MOS Capacitor MOSFET Devices. MOSFET s. INEL Solid State Electronics. Manuel Toledo Quiñones. ECE Dept. UPRM.

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1 INEL Solid State Electronics ECE Dept. UPRM 20th March 2006

2 Definitions MOS Capacitor Isolated Metal, SiO 2, Si Threshold Voltage qφ m metal d vacuum level SiO qχ 2 E g /2 qφ F E C E i E F E v qφ S semiconductor q S, q m : semiconductor and metal work functions. Electron affinity q Fermi potential q F : Proportional to doping type and level: P-type: F = + kt q ln N A n i N-type: F = kt q ln N D ni

3 Definitions MOS Capacitor Isolated Metal, SiO 2, Si Threshold Voltage qφ m metal d vacuum level SiO qχ 2 E g /2 qφ F E C E i E F E v qφ S semiconductor q S, q m : semiconductor and metal work functions. Electron affinity q Fermi potential q F : Proportional to doping type and level: P-type: F = + kt q ln N A n i N-type: F = kt q ln N D ni

4 Definitions MOS Capacitor Isolated Metal, SiO 2, Si Threshold Voltage qφ m metal d vacuum level SiO qχ 2 E g /2 qφ F E C E i E F E v qφ S semiconductor q S, q m : semiconductor and metal work functions. Electron affinity q Fermi potential q F : Proportional to doping type and level: P-type: F = + kt q ln N A n i N-type: F = kt q ln N D ni

5 Isolated Metal, SiO 2, Si Threshold Voltage Last equation (N-type): from textbook eq. (1.67) E F E V = ktln(n V =N E g A ) = 2 q F Eg = exp 2kT N V Eg = exp N A 2kT N A exp qf kt kt = N V exp q F = ktln NA n i Eg 2kT q F exp n i qf kt (1)

6 Isolated Metal, SiO 2, Si Threshold Voltage Oxide interface qϕ S (ϕ S >0) qϕ qφ F E C E i E F E v E g ' = 0 on bulk, by definition. ' S : surface potential. n and p: function of '(x) ' is positive when the bands bent downward. SiO 2 Si x

7 Isolated Metal, SiO 2, Si Threshold Voltage Oxide interface ' S < 0: accumulation of holes ' S = 0: flat-band condition qϕ qφ F E C F > ' S > 0: depletion of holes qϕ S (ϕ S >0) SiO 2 Si x E i E F E v E g F = ' S : midgap with n s = n p = n i (intrinsic concentration) ' S > F :inversion

8 Isolated Metal, SiO 2, Si Threshold Voltage '(x) can be obtained from Poisson s equation d 2 ' dx 2 = s (x) (2) s Depletion approximation: semiconductor is depleted to width w. Exact solution: In the bulk: s = qn A s = q (p(x) n(x) + N D N A ) (3) p bulk n bulk + N D N A = 0 N D N A = (p bulk n bulk ) = N A + n2 i N A (4)

9 Isolated Metal, SiO 2, Si Threshold Voltage '(x) can be obtained from Poisson s equation d 2 ' dx 2 = s (x) (2) s Depletion approximation: semiconductor is depleted to width w. Exact solution: In the bulk: s = qn A s = q (p(x) n(x) + N D N A ) (3) p bulk n bulk + N D N A = 0 N D N A = (p bulk n bulk ) = N A + n2 i N A (4)

10 Isolated Metal, SiO 2, Si Threshold Voltage Oxide interface E C qϕ S (ϕ S >0) qϕ qφ F E i E F E v E g Near the surface, E C and E V are bent down by q'. SiO 2 Si x

11 Isolated Metal, SiO 2, Si Threshold Voltage From chapter 1, the bulk electron concentration is EC E F n B = N C exp kt Due to band bending, E C! E C q' and, as we approach the Si SiO 2 interface, EC q' E F q' n(x) = N C exp = n B exp kt kt Using n B = n 2 i =N A, n(x) = n2 i N A exp q' kt (5)

12 Isolated Metal, SiO 2, Si Threshold Voltage For holes and p(x) = N V exp p B = N V exp EF E V kt EF (E V q') = p B exp kt p(x) = N A exp q' kt q' kt (6)

13 Isolated Metal, SiO 2, Si Threshold Voltage Equation 2 becomes 0 1 d 2 ' dx 2 = q s N q' Aexp {z kt } p(x ) q' ni 2 exp N A kt {z } n(x ) N A + n2 i N A {z } N D N A C A Or, d 2 ' dx 2 = q s N A exp q' kt 1 n 2 i N A exp q' kt! 1 (7)

14 Isolated Metal, SiO 2, Si Threshold Voltage Equation 7 can be integrated by multiplying both sides by d' dx dx. Let u = d' dx Then where Z x 1 d 2 ' dx 2 d' dx dx = Z b du dx = d 2 ' dx 2 g a dx = du g gu du g = 1 2 u2 j b a = 1 2 a = u(1) = d' dx (1) = 0 = ' (at the bulk) and at the surface, u(x) = d' dx. 2 d' j b a dx

15 Isolated Metal, SiO 2, Si Threshold Voltage Thus integration of the left-hand side of 7 yields d' 1 = dx 2 E 2 where E is the electric field. The right-hand side can be integrated straightforwardly to obtain d' dx 2 2kTN A = fexp( Si q' kt ) + q' kt 1 + n2 i NA 2 (exp( q' kt ) q' kt 1)g (8)

16 Isolated Metal, SiO 2, Si Threshold Voltage Surface charge: Q s = Si E s ; E s surface elect. field. B is our F ; s is our ' s.

17 Depletion Approximation Isolated Metal, SiO 2, Si Threshold Voltage Depletion region: kt q < ' < 2 F Keep only the q' kt on r.h.s. of equation 8, which becomes s 2qN A ' d' dx = Si or The solution is: Z ' ' s d' p ' = ' = ' s 1 Z x 0 s 2qN A dx x 2 w d Si

18 Isolated Metal, SiO 2, Si Threshold Voltage where w d = s 2 Si ' s qn A is the distance to which the band bending extends, or the depletion layer width. For ' s = 2 F, w d reaches its maximum value w dm. The total depletion layer charge is p Q d = qn A w d = 2Si qn A ' s

19 Strong Inversion MOS Capacitor Isolated Metal, SiO 2, Si Threshold Voltage For ' s > 2 F, the (ni 2 =N2 A )exp(q'=kt ) term becomes significant and must be taken into account. Equation 8 can then be approximated by v d' dx = u t 2kTN A Si q' kt + n2 i NA 2 exp which can only be integrated numerically. q' kt! (9) When strong inversion occurs, ' S = 2 F. The gate voltage at which this happens is called the threshold voltage V T.

20 Threshold Voltage MOS Capacitor Isolated Metal, SiO 2, Si Threshold Voltage When a gate voltage V g is applied, part of it appears as a potential drop accross the oxide. The other part appears as band bending in the silicon: V g = V ox + ' s The total charge per unit area induced in the silicon is the sum of the depletion and inversion layer charges, Q i and Q d. Thus V ox = Q s C ox = Q i + Q d C ox where C ox = ox t ox. Notice that Q s was given earlier in terms of ' s, so V g = Q s + ' s C ox is an implicit equation that can be solved for ' s.

21 Isolated Metal, SiO 2, Si Threshold Voltage Due to the work-function difference between the metal and semiconductor, a voltage V FB must be applied between gate and channel to flatten the bands. V FB is called the flat-band voltage. The effective gate voltage is V G V FB and the voltage across the gate oxide is V ox = (V G V FB ) ' S If, for simplicity, we assume that at the onset of strong inversion there is no inversion layer charge, then V ox C ox = ((V T V FB ) 2 F ) C ox = Q dm

22 Isolated Metal, SiO 2, Si Threshold Voltage Rearranging, V T = V FB + 2 F + Q dm C ox = V FB + 2 F + p 2F = V FB + 2 F + qn Aw dm C ox where = is known as the body factor. p qna 2 S C ox

23 Figure: MOSFET Structure

24 MOSFET Operation

25 Figure: Flat band condition.

26 Figure 2 shows the flat band condition. Observe that the Fermi level is constant in the y direction. This creates an energy barrier that prevents electrons from flowing between source and drain. As long as there is an energy barrier between the source and the drain there is no flow of electron. MOSFET is in cutoff.

27 To turn the MOSFET ON, a positive voltage that reduces the potential barrier between source and drain should be applied to the gate. The energy barrier at the surface is reduced, as seen in figure 3. Bands bent at the silicon surface, moving the conduction band toward the Fermi level.

28 Figure: Weak inversion layer.

29 Once the Fermi level is closer to the conduction band than to the valence band, the probability of occupancy of conduction band states is higher than that of the valence band states. The electron concentration becomes larger than the hole concentration. The inversion layer as been created. When ' S = 2 F, strong inversion is reached. Further increases in the gate voltage cause a quick increase in the electron concentration in the inversion layer. The additional gate voltage appears across the oxide and ' S remains constant.

30 Figure: Approximation used.

31 The gate voltage required for ' S = 2 F is the threshold voltage, V T. When a positive effective voltage V GS V FB is applied to the gate, holes are rejected from the region under the gate. The charge due to the uncompensated acceptor atom ions left behind is called the depletion region charge. The associated charge density, Q d, can be expressed in C=m 2.

32 The following approximation is used: the inversion-layer charge is neglected if the gate voltage is below V T ; under this conditions, the only uncompensated charge in the silicon substrate is Q d. the surface potential is pinned at ' S 2 F for V GS > V T. Increases beyond V T produce the inversion layer charge Q I, Q I = (V GS V T ) C ox

33 Situation is similar to two caps in series. One fraction of the effective applied voltage appears across the oxide capacitor, the other across the depletion region. The voltage drop across the oxide is given as the difference between the effective gate potential on one side of the oxide and the surface potential. Thus V GS V FB ' S = Q d + Q I C ox At the onset of strong inversion, the surface potential is ' S = 2 F, V GS = V T, and Q I = 0; thus and V T V FB 2 F j= Q d C ox V T = V FB + 2 F + Q d C ox

34 In chapter 2, we found that Q d = qn A w m = p 2s qn A (2 F ) In terms of the body factor, = voltage can be expressed as p 2sqN A C ox, the threshold V T = V FB + 2 F + p 2F This expression is valid if the substrate and the source are at the same potential. In figure 3, The situation for V DS = 0 is shown. There is no drain current and the quiescent point remains at the origin.

35 If a positive voltage is applied to the drain, the conduction band at the drain is lowered with respect to the source and electrons flow from source to drain. This is shown in figure 30. The drain current magnitude is determined by the amount of voltage applied to the drain and by the density of charge in the inversion layer. Since Q I = (V GS V T ) C ox the drain current can be expressed I D = (V GS V T )V DS where is a proportionality factor.

36 Figure: Triode region.

37 If a voltage is applied from source to substrate, the energy barrier seen by source s electrons is increased. The barrier between source/drain and substrate is increased by qv SB. See figure 1 and compare (a) with figure 3. As a consequence, the potential needs to be increased to ' S = 2 F + V SB. In the other hand, the gate-to-bulk voltage is increased to V GS + V SB V FB and the effect of V SB cancels.

38 Figure: Strong inversion, including the effect of the source-bulk potential.

39 The bulk voltage affects Q d through its dependence on the surface potential. when V SB = 0 under strong inversion Q d =C ox = p ' S = p 2F When V SB is applied, ' S = 2 F + V SB and Q d =C ox = p 2F + V SB The threshold voltage expression is also modified V T = V FB + 2 F + p 2F + V SB The change in threshold voltage is V T = V T (V SB ) V T (0) = (p 2F + V SB p 2 F )

40 Previous discussion assumes inversion region exists along the bulk at both source and drain sides. If a positive voltage is applied to the drain, electrons flow from source to drain. The positive drain voltage reduces the difference between the gate and drain voltages. There is a point at which V GD < V T and the inversion region disappears at the drain side. This is called channel pinch-off.

41 Figure: Saturation.

42 A depletion region is formed at the drain side. This depletion region, however, has little effect on the drain current The current is limited by the electron concentration at the pinch-off point. The electron concentration is controlled by the gate-to-source voltage. The drain current is controlled by the gate voltage. The MOSFET works like a voltage-controlled current source. This operating regime is called saturation region.

43 Very small devices are short-channel MOSFETs. In these devices the drain current saturates at smaller drain-to-source voltages. The small size of the device makes the electric field in the channel very large. Carrier mobility is reduced due to the large electric field. This can happen before channel pinch-off.

44 Figure: Types of MOSFET

45 Note: For p-channel MOSFETs use: V T = V FB 2 j F j p 2 j F j + j V SB j