MOS Fundamentals by nikeborome

VIEWS: 52 PAGES: 74

									MOSFETs




          ECE 663
A little bit of history..




                            ECE 663
        Operation of a transistor
   VSG > 0
   n type operation

                                                   VSG
                                              Gate               VSD
                                            Insulator
                                     Source Channel      Drain
                       More
                       electrons
                                           Substrate




Positive gate bias attracts electrons into channel
Channel now becomes more conductive
 Operation of a transistor


                                             VSG
                                        Gate               VSD
                                      Insulator
                             Source    Channel     Drain

                                      Substrate




Transistor turns on at high gate voltage
Transistor current saturates at high drain bias
Start with a MOS capacitor


                             VSG

                         Gate              VSD
                       Insulator
                Source Channel     Drain



                       Substrate
MIS Diode (MOS capacitor) – Ideal




                               ECE 663
                   Questions

 What is the MOS capacitance? QS(yS)
W

 What are the local conditions during inversion?   yS,cr

 How does the potential vary with position?   y(x)
 How much inversion charge is generated at the surface? Qinv(x,yS)

 Add in the oxide: how does the voltage divide?    yS(VG), yox(VG)
 How much gate voltage do you need to invert the channel? VTH

 How much inversion charge is generated by the gate? Qinv(VG)

 What’s the overall C-V of the MOSFET?            QS(VG)
Ideal MIS Diode n-type, Vappl=0


                            Assume Flat-band
                            at equilibrium
                      qfS


                               EC
                               EF
                               Ei
                                EV




                                  ECE 663
Ideal MIS Diode n-type, Vappl=0




                    Eg      
  fms    fm         yB   0
                    2q      




                                     ECE 663
Ideal MIS Diode p-type, Vappl=0




                            ECE 663
Ideal MIS Diode p-type, Vappl=0




                     Eg      
   fms    fm         yB   0
                     2q      




                                  ECE 663
          Accumulation




Pulling in majority carriers at surface
                                          ECE 663
But this increases the barrier
      for current flow !!




        n+   p   n+




                            ECE 663
Depletion




            ECE 663
                          Inversion




                                   yB




Need CB to dip below EF.
Once below by yB, minority carrier density trumps the intrinsic density.
Once below by 2yB, it trumps the major carrier density (doping) !

                                                             ECE 663
Sometimes maths can help…




                        ECE 663
           P-type semiconductor Vappl0




Convention for p-type: y positive if bands bend down   ECE 663
                         Ideal MIS diode – p-type

            ( Ei' EF ) / kT
np  ni e                        ni e ( E qyE
                                          i        F   ) / kT
                                                                 np0e qy / kT  np0ey

   CB moves towards EF if y > 0  n increases


p p  p p 0 e qy / kT  p p 0e y


   VB moves away from EF if y > 0  p decreases

                                                     q
                                              
                                                    kT
                                                                             ECE 663
      Ideal MIS diode – p-type




At the semiconductor surface, y = ys




              ns  np0ey s




              ps  pp0e y   s




                                       ECE 663
           Surface carrier concentration
                      ys                     ys
         ns  np0e             ps  pp0e
                                                     EC
• ys < 0 - accumulation of holes                          EF


• ys =0 - flat band

• yB> ys >0 – depletion of holes

• ys =yB - intrinsic concentration ns=ps=ni

• ys > yB – Inversion (more electrons than holes)


                                                     ECE 663
Want to find y, E-field, Capacitance

• Solve Poisson’s equation to get E field, potential based on
  charge density distribution(one dimension)

                                          d
                                           E
                 
               E   / k 0   /  s      1 D
                                          dx
                    dy
             
             E 
                    dx
                 d 2y
              2   /  s
                 dx

                                 
                ( x )  q(ND  N A  p p  n p )


                                                          ECE 663
• Away from the surface,  = 0
                    
             ND  N A  n p 0  p p 0

• and

            p p  n p  p p 0e y  n p 0e y


          d 2y
                 p p 0 (e y  1)  n p 0 (e y  1)
                  q
        
          dx 2    s




                                                            ECE 663
           Solve Poisson’s equation:


    d 2y
           p p 0 (e y  1)  n p 0 (e y  1)
            q
  
    dx 2    s


    E = -dy/dx

    d2y/dx2 = -dE/dx
               = (dE/dy).(-dy/dx)
               = EdE/dy


    d 2y
EdE/dy 2   p p 0 (e y  1)  n p 0 (e y  1)
            q
  
    dx      s
                                                      ECE 663
                         Solve Poisson’s equation:

• Do the integral:
• LHS:
                x       x2     dy
                 xdx     x
                0       2      dx
• RHS:
                     x              x
                           x
                     e          dx,  dx
                     0              0


• Get expression for E field (dy/dx):



               kT   qp p 0   y                           
                    2

                                e  y  1      e  y  1
                                                 np 0 y
 E    2
               
               q   2 s  
     field
                                                 pp0             

                                                             ECE 663
Define:

                          kT s      s        Debye Length
                   LD            
                          pp 0q 2
                                    qp p 0
                                                                   1
              n                                             
           y, p 0   e y  y  1  p 0 e y  y  1
                                                                    2
                                           n
         F
              pp0  
                                          pp0                 

Then:
                                                 E>0

                       2kT       np0                    y>0
          Efield        F  y,
                            
                                       
                      qLD        pp 0 
                                       
                                                  E<0
        + for y > 0 and – for y < 0                      y<0
                                                                    ECE 663
   Use Gauss’ Law to find
   surface charge per unit
   area



                  2kT        n 
Qs   s ES         y s , p 0 
                     F
                 qLD         pp0 




                                                                     1
                       2kT  y                                 
                           e    y s  1      e  y s  1
                                                                      2
                                               np 0 y
               Qs                    S              s


                      qLD                     pp0               
                                                                          ECE 663
Accumulation to depletion to strong Inversion

• For negative y, first term in F dominates – exponential
• For small positive y, second term in F dominates - y

                      n p 0 e y
• As y gets larger,              1   second exponential gets big
                        pp0



  yB = (kT/q)ln(NA/ni) = (1/)ln(pp0/√pp0np0)

   (np0/pp0) = e-2yB

   yS > 2yB
                                                            ECE 663
                 Questions

 What is the MOS capacitance? QS(yS)

 What are the local conditions during inversion?   yS,cr

How does the potential vary with position?   y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)

Add in the oxide: how does the voltage divide?   yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH

How much inversion charge is generated by the gate? Qinv(VG)

What’s the overall C-V of the MOSFET?            QS(VG)
         Charges, fields, and potentials

• Charge on metal = induced surface charge in semiconductor
• No charge/current in insulator (ideal)
         metal   insul semiconductor




                                       depletion
                                inversion




      QM  Qn  qN AW  QS                             ECE 663
 Charges, fields, and potentials




Electric Field      Electrostatic Potential




                                       ECE 663
                        Depletion Region




            Electric Field               Electrostatic Potential



              kT   qp p 0   y                           
                   2

                               e  y  1      e  y  1
                                                np 0 y
E    2
              
              q   2 s  
    field
                                                pp0             

                                                            ECE 663
            Depletion Region




Electric Field          Electrostatic Potential


           y = ys(1-x/W)2
          Wmax = 2s(2yB)/qNA
           yB = (kT/q)ln(NA/ni)            ECE 663
                 Questions

 What is the MOS capacitance? QS(yS)

 What are the local conditions during inversion?   yS,cr

 How does the potential vary with position?   y(x)
How much inversion charge is generated at the surface? Qinv(x,yS)

Add in the oxide: how does the voltage divide?   yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH

How much inversion charge is generated by the gate? Qinv(VG)

What’s the overall C-V of the MOSFET?            QS(VG)
Couldn’t we just solve
    this exactly?
        Exact Solution

            U = y
            US = yS
            UB = yB


  dy/dx = -(2kT/qLD)F(yB,np0/pp0)
           U
          dU/F(U) =  x/L
                         D
          US


F(U) = [eUB(e-U-1+U)-e-UB (eU-1-U)]1/2
          Exact Solution



       = qni[eUB(e-U-1) – e-UB(eU-1)]



             US
            dU’/F(U’,U ) =  x/L
                      B         D
            U


F(U,UB) = [eUB(e-U-1+U) + e-UB (eU-1-U)]1/2
Exact Solution

   NA = 1.67 x 1015




                      Qinv ~ 1/(x+x0)a
                      x0 ~ LD . factor
                 Questions

 What is the MOS capacitance? QS(yS)

 What are the local conditions during inversion?   yS,cr

 How does the potential vary with position?   y(x)
 How much inversion charge is generated at the surface? Qinv(x,yS)

Add in the oxide: how does the voltage divide?   yS(VG), yox(VG)
How much gate voltage do you need to invert the channel? VTH

How much inversion charge is generated by the gate? Qinv(VG)

What’s the overall C-V of the MOSFET?            QS(VG)
Threshold Voltage for Strong Inversion

• Total voltage across MOS structure= voltage across
  dielectric plus ys

                                          QS
     VT (strong _ inversion)  Vi  y S      2y B
                                          Ci
                               2 s y s (inv )
  QS (SI )  qN AWmax  qN A                    2 s qN A (2y B )
                                   qN A


                    2 s qN A (2y B )
              VT                     2y B
                          Ci


                                                            ECE 663
         Notice Boundary Condition !!
oxVi/tox = sys/(W/2) Before Inversion

After inversion there is a discontinuity in D due to surface Qinv

Vox (at threshold) = s(2yB)/(Wmax/2)Ci =

                        2 s qN A (2y B )
                  VT                     2y B
                              Ci




                                                           ECE 663
                Local Potential vs Gate voltage
         VG = Vfb + ys + (kstox/kox)√(2kTNA/0ks)[ys + eys-2yB)]1/2


                                  yox
                                        ys




Initially, all voltage drops across channel (blue curve). Above threshold,
channel potential stays pinned to 2yB, varying only logarithmically, so that
most of the gate voltage drops across the oxide (red curve).
  Look at Effective charge width

           ~Wdm/2




                                   ~tinv




Initially, a fast increasing channel potential drops across
increasing depletion width

Eventually, a constant potential drops across a decreasing
inversion layer width, so field keeps increasing and thus
matches increasing field in oxide
                 Questions

 What is the MOS capacitance? QS(yS)

 What are the local conditions during inversion?   yS,cr

 How does the potential vary with position?   y(x)
 How much inversion charge is generated at the surface? Qinv(x,yS)

 Add in the oxide: how does the voltage divide?    yS(VG), yox(VG)
 How much gate voltage do you need to invert the channel? VTH

How much inversion charge is generated by the gate? Qinv(VG)

What’s the overall C-V of the MOSFET?           QS(VG)
 Charge vs Local Potential
      Qs ≈ √(20kskTNA)[ys + eys-2yB)]1/2




Beyond threshold, all charge goes to inversion layer
     How do we get the curvatures?




Add other terms and keep
Leading term               EXACT
      Inversion Charge vs Gate voltage
   Q ~ eys-2yB), ys- 2yB ~ log(VG-VT)
   Exponent of a logarithm gives a linear variation of Qinv with VG

                  Qinv = -Cox(VG-VT)




Why Cox?
                 Questions

 What is the MOS capacitance? QS(yS)

 What are the local conditions during inversion?   yS,cr

 How does the potential vary with position?   y(x)
 How much inversion charge is generated at the surface? Qinv(x,yS)

 Add in the oxide: how does the voltage divide?    yS(VG), yox(VG)
 How much gate voltage do you need to invert the channel? VTH

 How much inversion charge is generated by the gate? Qinv(VG)

What’s the overall C-V of the MOSFET?           QS(VG)
                    Capacitance

                                           y
                                                  1
                        y     np0                
                    1 e     
                                      pp0 
                   
                               s
                                             e  s


         QS                            
    CD       S                                    
          y   2LD                     np0 
                             F  y S ,
                               
                                            
                                       pp0 
                                            

For ys=0 (Flat Band):
                                          x2 x3
       Expand exponentials….. e  1  x        ........
                                x

                                          2! 3!
                                       S
                   CD (flat _ band ) 
                                       LD

                                                         ECE 663
         Capacitance of whole structure

• Two capacitors in series:


     Ci - insulator

     CD - Depletion




    1 1   1                          Ci CD
                       OR      C
    C Ci CD                         Ci  CD

                            i
                       Ci 
                            d                 ECE 663
Capacitance vs Voltage




                         ECE 663
                 Flat Band Capacitance

• Negative voltage = accumulation – C~Ci
• Zero voltage – Flat Band

           V  0  y  0  C  CFB

                                         i
                                      d  LD
     1   1   1  1  1  s d   i LD      s
                               
    CFB Ci CD  i  s     i s          i
                d LD

                                  i
                      CFB        
                               d   LD
                                   i

                                   s




                                               ECE 663
                         CV
• As voltage is increased, C goes through minimum
  (weak inversion) where dy/dQ is fairly flat

• C will increase with onset of strong inversion

• Capacitance is an AC measurement

• Only increases when AC period long wrt minority
  carrier lifetime

• At “high” frequency, carriers can’t keep up – don’t
  see increased capacitance with voltage

• For Si MOS, “high” frequency = 10-100 Hz
                                                   ECE 663
CV Curves – Ideal MOS Capacitor




                                               i
                             C    '
                                             
                                          d   Wmax
                                  min              i

                                               s




                                         ECE 663
But how can we operate gate at
today’s clock frequency (~ 2 GHz!)
if we can’t generate minority
carriers fast enough (> 100 Hz) ?

                               ECE 663
MOScap vs MOSFET




                   ECE 663
                 MOScap vs MOSFET

               Gate                               Gate
             Insulator                          Insulator
             Channel
                                      Source    Channel     Drain



             Substrate                          Substrate



Minority carriers generated by         Majority carriers pulled in
RG, over minority carrier lifetime     from contacts (fast !!)
~ 100ms

 So Cinv can be << Cox if fast gate            Cinv = Cox
 switching (~ GHz)

                                                            ECE 663
                      Example Metal-SiO2-Si

   •   NA = 1017/cm3
   •   At room temp kT/q = 0.026V
   •   ni = 9.65x109/cm3
   •   s = 11.9x1.85x10-14 F/cm



                   N 
         4 s kT ln A     11.9 x8.85 x10 14 X 0.026 ln1017 9.65 x109 
                   ni  
Wmax            2                        1.6 x10 19 X 1017
               q NA

                         Wmax  10 5 cm  0.1mm



                                                              ECE 663
                            Example Metal-SiO2-Si
• d=50 nm thick oxide=10-5 cm
• i=3.9x8.85x10-14 F/cm
     i 3.9 x8.85 x10 14
Ci             5
                           6.9 x10 7 F / cm 2
    d         10

                    2kT  N A                  1017 
y s (inv )  2y B     ln      2 x0.026 x ln             0.84Volts
                     q    ni                  9.65 x109 


                 i        3.9 x8.85 x10 14
C   '
                                                 9.1x10 8 F / cm 2
            d   Wmax 5 x10 7  3.9 11.910 5
                
    min          i

                 s

 '
Cmin
      0.13
Ci


            qN AWmax          1.6 x10 19 x1017 x10 5
VTH                  2y B                 7
                                                        y s (inv )  0.23  0.84  1.07Volts
               Ci                   6.9 x10
                                                                                 ECE 663
    Real MIS Diode: Metal(poly)-Si-SiO2 MOS

• Work functions of gate and semiconductor are
  NOT the same

• Oxides are not perfect
   – Trapped, interface, mobile charges
   – Tunneling


• All of these will effect the CV characteristic and
  threshold voltage




                                                ECE 663
Band bending due to work function difference




                                    VFB  fms

                                         ECE 663
           Work Function Difference

• qfs=semiconductor work function =
  difference between vacuum and Fermi level
• qfm=metal work function
• qfms=(qfm- qfs)
• For Al, qfm=4.1 eV
• n+ polysilicon qfs=4.05 eV
• p+ polysilicon qfs=5.05 eV
• qfms varies over a wide range depending on
  doping


                                        ECE 663
ECE 663
SiO2-Si Interface Charges




                            ECE 663
Standard nomenclature for Oxide charges:

QM=Mobile charges (Na+/K+) – can cause
    unstable threshold shifts – cleanliness
    has eliminated this issue

QOT=Oxide trapped charge – Can be anywhere
    in the oxide layer. Caused by broken
    Si-O bonds – caused by radiation damage
    e.g. alpha particles, plasma processes,
    hot carriers, EPROM


                                       ECE 663
QF= Fixed oxide charge – positive charge layer
    near (~2mm) Caused by incomplete
    oxidation of Si atoms(dangling bonds)
    Does not change with applied voltage

QIT=Interface trapped charge. Similar in origin
    to QF but at interface. Can be pos, neg,
    or neutral. Traps e- and h during device
    operation. Density of QIT and QF usually
    correlated-similar mechanisms. Cure
    is H anneal at the end of the process.

Oxide charges measured with C-V methods
                                      ECE 663
Effect of Fixed Oxide Charges




                                ECE 663
ECE 663
                 Surface Recombination




Lattice periodicity broken at surface/interface – mid-gap E levels
Carriers generated-recombined per unit area



                                                          ECE 663
              Interface Trapped Charge - QIT

• Surface states – R-G centers caused by disruption of lattice
  periodicity at surface
• Trap levels distributed in band gap, with Fermi-type distributed:

                        
                       ND           1
                          
                       ND 1  g D e ( E   F   ED ) / kT




• Ionization and polarity will depend on applied voltage (above or
  below Fermi level
• Frequency dependent capacitance due to surface recombination
  lifetime compared with measurement frequency
• Effect is to distort CV curve depending on frequency
• Can be passivated w/H anneal – 1010/cm2 in Si/SiO2 system



                                                            ECE 663
Effect of Interface trapped charge on C-V curve




                                          ECE 663
a – ideal
b – lateral shift – Q oxide, fms
c – distorted by QIT




                     ECE 663
           Non-Ideal MOS capacitor C-V curves

• Work function difference and oxide charges shift CV curve in
  voltage from ideal case

• CV shift changes threshold voltage

• Mobile ionic charges can change threshold voltage as a function of
  time – reliability problems

• Interface Trapped Charge distorts CV curve – frequency
  dependent capacitance

• Interface state density can be reduced by H annealing in Si-Si02

• Other gate insulator materials tend to have much higher
  interface state densities
                                                            ECE 663
                       All of the above….

• For the three types of oxide charges the CV curve is shifted
  by the voltage on the capacitor Q/C

                                    1 1 d           
          VFBoxide _ ch arg e            x( x )dx 
                                   Ci  d 0
                                                     

• When work function differences and oxide charges are
  present, the flat band voltage shift is:


                                    Qf    Qm  Qot 
                  VFB  fms 
                                            Ci



                                                          ECE 663
 Some important equations in the
 inversion regime (Depth direction)

VT = fms + 2yB + yox
                                          Gate
                                        Insulator

yox = Qs/Cox                     Source Channel     Drain


Qs = qNAWdm                            Substrate

Wdm = [2S(2yB)/qNA]
                                x
VT = fms + 2yB + ([4SyBqNA] - Qf + Qm + Qot)/Cox



Qinv = Cox(VG - VT)

								
To top