcitation）. This is accompanied by the
emission of a photon (radiative relaxation/spontaneous emission) or by a transfer
of energy to another particle
What is meant by de-excitation?
to cause (an electron) to fall from an excited energy level to a lower energy
level.
Negative Differential Resistance (NDR)
With gallium arsenide (GaAs) and indium phosphide (InP) are the compound
semiconductor material, (which shows negative differential resistance ), the high-field
behaviour of the drift-velocity does not follow this pattern of saturation
but actually‫ في الواقع‬the drift-velocity reaches a peak and then starts to decrease.
Example of negative differential resistance (dV=dI), showing drift velocity
(proportional to current I) against the magnitude of electric field (proportional to
voltage V ). Between points A and C, dV=dI is positive, negative between points C
and D, then positive again thereafter.
This is because the band structure in these materials has more than one local
minima in the conduction band. Figure 2. shows a simplified conduction band
diagram for GaAs.
The lowest conduction band minima is at the point ( k =0). At this point the
electrons have a low effective mass and a high mobility.
A second local minimum occurs at the edge the Brillouin zone and is 0.36 eV
higher.
 This minimum has a higher effective mass and therefore a lower
mobility. At low electric-fields, nearly all of the conduction electrons will
occupy the lower minimum at k =0 with the lesser effective mass and
high mobility.
 If the electric-field increased gradually to about 3.2 kV cm -1 , the
electrons will have enough energy to occupy energies in the second
minima with its associated higher effective mass and lower mobility.
 As the field is increased further, the proportion of electrons with the
lower mobility increases and the drift-velocity will continue to decrease
until all the electrons share the lower mobility and the drift-velocity will
level off. This drop in velocity with increasing field was suggested by
Ridley and Watkins in 1961 and discovered in 1963 by Gunn, after
whom the effect was named.
Figure 2. Simplified band structure of GaAs. When the electrons acquire at least
0.36 eV they have enough energy to transfer to the satellite valley ‫وادي األقمار‬
‫الصناعية‬. This transfer of electrons is the cause of negative differential resistance
(NDR).
Fig. 1.2. (a) The Brillouin zone of a face-centered cubic crystal, showing the  , X
and L symmetry points. (b) Schematic band-structure of a direct band-gap
semiconductor (gallium arsenide GaAs) showing the  , X and L valleys. The
energy is chosen to be to zero at the edge of the valence band.
Basic Parameters
Energy gap
1.424 eV
Energy separation (EΓL) between Γ and L valleys
0.29 eV
Energy separation (EΓX) between Γ and X valleys
0.48 eV
Energy spin-orbital splitting
0.34 eV
Intrinsic carrier concentration
2.1·106 cm-3
Intrinsic resistivity
3.3·108 Ω·cm
Effective conduction band density of states
4.7·1017 cm-3
Effective valence band density of states
9.0·1018
Why do we need Brillouin zone?
The Brillouin zones are used to describe and analyze the electron energy in the
band energy structure of crystals
Brillouin
what does a wave vector K mean
In physics, a wave vector (also spelled wavevector) is a vector which helps
describe a wave. Like any vector, it has a magnitude and direction
Negative differential resistance
One of the more interesting in high-field transport is that of negative differential
resistance. At low field, conduction generally tends to be Ohmic, following the
relation V = IR for the applied voltage, V , current I and resistance R.. More
generally, we may take the derivative dV/dI = R, so that R may now be referred to
as the differential resistance. We find that under certain circumstances ‫ ظرضو‬R
may become negative.
A particular example is shown in Fig. 1.3. The graph is labeled as drift velocity, vD
against (the magnitude) of the applied field E. We could have labeled the axes
with current I along the ordinate and voltage V along the abscissa. Between point
A (zero voltage, zero current) and C, we have the familiar situation of a positive
differential resistance. However, between C and D, R becomes negative (this
would have been more obvious if we had plotted V along the ordinate (y-axis
(vertical) coordinate and I along the abscissa (x -axis (horizontal) coordinate. Such
a situation may sometimes give rise to oscillations in the current.
Fig. 1.3. Example of negative differential resistance (dV=dI), showing drift velocity
(proportional to current I) against the magnitude of electric field (proportional to
voltage V ). Between points A and C, dV=dI is positive, negative between points C
and D, then positive again thereafter.
The current-voltage or IV curve of a device with negative differential resistance.
The current increases with voltage (positive resistance), then decreases with
voltage in the region of negative resistance, then increases with voltage again.
Transferred Electron Devices (Gunn Diodes)
•The Gunn diode is formed by adding metal contacts at the ends of a doped semiconductor
material, which shows negative differential resistance (usually GaAs) as shown in Figure 3 (a).
•When biased in the negative differential resistance region, random perturbations‫ اضطرابات‬in
the net charge as a result of thermal motion of electrons cause the formation of charge
domains(an area with low electron mobility is created) which travel with the electricfield until they are absorbed at the anode of the device.
• Figure 3 explains the formation and growth of these Gunn domains. Initially, the device is
biased at E 0 , which is within the negative differential resistance regime.
ً
.‫ والذي يقع ضمن نظام المقاومة التفاضلية السلبية‬، E 0 ‫متحيزا عند‬
‫ يكون الجهاز‬، ‫في البداية‬
• A local perturbation in the net charge density causes a build up of negative charge (A) with
an associated region to the right of the perturbation where there are fewer charge carriers (B)
as indicated in Figure 3 (b).
•From the electric-field - drift-velocity characteristic, Figure 3 (c), it is clear that in the negative
differential resistance regime, the drift-velocity is higher for regions E A and lower at E B .
•Therefore, the electrons pile up‫تتجمع‬
as they travel with the electric-field and the domain
grows as shown by C and D in Figure 3(b).
•The domain will continue to grow until the drift-velocity of the electrons is the same. This
occurs at v c for E C and E D .
•With drift-velocities of the order of 10
7
cm s -1 and device length of a few microns, the
frequency of theses devices is of the order of 10 10 Hz or tens of Gigahertz, which is in the
microwave frequency.
•In practice, with this simple device, the frequency of operation is limited by a 'dead-zone'
near the cathode in which no Gunn domains form.
•Thus the device length is constrained by the incorporation of the dead zone.
• More advanced designs include heterostructures or a notched doping profile to decrease
the length of the dead zone.
Figure 3. (a) Gunn diode structure. (b) Net Charge density in a Gunn Domain. (c) Electric-field
drift-velocity characteristics.
What is bias mean in electronics?
In electronics, 'biasing' usually refers to a fixed DC voltage or current applied to a
terminal of an electronic component such as a diode, transistor or vacuum tube in
a circuit in
Working principle of gunn diode
Gunn diode’s principle of operation is based on the Gunn effect. In some materials
(such as GaAs and InP), after reaching a threshold level by an electric field in the
material, the electrons mobility decreases simultaneously, while electric field
increases producing negative resistance. When the electric field intensity of
Gallium Arsenide crystal reaches its critical value at the negative “electrode”, an
area with low electron mobility is created (domain of a strong electric field).
Area moves with the average speed of electrons towards the positive “electrode”.
When area contacts with the positive “electrode” at the negative electrode, a cyclic
formation of the area of low electron mobility and high electric field start to recreate.Due
to cyclical
phenomenon, oscillations
are
generated,
which frequency can reach up to 100 GHz.
Transferred electrons
Several mechanisms for the occurrence of an NDR are known. One of the best
known is that due to transferred electrons , in which hot electrons in an initial valley are
scattered into a higher lying valley with a larger effective mass. At low fields most of the
electrons will be in the smaller effective mass valley. As the electron temperature is
increased with the application of a higher electric field, an increasing number of electrons
become resonant with higher lying valleys and inter-valley scattering into these will start
to become significant. The electrons in these higher valleys see a larger effective mass and
so have a lower mobility than those in the lower valley. As the population of these slower
moving electrons builds up, the overall mobility of the entire electron population may then
begin to decrease, leading to a reduction of current and, hence, a negative differential
resistance
Energy band structure of GaAs, electrons in Γ valley have light mass and high mobility;
those in the L valley have heavy mass and low mobility; the two valleys separated by a
small energy gap of 0.31 eV.
Avalanche Breakdown
In high electric fields, an electron in the conduction band can gain enough energy
to lift an electron from the valence band into the conduction band, thus generating
one free electron in the conduction band and one free hole in the valence
band. This process is known as impact ionization.At sufficiently high fields, the
electrons hole pairs acquire enough energy to ionise atoms in the lattice and the
electric-field accelerates these electrons and holes to ionise more atoms in a
process known as impact ionisation . In this way, a few electrons can lead to a
runaway chain reaction ‫ تفءعر تسسسرسي مرء‬which cause a breakdown in the material.
This is known as avalanche breakdown.as shown in FiG.
Figure: illustrates Impact ionization causing avalanche current in existence
of high electric field where a number of free carriers are multiplied in each
step, quickly multiplying and causing a breakdown in the material. This is
known as avalanche breakdown..
The critical voltage at which breakdown occurs V br is given by
Where E br is the electric-field at which breakdown occurs,

s
is the relative
permittivity, q is the electron charge in eV and N d the doping density. At T=300 K,
the breakdown field are given below for various common semiconductors. Using
(5) we plot the breakdown voltage against doping density.
Figure 4. Breakdown voltage against doping density at T=300 K.