2010 Semiconductor Devices

Final Exam [Solution]

Pf. Sung June Kim

Test Date: 16-Dec-2010

Student Number:                                Name:

Physical Constants

Symbol	Name	Value
q	Electronic charge (magnitude)	[eqn: 1.60 TIMES 10 ^{-19} coul]
[eqn: varepsilon _{0}]	Permittivity of free space	[eqn: 8.85 TIMES 10 ^{-14} `farad/cm]
k	Boltzmann constant	[eqn: 8.617 TIMES 10 ^{-5} eV/K]
h	Planck constant	[eqn: 6.63 TIMES 10 ^{-34} joule.sec]
[eqn: m _{0}]	Electron rest mass	[eqn: 9.11 TIMES 10 ^{-31} kg]
kT	Thermal Energy	[eqn: 0.0259`eV`(T=300K)]
kT/q	Thermal voltage	[eqn: 0.0259`V`(T=300K)]
[eqn: K _{S}]	Dielectric constant of Si	[eqn: 11.8`(T=300K)]
[eqn: K _{O}]	Dielectric constant of SiO2	[eqn: 3.9`(T=300K)]

Total: 5 problems, 6 pages

Given time: 1h 30mins

1. (BJT Fundamentals)

Fig P1.1

(a) Fig P1.1 shows schematically components of electron and hole flows in BJT. Briefly explain each current component ①~⑤ (5 pts).

sol)
① small component due to injection of hole
② Linking current
③ small component due to injection of hole from base to collector
④ small component due to injection of electron from base
⑤ Back injection current

(b) In the backward connection of BJT with [eqn: V _{CB} `>0] and [eqn: V _{EB} `<0] the device exhibits a lower gain. Explain why the backward connection leads to a lower gain to base-width modulation. (5 pts).

sol)

(i) The efficiency of the injecting collector is far below that of the emitter and [eqn: alpha _{dc} = gamma a _{T}] is thereby degraded. The E-B junction has [eqn: N _{E} >> N _{B}], making hole injection into the base much greater than electron injection from the base into the emitter under active mode biasing. On the other hand, [eqn: N _{C} <N _{B}], giving rise to significant electron injection from the base into the collector under "backward" (inverted) operation.

2. (MOS FUNDAMENTALS) Complete the following table making use of the ideal-structure C-V characteristic and energy band diagrams.

Fig P2

For each of the biasing conditions named in the table, employ letters (a-e) to identify the corresponding bias point on the ideal MOS-C C-V characteristic. Likewise, draw to identify the each band diagram associated with each of the biasing conditions. (10 pts)

Sol)

Bias Condition	Capacitance ( a - e )	Band Diagram
Inversion	e
Depletion	c
Flat-band	b
[eqn: V _{G} =V _{T}]	d
Accumulation	a

3. (MOSFETS-THE ESSENTIALS) Refer to Fig P.3 for this problem, consider the characteristics plotted in the figure to be experimental characteristics of a MOSFET under test .

Fig P. 3

n-channel MOSFET

[eqn: x _{o} =0.05 mu m]

[eqn: N _{A} =10 ^{15} /cm ^{3}]

[eqn: bar{mu } _{n} =550cm ^{2} /Vsec]

[eqn: L=7 mu m# Z=100` mu m]

[eqn: T=300K]

(a) Making use only of the plotted characteristics, estimate [eqn: g _{d}] if the quiescent operating point of the MOSFET is [eqn: V _{G} =5V,`V _{D} =0]. Indicate how you arrived at your answer. (5 pts)

sol) Since [eqn: g _{d} == Partial I _{D} / Partial V _{D`|V _{G}}] , it follows that [eqn: g _{d}]can be deduced from the slope of the appropriate [eqn: V _{G} =constant`I _{D} -V _{D} `characteristic] evaluated at the [eqn: V _{D}] operating point. Specifically, the slope of the [eqn: V _{G} =5V] characteristic in Fig P.3 evaluated at [eqn: V _{D} =0] yields

∴ [eqn: g _{d} =(3.0 TIMES 10 ^{-3} A)/(2.3V)=1.3 TIMES 10 ^{-4} S]

(b) Making use only of the plotted characteristics, determine the threshold voltage ([eqn: V _{T}]) of the MOSFET. Use the output conductance in triode (linear) region. Explain how you arrived at your answer. (5 pts).

sol) Since [eqn: V _{T} =V _{T} ']for the given device (the characteristics were generated assuming an ideal device),

[eqn: PHI _{F} = {kT} over {q} ln(N _{A} /n _{i} )=(0.0259)ln(10 ^{15} /10 ^{10} )=0.298V]

and

[eqn: V _{T} =2 PHI _{F} + {K _{S} `x _{o}} over {K _{O}} sqrt {{4qN _{A}} over {K _{S} `` varepsilon _{0}} PHI _{F}}# ```````````=`2(0.298)`+` {(11.8)(5 TIMES 10 ^{-6} )} over {(3.9)} sqrt {{(4)(1.6 TIMES 10 ^{-19} )(10 ^{16} )} over {(11.8)(8.85 TIMES 10 ^{-14} )} (0.298)} =4.11 TIMES 10 ^{-3} `S]

(c) From the I-V, determine the approximate value of the transconductance([eqn: g _{m}]) when [eqn: V _{G} =5V,`V _{D} =4V]. Band on that [eqn: g _{m}], what is the short circuit unity-gain cutoff frequency of the MOSFET? Indicate how you arrived at your answer (10 pts).

sol) Inspecting the device characteristics, we find the MOSFET is saturation biased when [eqn: V _{G} =5V,`V _{D} =4V]. When biased above pinch-off, [eqn: g _{m} =(Z bar{mu _{n}} C _{O} /L)V _{Dsat}] using either the square-law or bulk-charge entry. [eqn: V _{D,sat} CONG 4V] when [eqn: V _{G} =5V]. Also,

[eqn: C _{O} = {K _{O} `` varepsilon _{0}} over {x _{o}} = {(3.9)(8.85 TIMES 10 ^{-14} )} over {(5 TIMES 10 ^{-6} )} =6.90 TIMES 10 ^{-8} F/cm ^{2}]

Thus,

∴ [eqn: g _{m} = {Z mu _{n} C _{O}} over {L} V _{D,sat} = {(100)(550)(6.90 TIMES 10 ^{-8} )(4)} over {7} =2.17 TIMES 10 ^{-3} S]

[eqn: f= {bar{mu _{n}} V _{D,sat}} over {2 pi L ^{2}} = {(550 TIMES 10 ^{-4} ) TIMES 4} over {2 pi TIMES (7 TIMES 10 ^{-6} ) ^{2}} =0.715GHz]

4. (FIELD EFFECT DEVICES) The energy band diagram for a p-Si/SiO2/n-Si (SOS) capacitor under flat-band conditions is given below. To achieve the pictured state, there must of course be a non-zero voltage applied to the SOS-C gate. The SOS-C is ideal except for a non-zero workfunction difference.

([eqn: T=300K,`N _{A} (p-side)=10 ^{16} /cm ^{3} ,`N _{D} (n-side)=10 ^{16} /cm ^{3} ,`n _{i} =10 ^{10} /cm ^{3} ,`x _{o} =4 TIMES 10 ^{-6} cm,`and`A _{G} =10 ^{-3} cm ^{2}])

Fig P. 4

(a) What is the voltage being applied to the p-Si/SiO2/n-Si SOS-C to achieve the pictured flat-band condition? Give both the polarity and magnitude of [eqn: V _{G}] (10 pts).

sol)
[eqn: V _{G} = {1} over {q} (E _{FN} -E _{FP} )= {1} over {q} [(E _{FN} -E _{i} )+(E _{i} -E _{FP} )]= {kT} over {q} [ln( {N _{D}} over {n _{i}} )+ln( {N _{A}} over {n _{i}} )]=2(0.0259)ln(10 ^{16} /10 ^{10} )=0.7152V]

(b) Sketch the energy band diagram and the associated block charge diagram for the SOS-C when a large negative gate voltage is applied to the device. Add descriptive words to your sketches as necessary to forestall a misinterpretation of your answer (10 pts).

sol)

(c) Make a sketch of the high-frequency [eqn: C-V _{G}] characteristic to be expected from the SOS-C described in this problem. Explain how you arrived at your [eqn: C-V _{G}] sketch (10 pts).

sol) When [eqn: V _{G} >0], both semiconductor components are accumulated. Thus C approaches [eqn: C _{O}] at large positive gate biases. When [eqn: V _{G} <0], the two semiconductors components first deplete and then invert. Inversion occurs at the same bias voltage for the two sides of the SOS-C because [eqn: N _{A} (p-side)=N _{D} (n-side)]. The high-frequency capacitance is therefore expected to smoothly decrease to [eqn: C _{min}]at large negative biases. As sketched below, the deduced characteristic should look very similar to a standard n-bulk high-frequency MOS-C C-V curve.

5. (NONIDEAL MOS) The [eqn: C _{G} -V _{G}] ([eqn: V _{D} =0]) characteristic derived from an n-channel MOSFET is pictured in Fig P. 5.

Fig P. 5

(a) What is the threshold voltage, [eqn: V _{T}] , of the transistor? If you want to increase the threshold voltage over zero, what would you do for this adjustment? Explain. (5 pts).

sol) [eqn: V _{T} =-1V]. In progressing toward positive biases, the C-V curve increases toward [eqn: C=C _{O}] at roughly the inversion-depletion transition point. [eqn: V _{G}] at the inversion-depletion transition point is the threshold voltage in MOSFETs.

(b) Given [eqn: PHI _{MS} =-1V], QIT = 0, and the fact that the device is stable under bias-temperature stressing, how do you explain the observed flat band voltage of [eqn: -2.2V] ? (10 pts)

sol) Since the device is stable under bias-temperature stressing, one concludes that [eqn: Q _{M} =0]. With [eqn: Q _{M} =0], QIT = 0, and [eqn: PHI _{MS} =-1V] unable to account for the entire flat-band shift of [eqn: -2.2V], the residual shifting can only be attributed to the fixed charge.

(c) If the MOSFET is doped such that [eqn: PHI _{F} =0.3V], what substrate bias ([eqn: V _{BS}]) must be applied to attain a [eqn: V _{GS``|`at`threshold} `=1V] ? (10 pts)

sol)

[eqn: V _{BS} =2 PHI _{F} [1-(1+ {TRIANGLE V _{T} prime } over {V _{T} prime -2 PHI _{F}} ) ^{2} ]]

Noting

[eqn: V _{GS``|at`threshold} `-`V _{T} =V _{GS``} prime _{|at`threshold} -V _{T} prime = TRIANGLE V _{T} prime =2V]

and

[eqn: V _{T} prime =V _{T} -V _{FB} =`-1.0V`+2.2V`=1.2V]

direct substitution into the [eqn: V _{BS}] expression then yields

[eqn: V _{BS} =0.6[1-(1+ {2} over {1.2-0.6} ) ^{2} ]=`-10.7V]

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