ECE66: Solid State evices Lecture 3 MOSFET I- Characteristics MOSFET non-idealities Gerhard Klimeck gekco@purdue.edu Outline 1) Square law/ simplified bulk charge theory ) elocity saturation in simplified theory 3) Few comments about bulk charge theory, small transistors 4) Flat band voltage - What is it and how to measure it? 5) Threshold voltage shift due to trapped charges 6) Conclusion Ref: Sec. 16.4 of SF Chapter 18, SF
Post-Threshold MOS Current ( G > th ) I W = S eff L µ ch Q i ( ) d Formula overview derivation to follow 1) Square Law ) Bulk Charge 3) Simplified Bulk Charge [ ] Q ( ) = C ) i G G T 4) Eact (Pao-Sah or Pierret-Shields) i G G T ( φ + ) qε SiN A B Q ( ) i = CG G FB ψ B C O Q ( ) = C ) [ m ] 3 Effect of Gate Bias S G GS W W M n+ n+ P B y BI GS > T ψ B No source-drain bias Gated doped or p-mos with adjacent n + region a) gate biased at flat-band b) gate biased in inversion A. Grove, Physics of Semiconductor evices, 1967. 4
The Effect of rain Bias band diagram for an n-mosfet a) device b) equilibrium (flat band) c) equilibrium (ψ S > ) d) non-equilibrium with G and > applied epletion very different in source and drain side Alam Gate voltage ECE-66 must ensure S9channel formation=> LARGE 5 SM. Sze, Physics of Semiconductor evices, 1981 and Pao and Sah. F N Effect of a Reverse Bias at rain W W M R R R GS > T ( R ) BI + R ψ S = ψ B + R Gated doped or p-mos with adjacent, reverse-biased n + region a) gate biased at flat-band b) gate biased in depletion c) gate biased in inversion A. Grove, Physics of Semiconductor evices, 1967. 6
S G n+ n+ G = = P B G = > Inversion Charge in the Channel Band diagrams (Gate Channel - rain) Q = C ( ) i G th + qn ( W ( ) W ( = ) ) B A T ue to drain bias, additional gate voltage (as compared to threshold in MOSCAP) is now needed to invert the channel throughout its length G > > T B Body potential = constant, n+ region potential lowered 7 Inversion Charge at one point in Channel G G B th qn AWT ( = ) = φf C * th qn AWT ( ) = ( φf + ) C ( W ) W = ) * qn A T ( T ( ) th = th + C Q C * i = ( G th ) Threshold voltage in the presence of drain bias 8
Apprimations for Inversion Charge ( ) Q = C ( ) + qn W ( ) W ( = ) i O G th A T T ( φ + ) κ ε ( φ ) = C ( ) O G th + qκ Sε on A q S on B A B Apprimations: Q C ( ) i G th Q C ( m ) i G th Square law apprimation Simplified bulk charge apprimation 9 The MOSFET S G S = > n+ n+ P F n = F p = E F B F n = F p q F n increasingly negative from source to drain (reverse bias increases from source to drain) 1
Elements of Square-law Theory oltage and hence inversion charge vary spatially G > y GCA : E y << E G G [ ] Q ( y) = C m ( y) i G th 11 Charge along the channel. n ( E F ) β ( E Fp ) β p = NCe C n = N e C p n
Charge along the channel n ( E F ) β ( E Fp ) β p = NCe C n = N e C G > epletion into the channel. n C n = N e C ( E F ) β p ( E F ) β p = N e C G N A n W T ( =) W T ( )
epletion into the channel G > epletion Another view of Channel Potential Source rain N+ N+ P-doped F P F P F N F N F P F N E C E F E 16
d J1 = Q1 µ E1 = Q1 µ dy d J = Q µ E = Q µ dy d J3 = Q3 µ E3 = Q3 µ dy d J4 = Q4 µ E4 = Q4 µ dy Jidy = µ i = 1, N i = 1, N J µ dy = C ( m ) d i = 1, N Q d i G th = ( G th ) Lch 1 3 4 µ C J m Q Square Law Theory G > Q 1 Q Q 3 Q 4 17 Square Law or Simplified Bulk Charge Theory µ C I = W ( G th ) m Lch di * = = ( G th ) m, sat = ( G th ) m d I SAT = ( GS T )/ m GS W µ Co I = ml ( ) G T ch Unphysical, since current does not decrease with increase in d µ C J = ( G th ) m Lch W I = µ C ( ) L o G T S Region of validity for the epression for currents 18
Why square law? And why does it become invalid SAT = ( GS T )/ m W µ Co I = ml ( ) G T ch Q C ( m ) i G th G > I GS Q S This situation doesn t arise since electrons travelling from left to right are swept into the drain under the effect of the reverse bias applied 19 Linear Region (Low S ) µ C I = W ( G th ) m Lch I W I = µ C ( ) o G T Lch = R S CH Slope gives mobility S small Actual Mobility degradation at high GS Subthreshold Conduction T Intercept gives T GS Can get T also from C-
Outline 1) Square law/ simplified bulk charge theory ) elocity saturation in simplified theory 3) Few comments about bulk charge theory, small transistors 4) Flat band voltage - What is it and how to measure it? 5) Threshold voltage shift due to trapped charges 6) Conclusion Ref: Sec. 16.4 of SF Chapter 18, SF 1 elocity vs. Field Characteristic (electrons) velocity cm/s ---> elocity saturates at high fields because of scattering 1 7 υ = µe 1 4 υ = υ sat Electric field /cm ---> υ µ E d = 1/ 1 + ( E Ec) µ E υd = 1 + ( E E ) υ µe d, sat = This epression can be used to re-derive the epression for current since since mobility is now, in principle, a function of distance c c
Recap - derivation for MOSFET current G > d J1 = Q1 µ 1E1 = Q1 µ 1 dy d J = Q µ E = Q µ dy d J3 = Q3 µ 3E3 = Q3 µ 3 dy d J 4 = Q4 µ 4E4 = Q4µ 4 dy 1 3 4 Jidy = µ ( y) i= 1, N i= 1, N Q d i 3 elocity Saturation dy = 1, G th i N E µ 1 + Ec J = C ( m ) d G > 1 L ch d m dy 1 + = C ( G th ) c dy J µ E L ch S J m J dy + d = C ( G th ) Ec µ C m J = ( ) G th Lch + E c Q υ v sat 4
Significance of the new epression µ C m J = ( ) G th Lch + E c At very small channel lengths and high drain biases, the current epression becomes independent of the channel length In the linear region in the I- d characteristics, you have a resistance that doesn t depend on the length of the channel 5 Calculating SAT di d S = I µ C W L o = ( G th ) m + ch Ec Take log on both sides and then set the derivative to zero. SAT = ( ) / m G th 1+ 1+ µ o ( G th ) mυ sat L ch < GS T ( ) m 6
µ C m J = ( ), sat G th, sat, sat L + ch EC elocity Saturation in short channel devices, sat I This epression can be derived by plugging in the value of d,sat for the short channel regime ( ) I = WC υ sat G T GS S 7 Linear Law Epression at the limit of L --> SAT ( G th ) ( ) / m = 1 + 1 + µ mυ L G th sat ch ( ) υ L mµ SAT sat ch G th ( ) I = W C υ SAT sat G th ( ) I = W C υ SAT sat G th ( ) ( ) 1+ µ mυ L 1 G th sat ch 1+ µ mυ L + 1 G th sat ch Complete velocity saturation Current independent of L 8
Signature of elocity Saturation I I GS GS ( ) W G I = µ C L m ch S S th ( ) I = Wυ C sat G th Can pull out ide thickness from eperimental curves How? 9 I and ( GS - T ): In practice.. I I α ( = ) ~ ( ) G th GS 1 < α < S Complete velocity saturation Long channel 3
Outline 1) Square law/ simplified bulk charge theory ) elocity saturation in simplified theory 3) Few comments about bulk charge theory, small transistors 4) Flat band voltage - What is it and how to measure it? 5) Threshold voltage shift due to trapped charges 6) Conclusion Ref: Sec. 16.4 of SF Chapter 18, SF 31 Apprimations for Inversion Charge ( ) Q = C ( ) + qn W ( ) W ( = ) i O G th A T T ( ) ( ) = C ( ) + qκ ε N φ + qκ ε N φ O G th S o A B S o A B Apprimations: Q C ( ) i G th Q C ( m ) i G th Square law apprimation Simplified bulk charge apprimation One could substitute the epression for Q i above eplicitly instead of using m to simplify the equation, resulting in a more complete bulk charge epression 3
J µ dy = C ( ) d + [...] d i= 1, N O G th Complete Bulk-charge Theory Additional dependent terms abstracted into m previously J µ L ch dy = C ( ) d + [...] d J O G th 3 / µ C 4 3 qn AWT = ( G th ) φf 1+ 1+ Lch 3 CO φ F 4φ F (Eq. 17.8 in SF). Eplicit dependence on bulk doping 33 elocity Overshoot υ µ n Average velocity (cm/s). 1 7 1.5 1 7 1. 1 7 5. 1 6 ( E)E 1 3 /cm 1 5 /cm 1 3 /cm υ sat. 1.5 1 1.5 Position (µm).35.3.5..15.1.5 Kinetic energy per electron (e) alid for bulk semiconductors, not valid at top of the barrier 34
elocity Overshoot in a MOSFET 35 Intermediate Summary 1) elocity saturation is an important consideration for short channel transistors (e.g., =1, L ch =nm). Therefore, α ~ 1 for most modern transistors. ) Bulk charge theory eplains why MOSFET current depends on substrate (bulk) doping. In the simplified bulk charge theory, doping dependence is encapsulated in m. 3) Additional considerations of velocity overshoot could complicate calculation of current. 4) Good news is that for very short channel transistors, electrons travel from source to drain without scattering. A considerably simpler Lundstrom theory of MOSFET applies. 36
Outline 1) Square law/ simplified bulk charge theory ) elocity saturation in simplified theory 3) Few comments about bulk charge theory, small transistors 4) Flat band voltage - What is it and how to measure it? 5) Threshold voltage shift due to trapped charges 6) Conclusion I ( = ) ~ ( G th ) α 1 < α < Ref: Sec. 16.4 of SF Chapter 18, SF th γ MQM QF QIT ( φs ) = th, ideal + φms C C C O O O 37 (1) Idealized MOS Capacitor In the idealized MOS capacitor, the Fermi Levels in metal and semiconductor align perfectly so that at zero applied bias, the energy bands are flat acuum level y Substrate (p) χ i χ s Φ m E C Recall that Q = C ( ) i G th, ideal E F th, ideal Q = ψ s B C ψ s = φ F metal insulator p semiconductor 38
Potential, Field, Charges χ i χ s Φ m E bi = ρ No built in potential, fields or charges at zero applied bias in the idealized MOS structure 39 Real MOS Capacitor with Φ M < Φ S E AC Note the difference Φ M = qφ m χ S Φ S qψ S > E C E F E C E F E E In reality, the metal and semiconductor Fermi Levels are never aligned perfectly when you bring them together there is charge transfer from the bulk of the semiconductor to the surface so that we have alignment o we need to apply less or more G to invert the channel? 4
G = Physical Interpretation of Flatband oltage The Flatband oltage is the voltage applied to the gate that gives zero-band bending in the MOS structure. Applying this voltage nullifies the effect of the built-in potential. This voltage needs to be incorporated into the idealized MOS analysis while calculating threshold voltage = φ = < FB ms bi E C E F G = FB < E C E E F E bi = φ ms > + ψ S = flat band 41 How to Calculate Built-in or Flat-band oltage The presence of a flatband voltage lowers or raises the threshold voltage of a MOS structure. Engineering question Is it desirable to have a metal having a work function greater or less than the electron affinity+(ec-ef) in the semiconductor? q = q ( χs E ) = + Φ bi g p FB φ MS M q bi acuum level χ s Therefore, Q = C ( ) i G th Φ m E C E F E th Q B = φ F C FB 4
Measure of Flat-band shift from C- Characteristics The transition point between accumulation and depletion in a non-ideal MOS structure is shifted to the left when the metal work function is smaller that the electron affinity +(Ec-Ef). At zero applied bias the semiconductor is already depleted so that a very small positive bias inverts the channel. The flatband voltage is the amount of voltage required to shift the curve such that the transition point is at zero bias. C/C Ideal th Actual G th 43 Outline 1) Square law/ simplified bulk charge theory ) elocity saturation in simplified theory 3) Few comments about bulk charge theory, small transistors 4) Flat band voltage - What is it and how to measure it? 5) Threshold voltage shift due to trapped charges 6) Conclusion Ref: Sec. 16.4 of SF Chapter 18, SF γ Q Q Q ( φ ) = + φ M M F IT s C C C th th, ideal MS 44
() Idealized MOS Capacitor acuum level y Substrate (p) χ i χ s Φ m Recall that Q = C ( ) i G th, ideal Q = E C E F E th, ideal Q = ψ s B C ψ s = φ F metal insulator p semiconductor 45 istributed Trapped charge in the Oide O X O E C QM = ρ( ) In the absence of charges in the ide, the field is constant (d/d = constant). The presence E of a charge distribution inside F the ide changes the field inside the ide and effectively traps field lines comping from the gate. As a result, depending on E the ρ polarity of charges in the ie, the threshold voltage Mis modified. γ M = ρ d ( ) d ( ) d th QF = ψ S γ C M Q C M m represents the centroid of the charge distribution one can think of this as replacing the entire distribution with a delta charge at this point 46
An Intuitive iew Reduced gate charge Ideal charge-free ide Bulk charge -E -E Interface charge -E 47 Gate oltage and Oide Charge G = +ψ s Kirchoff s Law balancing voltages d de ρ( ) = = d d κ ε E ( ) E ( ) de = d d =E ( ) E ( ) =E ( ) ρ ( ')d ' κ ε ρ( ') d' κ ε Known from boundary conditions in semiconductor and continuity of E = κ S κ E S ( ) = κ S κ E S ( ) ρ d ( ')d ' κ ε ρ ( )d κ ε -E 48
Gate oltage and Oide Charge S ( ) = κ ρ E S ( ) κ o κε = ψ ( = φ ) + th s F κ S 1 = ψ s = φf ) + E S ( ρ κ C κ S 1 = E S κ C ( ) ( ) d d ( ) ρ ( ) d 1 = th, ideal ( ) d C Q = th, ideal C ρo M γ M 49 Interpretation for Bulk Charge C/C 1 1 th = th, ideal ρ( ) δ ( 1 ) d C = th, ideal o QM ( 1 ) C o New T Ideal T G 5
Interpretation for Interface Charge C/C * 1 th = th ρ( ) δ ( o ) d C o * Q = th C F o New T Ideal T G 51 Time-dependent shift of Trapped Charge C/C E 1 = Q ( ) δ ( ( t)) d th th, ideal 1 C ( t) Q ( ) C 1 = th, ideal Ideal T G Sodium related bias temperature instability (BTI) issue 5
Bias Temperature Instability (Eperiment) M O S - --- - + + + (-) biases + M O S + (+) biases + + - ρ ion ρ ion.1 o o.9 o o 53 Conclusion 1) Non-ideal threshold characteristics are important consideration of MOSFET design. ) The non-idealities arise from differences in gate and substrate work function, trapped charges, interface states. 3) Although nonindeal effects often arise from transistor degradation, there are many cases where these effects can be used to enhance desirable characteristics. 54