Download Homework 8

Solid State Device [100 points]
Homework #8
8.1. Ebers-Moll model
Starting with the Ebers Moll model for active mode operation (VEB > 0, VCB < 0, exp(qVCB/kT)
<< 1), derive
(a)
(b)
(c)
(d)
the common base input characteristic IE(VEB, VCB)
the common base output characteristic IC(VCB, IE)
the common emitter input characteristic IB(VEB, VEC)
the common emitter output characteristic IC(VEC,IB)
Show all algebraic steps [40]
8.2. MOSFET Threshold Voltage
In an ideal MOSFET, the saturation current is 25 A at 1V gate bias, and 150 A at 3V gate bias.
What is the threshold voltage? [5]
8.3. MOSFET Inversion [30]
The equation for the MOS potential worked out in class is
d/dx = ±√2(kT/q)/LD. F(,np0/pp0),
where F(,np0/pp0) = [(e- +  - 1) + (np0/pp0)(e- -1)]1/2
All the physics of accumulation, depletion and inversion regions is carried in this
equation. Let us try to extract them one at a time through simplifications.
(i) First let’s try to work out the depletion region physics. In the depletion region, which
sign in the front of the expression should we use? Present your argument to explain this
[5]
(ii) Let us just focus on the depletion term, which is ~ inside the bracket and is the
dominant term for small positive bias. Drop all other terms, and simplify the equation,
with boundary condition (0) = s to extract the position dependence of the potential
in the depletion region. This should look like the plots of the potential variation across
the depletion region that we worked out in class [10]
(iii) Let us also try to extract the depletion capacitance. Using CD = dQs/ds, where Qs is
related to surface electric field Es using Gauss’ Law, simplify the expression by picking
out only the ‘depletion term’ from the F function above. Show that the depletion
capacitance is the same expression as you would expect from simple physical arguments
[5]
(iv) Now let’s go to the inversion region. In the inversion region where you work with
only the (np0/pp0)e term, which can be rewritten as eThe left side can be
written as d(-2B)/dx. Thus, we have a differential equation describing how fast (2B) varies.
Solve this equation with boundary condition (0) = s. From the resulting (x), calculate
d/dx and thus the surface charge density Qs, which is now primarily the inversion
charge. This tells us how fast the inversion charge varies with position into the
semiconductor. Simplify the expression as far as possible [10]
8.4. MOSFET I-V
An 𝐼𝐷 − 𝑉𝐷 characteristic derived from an ideal NMOSFET is pictured below. Note that
𝐼𝐷,𝑠𝑎𝑡 = 10−3A and 𝑉𝐷,𝑠𝑎𝑡 = 5𝑉 for the given characteristic. Answer the following
questions using the square law theory and the figure.
1. Carefully sketch the inversion layer and depletion region inside the MOSFET
corresponding to the point labeled (1) in the figure. Show and label all parts of the
figure. [5]
2. Given a threshold voltage of 𝑉𝑇 = 1V, what is the gate voltage one must apply to
the MOSFET gate to obtain the pictured characteristic? [3]
3. If 𝑡𝑜𝑥 = 0.1μm, what is the inversion layer charge/cm2 at the drain end of the
channel when the MOSFET is biased at point (2) on the characteristic? [3]
4. Suppose the gate voltage is readjusted so that 𝑉𝐺 − 𝑉𝑇 = 3V. For the new
condition, determine 𝐼𝐷 if 𝑉𝐷 = 4V [3]
5. Determine 𝑔𝑑 if the quiescent operating point of the MOSFET is point (3) on the
pictured characteristic [3]
6. Determine 𝑔𝑚 if the quiescent operating point of the MOSFET is point (3) on he
pictured characteristic [3]
7. If 𝑉𝐷 =0 (i.e. the drain is shorted to the source and back), sketch the general shape
of the 𝐶𝐺 (gate capacitance) versus 𝑉𝐺 (gate voltage) characteristic to be expected
from the MOSFET [5]