• The source and body may not be at the same potential. The curve represented as EFi is the Fermi level from the p substrate through the reverse biased source–substrate junction to the source contact.
• The space charge region width under the oxide increases from the original xd @ threshold value when a VSB  > 0 is applied.  Since the maximum depletion region depth Xd was due to 2Ef it is now due to 2Ef + Vsb and the maximum Xd is bigger before Vt is reached.

@Threshold with:   

[pmath] V_SB = 0[/pmath]

[pmath] Q_Space = qN_A x_d_th [/pmath]

With Vsb applied

[pmath] V_SB = V_SB[/pmath]

[pmath] Q_Space = qN_A x_d_V_SB [/pmath]

And thus

[pmath] Delta Q_Space = qN_A [ x_d_V_SB – x_d_th ] [/pmath]

Substituting from the identity table below:

[pmath] Delta Q_Space = {sqrt{2}}qN_A ( sqrt { {2 phi_s + V_SB}/{Phi_s} } – sqrt { {2 phi_s}/{Phi_s} } ) [/pmath]

Factoring

[pmath] Delta Q_Space = {sqrt{2}qN_A} / sqrt { {Phi_s} } ( sqrt { { 2 phi_s + V_SB } } – sqrt { { 2 phi_s } } ) [/pmath]

The change in the threshold voltage is

[pmath] Delta V_T = – {  { Delta }{Q_Space}}/{C_ox}  [/pmath]

[pmath] Delta V_T = {{sqrt{2}}{qN_A }}/{C_ox circ sqrt { Phi_s }}[sqrt { 2 phi_s + V_SB  } – sqrt { 2 phi_s } ] [/pmath]

And here we find the body effect factor.  It is defined as:

[pmath] gamma ={sqrt{2}{qN_A }}/{C_ox circ sqrt { {Phi_s} }} [/pmath]

 

The following is my work and needs to be verified

[pmath] gamma ={sqrt{2}{qN_A / varepsilon_s}{varepsilon_s}}/{C_ox circ sqrt { {Phi_s} }} [/pmath]

Substituting Identity

[pmath] gamma ={sqrt{2}{{Phi_s}{varepsilon_s} } }/{{C_ox}{sqrt{ {Phi_s}} }} [/pmath]

Since:

[pmath] 1/t_ox = {{C_ox}/{varepsilon_s}} [/pmath]

The following simple expression results

[pmath] gamma = t_ox {{varepsilon_s}/{varepsilon_ox}} {sqrt{2 Phi_s }} [/pmath]

 

Identity Table

[pmath] Phi = {qN_a / varepsilon_s} [/pmath]	[pmath] E_surface = sqrt { {2 phi_s Phi_s} } [/pmath]
[pmath]  x_d = sqrt { {2 phi_s } / {Phi_s} } [/pmath]	[pmath] E_surface = Phi_s x_d [/pmath]

 

 

Research Links

• Introduction to Solid State Devices: Chapter 10   See page 6, Substrate Bias & Body Effect coefficient page 49  ( The whole book is here )
• PHP Math Publisher Syntax