11 NDR and Gunn Effect (Transferred Electron) devices
Gunn Diodes represent an example of negative differential resistance (NDR) devices
Why achieving the NDR is so attractive?
NDR
Load
Power dissipated in the diode = I2 x Rd < 0
The NDR diode can serve as an amplifier or oscillator
without any external circuits providing a feedback
The NDR can be easily achieved in ANY semiconductor diode
− EG
N v N c e 2kT
n=
R=
EG
L 2 kT
e
1 L
1
=
qnµ S q( N v N c ) µ S
EG
= B e 2 kT ; where B =
L
q( N v N c ) µ S
1
When the current flows through the sample, the sample
temperature increases due to Joule heating
∆T = RT P = RT I.V;
T = T0 + ∆T = T0 + RT I.V;
Therefore,
EG
EG
EG
R = B e 2kT = B e 2 k (T0 + ∆T ) = B e 2 k (T0 + RT IV )
V = IR = I B e
EG
2 k (T0 + RT IV )
α
= IB e (T0 + RT IV )
The NDR due to a self-heating
R (a.u.)
R=
EG
B e 2 kT
=B e
EG
2 k (T0 + ∆T )
=Be
EG
2 k (T0 + RT IV )
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006
P (a.u.)
At low powers the resistance does not depend on the power
The NDR due to a self-heating
EG
α
V = IR = I B e 2k (T0 + RT IV ) = IB e (T0 + RT IV )
0.05
25000
0.045
0.04
20000
15000
I(a.u.)
R (a.u.)
0.035
10000
0.03
NDR
0.025
0.02
0.015
5000
0.01
0.005
0
0
0
0.5
1
IV (a.u.)
1.5
2
0
10
20
30
V (a.u)
At high powers the temperature rise increases the concentration and the resistance
decreases
40
Load-line method for electric circuit with NDR
0.25
Current - Voltage
Current, A
0.2
t=27 C
0.15
0.1
Solution: the
lowest current
after turn on
0.05
0
0
5
10
15
Voltage, V
20
25
Thermistor with NDR as a temperature sensitive switch
(Fire Alarm)
0.25
Solution
at 77
C VERY HIGH
Current
- Voltage
CURRENT – FIRE ALARM!
Current, A
0.2
t=27 C
t=77 C
t=127 C
0.15
0.1
0.05
0
0
Solution at 27 C – very
low current (<1 mA)
5
10
15
Voltage, V
20
25
Gunn Effect (Transferred Electron) NDR devices
First Observation
n-GaAs
2 mm
-
+
I
“Noisy” current
V
J. B. Gunn, 1963
11 Gunn Effect (Transferred Electron) devices
First Observation
n-GaAs
2 mm
-
+
I
~20 ns
I
t
V
J. B. Gunn, 1963
v = 0.2 cm/20 ns
= 107 cm/s
Ridley-Watkins-Hilsum-Gunn Effect
B.K. Ridley, 1963:
“Domain instability should occur in a semiconductor sample with a
negative differential resistance”
In GaAs, InP and other III-V compounds, the differential
mobility may become negative at high electric fields
∂v
µd =
<0
∂F
So does the differential conductivity:
σd = qnµd < 0
The negative slope of the v vs. F characteristic develops as a
consequence of the intervalley transition of electrons from the
central Γ valley of the conduction band into the satellite valleys.
When the electric field is low, electrons are primarily located in the central valley of the
conduction band. As the electric field increases, many electrons gain enough energy
from the electric field for the intervalley transition into the satellite valleys.
The electron effective mass in the L and X valleys of the conduction band is much greater
than in the Γ valley. Also, the intervalley transition is accompanied by an increased
intervalley scattering.
These factors result in the decrease of the electron velocity in high electric fields
m2 >> m1
m1
The mechanism of negative differential mobility in GaAs and other III-Vs
When the electric field is low, practically all electrons are in the lowest minimum
of the conduction band and the electron drift velocity v is given by
v1 = µF
where µ is the low-field mobility and F is the electric field.
In a higher electric field electrons are "heated" by the field and some carriers may
have enough energy to transfer into upper valleys where the electron velocity is
v2 ~ vs
Here vs is the saturation velocity.
The current density is given by
j = qv1(F)n1 + qv2(F)n2
where n1 is the electron concentration in the lowest valley and n2 is the electron
concentration in the upper valley:
n1 + n2 = no, where no ~ ND.
The mechanism of negative differential mobility in GaAs and other III-Vs
We can define an average drift velocity of all the electrons, vs as
or
In order to find the electric field dependence of v, we need to know
the n2(F) dependence
The fraction of electrons in the upper valleys p = n2/no can be approximate as
p=
A ( F / FS ) t
1 + ( F / FS ) t
where Fs = vs/µ , A ~ 0.6 and t ~ 4 for GaAs
The exact expressions for GaAs are:
An important results that follows from v(F) dependence:
is that the differential mobility,
becomes NEGATIVE if:
This occurs if the electric field exceeds the critical value FP ~ FS.
For GaAs, FP ~ 3.5 kV/cm
This approximation gives a good agreement with Monte Carlo
simulations and experimental data
µ
Instabilities in the samples with bulk NDR
The negative differential resistance may lead to a growth of small
fluctuations in the space charge in a sample.
A simplified equivalent circuit may be presented as a parallel combination
of the differential resistance
L
Rd =
qnµd S
and the differential capacitance:
The RC time constant:
Cd =
εS
Remember, ε = εε0 here!
L
τ d = Rd Cd =
ε
qN dµd
This time constant is called the Maxwell differential dielectric relaxation time.
Instabilities in the samples with bulk NDR
τ d = Rd Cd =
ε
qN dµd
In a material with a positive differential conductivity a space
charge fluctuation, ∆Q, decays exponentially with time:
∆Q = ∆Q(0).exp(-t/τd)
where ∆Q(0) is the magnitude of the fluctuation at t = 0.
When the differential conductivity is negative the space charge
fluctuation may actually grow with time
∆Q = ∆Q(0)exp(t/τd)
Let the average field in the sample be greater than FP:
1. The sample has a fluctuation of electron concentration;
2. This fluctuation leads to an electric field fluctuation:
2
3
∂F q( n − n0 )
=
ε
∂x
Remember, ε = εε0 here!
1
4
3. In the higher-filed region the electrons “slow down”
4. These slow electrons INCREASE the original concentration fluctuation
The fluctuation develops in such a way that the accumulation layer remains “behind”
and the “front edge” is depleted with electrons
2
1
3
4
>
At the same time the entire fluctuation drifts towards the positive contact (the anode)
with the velocity of the “slow” electrons, i.e. vs
If the sample is long enough, the fluctuation develops into a “high-field domain”
What time is needed to develop a high-field domain?
The characteristic time of the space charge growing is ~ 3τd:
τ d = Rd Cd =
ε
qN dµd
During the time of 3τd the domain travels the distance:
Ltr ~ vs.3τd
Therefore the instability occurs if
L> 3vs. τd
This leads to the so-called Kroemer criterion for the Gunn instabilities:
Remember, ε = εε0 here!
For GaAs µd ~ 0.07 m2 /V-s
Derive the Kroemer criterion
When the sample parameters meet the Kroemer criterion, a high-field domain
periodically develops at the cathode side, drifts towards the anode and
dissolves there.
• There is always ONE and ONLY ONE domain propagating in the sample if
the applied voltage is above the threshold and constant.
• The current decreases with the domain formation and increases when the
domain dissipates.
• The oscillation frequency (the “transit time” frequency), fT ~ vs/L
The NDR and domain formation mechanisms explain Gunn’s observations
Find the diode length and the required doping level for GaAs sample to
have Gunn instability with the domain transit time of 100 ps
For GaAs µd ~ 0.07 m2 /V-s
and vs ~ 105 m/s