MOS electrotatic: Quantitative analyi In thi cla, we will Derive analytical expreion for the charge denity, electric field and the electrotatic potential. xpreion for the depletion layer width Decribe delta depletion olution Derive gate voltage relationhip Gate voltage required to obtain inverion
lectrotatic potential, (x) Define a new term, (x) taken to be the potential inide the emiconductor at a given point x. [The ymbol intead of V ued in MOS work to avoid confuion with externally applied voltage, V] ( x) [ i (bulk) i q ( x)] Potential at any point x S q [ i (bulk) i (urface)] Surface potential F q [ i (bulk) F ] F related to doping concentration F > 0 mean p-type F < 0 mean n-type
lectrotatic parameter S i poitive if the band bend downward S F at the depletion-inverion tranition point 3
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xample Conider the following F and S parameter. Indicate whether the emiconductor i p-type or n-type, pecify the biaing condition, and draw the energy band diagram at the biaing condition. (i) F kt/q; S kt/q F + kt/q mean that i F in the emiconductor i kt (a poitive value); So, p-type. N A n i exp [( i F ) / kt] S kt/q mean i (bulk) i (urface) kt; i.e. the band bend downward near the urface. C kt i F V 5
xample (continued) (ii) F 9 kt/q; S 8 kt/q here F 9 kt/q mean [ i (bulk) F ] 9 kt; i.e., i F. Thu the emiconductor i n-type. i below S 8 kt/q mean that i (bulk) i (urface) 8 kt; So band bend upward near the urface. The urface i inverted ince the urface ha the ame number of hole a the bulk ha electron. C -9kT i F V 6
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Delta-depletion olution Conider p-type ilicon Accumulation condition V G < 0 M O S p- The accumulation charge are mobile hole, and appear cloe to the urface and fall-off rapidly a x increae. Aume that the free carrier concentration at the oxideemiconductor interface i a δ- function. Accumulation of hole x Charge on metal Q M Charge on emiconductor (charge on metal) Q Accumulation Q M 8
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Delta depletion olution (cont.) Conider p-type, depletion condition Apply V G uch that < F Charge in are immobile ion - reult in depletion layer imilar to that in pn junction or Schottky diode. V G > 0 M O S p- W q N A A W Q M ( ) (+) If urface potential i (with repect to the bulk), then the depletion layer width W will be qn A S and qna W Q M At the tart of inverion, F and W W w T qna Depletion of hole d/dx qn A / i x F /
Depletion layer width, W and -field For a p + n junction, or a MS (n-) junction, the depletion layer width i given by: / Where V W Vbi qn bi i related to the amount of band bending. V bi in Volt i numerically D equal to the amount of band bending in ev. max qn qn D W D V bi / For MOS, the ame equation applie, except that V bi i replaced by. max (in ) qn D n-type / or qn A p-type /
Delta depletion olution (cont.) Conider p-, trong inverion. Once inverion charge appear, they remain cloe to the urface ince they are mobile. Any additional voltage to the gate reult in extra Q M in gate and get compenated by extra inverion electron in emiconductor. V G >>0 Q M M O S w p- Depletion of hole Inverion electron: δ-function-like So, depletion layer doe not have to increae to balance the charge on the metal. lectron appear a δ-function near the urface. Maximum depletion layer width W W T 3
Gate voltage relationhip Applied gate voltage will be equal to the voltage acro the oxide plu the voltage acro the emiconductor. Conider p-type. V G ox + Semi Semi (x 0) (bulk) S ox x ox ox No drop in the metal!! V G > 0 M O S p- ox Semi (From what we learned in N 35, the boundary condition tate that a ( D D ) ρ nce the interface doe not have any charge up to inverion, we can ay that D ox D i 0 or ox ox ox ( / ox ) 4
Gate voltage relationhip (cont.) A F A A A 0 for / / qn qn qn W qn < < 5 F A ox ox ox ox ox ox G 0 for + + + / qn x x x V