ECE315 / ECE515
Lecture-2 Date: 06.08.2015
• NMOS I/V Characteristics • Discussion on I/V Characteristics • MOSFET – Second Order Effect
ECE315 / ECE515 NMOS I-V Characteristics
Gradual Channel Approximation: Cut-off → Linear/Triode → Pinch-off/Saturation Assumptions:
• VSB = 0
• VT is constant along the channel • 퐸푥 dominates 퐸푦 => need to consider current flow only in the 푥 −direction
• Cutoff Mode: ퟎ ≤ 푽푮푺 ≤ 푽푻 IDS(cutoff) =0
This relationship is very simple—if the MOSFET is in cutoff, the drain current is simply zero ! ECE315 / ECE515
Linear Mode: 푽푮푺 ≥ 푽푻, ퟎ ≤ 푽푫푺 ≤ 푽푫(푺푨푻) => 푽푫푺 ≤ 푽푮푺 − 푽푻 • The channel reaches the drain.
Gate VS=0 VDS
Oxide
Source Drain (p+) (p+) n+ n+ y Channel x x=0 x=L Substrate (p-Si) Depletion region
VB=0
• 푉푐(푥): Channel voltage with respect to the source at position 푥. • Boundary Conditions: 푉푐 푥 = 0 = 푉푆 = 0; 푉푐 푥 = 퐿 = 푉퐷푆 ECE315 / ECE515
Linear Mode (Contd.)
푄푑: the charge density along the direction of current = W퐶표푥[푉퐺푆 − 푉푇]
where, W = width of the channel and WCox is the capacitance per unit length
yd Drain end dx Channel Source end
• Now, since the channel potential varies from 0 at source to VDS at the drain
푄푑 푥 = 푊퐶표푥 푉퐺푆 − 푉푐(푥) − 푉푇 , where, Vc(x) = channel potential at 푥.
• Subsequently we can write: 퐼퐷 푥 = 푄푑 푥 . 푣 , where, 푣 = velocity of charge (m/s)
ECE315 / ECE515
Linear Mode (Contd.)
푣 = 휇푛퐸 ; where, 휇푛 = mobility of charge carriers (electron) dV E = electric field in the channel given by: Ex () dx dV Therefore, I()() x WC V V x V D ox GS c T n dx
• Applying the boundary conditions for 푉푐(푥) we can write:
xL VV DS IxI().(). Idx WCV VxV dV D D D ox GS T n xV00 1 W • Simplification gives the drain current 2 ICVVVVD n ox2 GS T DS DS in linear mode as: 2 L
1 W 2 Then, ICVVD,max n ox GS T 2 L
The 퐼퐷,푚푎푥 occurs at, VDS = VGS-VT, [how/why?] called overdrive voltage ECE315 / ECE515
Linear Mode (Contd.)
1 W ICVVVV 2 2 D2 n oxL GS T DS DS Observations:
• ID is dependent on constant of technology (μnCox), the device dimensions (W and L), and the gate and drain potentials with respect to the source
• For VDS << 2(VGS-VT), ID can be approximated as: W ICVVV Linear function of V D n oxL GS T DS DS
Thus, for small values of VDS the drain current can be thought of as a straight line → implying that the path from source to drain can be represented by a linear resistor → support of earlier assumption ECE315 / ECE515
Linear Mode (Contd.) MOSFET transistor operate as a VDS 1 RDS resistor whose value is controlled by I W D CVV n oxL GS T overdrive voltage ECE315 / ECE515
Pinch-off point (Edge of Saturation): 푽푮푺 ≥ 푽푻, 푽푫푺 = 푽푫(푺푨푻) • The channel just reaches the drain but with zero inversion at the drain • Electrons start to drift from the channel to the drain
1 W 2 • The drain current is given by: IICVV D D,max 2 n oxL GS T
Saturation Mode: 푽푮푺 ≥ 푽푻, 푽푫푺 ≥ 푽푮푺 − 푽푻
• After pinch-off, the ID saturates i.e, is relatively independent of VDS
1 W 2 ICVV D, sat2 n oxL GS T ECE315 / ECE515
Saturation Mode (Contd.) -4 x 10 VDS = VGS - VT 6
5 VGS= 2.5 V Linear/ Saturation
4 Resistive
(A)
3 VGS= 2.0 V Quadratic
D I Relationship 2 VGS= 1.5 V
1 VGS= 1.0 V On the right of this 0 parabolic curve 0 0.5 1 1.5 2 2.5 VDS (V)
1 W 2 ICVV 1 W D, sat2 n oxL GS T ICVVVV 2 2 D2 n oxL GS T DS DS ECE315 / ECE515 Saturation Mode (Contd.) • MOSFET can be used as current source connected between the drain and the source, controlled by VGS.
• MOSFET in saturation mode → produces a current regulated by 푉퐺푆 → imperative to define a figure of merit (FOM) that identifies the effectiveness with which the MOSFET can convert voltages in currents →
the FOM in this scenario is called “transcodunctance (gm)”. • Defined as the change in the drain current divided by the change in the gate-source voltage.
IWD gm n C ox V GS V T VLGS VDS ,. const ECE315 / ECE515
Transconductance (gm) • In essence, gm represents the sensitivity of the device. For a high gm, a small change in VGS results in a large change in ID.
• Other formulations of gm:
W 2ID gCmn ox2. I D gm L VVGS T
• Behavior of gm as a function of one parameter while other parameters remain fixed.
Home Assignment # 0
Can we define gm in the triode/linear region? Due by 12.08.2015 ECE315 / ECE515
Channel Resistance for Small VDS • Recall voltage 푉퐷푆 will be directly proportional to 퐼퐷, provided that: 1. A conducting channel has been induced. 2. The value of 푽푫푺 is small. Note for this situation, the MOSFET will be in triode region.
→ Recall also that as we increase the value of 푉퐷푆, the conducting channel will begin to pinch off—the current will no longer be directly proportional to 푉퐷푆.
• Specifically, there are two phenomena at work as we increase 푽푫푺 while in the triode region:
1. Increasing 푉퐷푆 will increase the potential difference across the conducting channel → leads to proportional increase in 퐼퐷. 2. Increasing 푉퐷푆 will decrease the conductivity of the induced channel → leads to decrease in 퐼퐷. ECE315 / ECE515
Channel Resistance (contd.)
• There are two physical phenomena at work as we increase 푉퐷푆, and there are two terms in the triode drain current equation!
1 W ICVVVV 2 2 D2 n oxL GS T DS DS WW1 ICVVVCV 2 III D n oxLL GS T DS2 n ox DS DDD12
W 1 W Where: ICVVV ICV 2 D1 n oxL GS T DS D2 2 n oxL DS ECE315 / ECE515
W ICVVV 퐼퐷1 D1 n oxL GS T DS
We note that this term is directly proportional to 푉퐷푆 — if 푉퐷푆 increases 10%, the value of this term will increase 10%. Note that this is true 푉 − 푉 푉 regardless of the magnitude of 푉퐷푆! 퐺푆 푇 퐷푆
→ It is evident that this term describes the first phenomenon:
1. Increasing 푉퐷푆 will increase the potential difference across the conducting channel → leads to proportional increase in 퐼퐷. In other words, this first term would accurately describe the relationship between 퐼퐷 and 푉퐷푆 if the MOSFET induced channel behaved like a resistor! ↔ it means the second term doesn’t allow it to behave like a perfect resistor. ECE315 / ECE515
1 W 퐼퐷2 ICV 2 D2 2 n oxL DS 푉퐺푆 − 푉푇 푉퐷푆
It is apparent that 퐼퐷2 is not directly proportional to 푉퐷푆, but instead proportional to 푉퐷푆 squared!!
Moreover, the minus sign means that as 푉퐷푆 increases, 퐼퐷2 will actually decrease! This behavior is nothing like a resistor—what the heck is going on here?? → This second term essentially describes the result of the second phenomena:
2. Increasing 푉퐷푆 will decrease the conductivity of the induced channel → leads to decrease in 퐼퐷. ECE315 / ECE515
• Now let’s add the two terms 퐼퐷1 and 퐼퐷2 together to get the total triode drain current 퐼퐷: It is apparent that the second term
퐼퐷 퐼퐷2 works to reduce the total drain current from its “resistor-like” value 퐼퐷1 퐼퐷1. This of course is physically due to the reduction in channel conductivity as 푉퐷푆 increases. 퐼퐷 Q: But look! It appears to me that for 푉퐺푆 − 푉푇 푉퐷푆 small values of 푉퐷푆, the term 퐼퐷2 is very small, and thus 퐼퐷 ≈ 퐼퐷1 (when 푉퐷푆 is 퐼퐷2 small)!
A: Absolutely true! Recall this is consistent with our earlier discussion about the induced channel—the channel conductivity begins to significantly degrade only when 푉퐷푆 becomes sufficiently large! ECE315 / ECE515
Channel Resistance (contd.) W • Thus, we can conclude: IICVVV For small 푉 D D1 n oxL GS T DS 퐷푆 • Moreover, we can (for small 푉퐷푆 ) 1 R For small approximate the induced channel as DS W 푉퐷푆 nCVV ox GS T 푽푫푺 a resistor 푅퐷푆 of value 푅퐷푆= : L 퐼퐷 Q: I’ve just about had it with this “for small 푉퐷푆” nonsense! Just how small is small? How can we know numerically when this approximation is valid?
A: Well, we can say that this approximation is valid when 퐼퐷2 is much smaller than 퐼퐷1 (i.e., 퐼퐷2 is insignificant). ECE315 / ECE515
Channel Resistance (contd.) • Mathematically, we can state as: IIDD21
1 WW2 CVCVVV VVDSGS2 V T 2 n oxLL DS n ox GS T DS
Thus, we can approximate the induced channel as a resistor 푅퐷푆 when 푉퐷푆 is much less than the twice the excess gate voltage. 1 R DS W For CVV n oxL GS T
Q: There you go again! The statement 푉퐷푆 ≪ 2(푉퐺푆 − 푉푇) is only slightly more helpful than the statement “when 푉퐷푆 is small”. Precisely how much smaller than twice the excess gate voltage must 푉퐷푆 be in order for our approximation to be accurate? ECE315 / ECE515
Channel Resistance (contd.)
A: We cannot say precisely how much smaller 푉퐷푆 needs to be in relation to 2(푉퐺푆 − 푉푇) unless we state precisely how accurate we require our approximation to be! • For example, if we want the error associated with the approximation 퐼퐷 ≈ 퐼퐷1 to be less than 10%, we find that we require the voltage 푉퐷푆 to be less than 1/10 the value 2(푉퐺푆 − 푉푇). 2VV VV In other words, if: V GS T GS T DS 10 5 I • we find then that 퐼 is less than 10% of 퐼 : I D1 퐷2 퐷1 D2 10 This 10% error criteria is a typical “rule-of thumb” for many approximations in electronics. However, this does not mean that it is the “correct” criteria for determining the validity of this (or other) approximation. ECE315 / ECE515
Channel Resistance (contd.) • For some applications, we might require better accuracy. For example, if we require less than 5% error, we would find that: VV V GS T DS 10 It is important to note that we should use these approximations when we can—it can make our circuit analysis much easier!
See, the thing is, you should use these approximations whenever they are valid. They often make your circuit analysis task much simpler ECE315 / ECE515 Second Order Effect - Channel Length Modulation • I have been saying that for a MOSFET in saturation, the drain current is independent of the drain-to-source voltage 푉퐷푆 i.e. 1 W 2 ICVV In reality, this is only approximately true! D2 n oxL GS T • Let us look at operation of NMOS in saturation mode: Observations: • The pinch-off point moves towards the source with the
increase in VDS • Channel length reduces • Channel resistance decreases
This modulation of channel length (L) by VDS is known as channel-length modulation, and leads to slight dependence of
퐼퐷 on VDS. ECE315 / ECE515
Second Order Effect - Channel Length Modulation (contd.)
1 W 2 • The drain current in saturation mode is: ICVV D, sat2 n oxL GS T
The decrease in channel length with L actually varies with V increase in VDS essentially increases the DS drain current ID 1 1 1 1 1 1 1 If ΔL = L-L then: . . .(1 V ) 1 LLLLLVL L 1 DS 1 1 DS L λ: channel length modulation coefficient (usually less than 0.1) • Therefore the drain current in saturation mode becomes:
1 W 2 ICVVV (1 ) D, sat2 n oxL GS T DS ECE315 / ECE515 Second Order Effect - Channel Length Modulation (contd.)
1 W 2 Where the value λ is a MOSFET device ICVVV (1 ) D, sat2 n oxL GS T DS parameter with units of 1/V (i.e., V-1). Typically, this value is small (thus the dependence on 푉퐷푆 is slight), ranging -1 -4 from 0.005 to 0.02 V . x 10 6 VGS= 2.5 V
5 VDS = VGS - VT
4 VGS= 2.0 V
(A)
3
Quadratic D I Linear/ Saturation Relationship Resistive 2 VGS= 1.5 V Discontinuities 1 VGS= 1.0 V
0 0 0.5 1 1.5 2 2.5 VDS (V) ECE315 / ECE515 Second Order Effect - Channel Length Modulation (contd.)
• Often, the channel-length modulation parameter λ is expressed 1 푉 = as the Early Voltage VA, which is simply the inverse value of λ. 퐴 λ • The parameter VA is set at the time of fabrication and hence the circuit designers can’t alter it at circuit/system design stage. ECE315 / ECE515
Second Order Effect - Channel Length Modulation (contd.)
• The drain current for a MOSFET in saturation can likewise be expressed as: 1 W 2 1 WV2 DS ICVVVD n ox GS T (1 DS ) ICVVD n ox GS T 1 2 L 2 LVA
• Now, let’s define a value IDI, which is simply the drain current in saturation if no channel- 1 W 2 ICVVDI n ox GS T length modulation actually occurred—in other 2 L words, the ideal value of the drain current:
• Thus, we can alternatively write the drain current VDS IID DI 1 in saturation as: VA
This explicitly shows how the drain current behaves as a function of voltage 푉퐷푆. ECE315 / ECE515 Second Order Effect - Channel Length Modulation (contd.)
푉퐷푆 We can interpret the value 푉퐴 as VDS IID DI 1 the percent increase in drain current 퐼 V 퐷 A over its ideal (i.e., no channel length modulation) saturation value • Now, let’s introduce a third way (i.e. in addition to, λ 1 VA and V ) to describe the “extra” current created by ro A II channel-length modulation. Define the Drain Output DI DI Resistance 풓풐: • Using this definition, we can write the saturation drain current expression as: 1 WV VDS DS II1 ICVVD n ox GS T D DI 2 Lr VA o Thus, we interpret the “extra” drain current (due to channel length
modulation) as the current flowing through a drain output resistor ro. ECE315 / ECE515 Second Order Effect - Channel Length Modulation (contd.)
Finally, there are three important things to remember about channel- length modulation:
• The values λ and VA are MOSFET device parameters, but drain output resistance 푟표 is not (푟표 is dependent on 퐼퐷퐼). • Often, we “neglect the effect of channel-length modulation”, meaning 2 that we use the ideal case for saturation: 퐼퐷 = 퐼퐷퐼 = 휇푛퐶표푥(푉퐺푆−푉푇) . Effectively, we assume that λ = 0, meaning that 푉퐴 = ∞ and 푟표 = ∞ (i.e., not 푉퐴 = 0 and 푟표 = 0). • The drain output resistance 푟표 is not the same as channel resistance 푅퐷푆. The two are different in many, many ways. ECE315 / ECE515 Second Order Effect - Channel Length Modulation (contd.)
• For a longer channel length, the relative change in L and therefore in ID for a given change in VDS is smaller. • To minimize channel length modulation, smaller length transistors should be avoided.
Q: Any idea about limitation of long channel devices?