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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

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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 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<VDSAT VGS>VT 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.) 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 1 W 2 Thus, for small values of V theICVVVVD drain n current ox2 can GS be T DS DS DS 2 L 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 gm 2. n C ox 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 퐼퐷1 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: W ICVVV 1. Increasing 푉D퐷푆1 nwill ox Lincrease GS T the DS 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 VVV2 nCVCVVV ox DS n ox GS T DS DS GS T 2 LL 1 RDS Thus, we can approximate the induced channel as Wa resistor 푅퐷푆 when nCVV ox GS T 푉퐷푆 is much less than the twice the excess gateL voltage. For 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).
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