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									               Lecture 7

               OUTLINE
• Poisson’s equation
• Work function
• Metal-Semiconductor Contacts
  – Equilibrium energy band diagrams
  – Depletion-layer width

    Reading: Pierret 5.1.2, 14.1-14.2; Hu 4.16
                     Poisson’s Equation
                                                                        area A
Gauss’ Law:
    s ( x  Dx) A   s ( x) A  DxA
                                                           E(x)        E(x+Dx)
                                                                  Dx

   ( x  Dx)   ( x)                             s : permittivity (F/cm)
               Dx          s                         : charge density (C/cm3)


  d  
     
       
  dx   s



EE130/230M Spring 2013          Lecture 7, Slide 2
Charge Density in a Semiconductor
• Assuming the dopants are completely ionized:

                          = q (p – n + ND – NA)




EE130/230M Spring 2013          Lecture 7, Slide 3
                         Work Function
                         E0: vacuum energy level




  FM: metal work function             FS: semiconductor work function


EE130/230M Spring 2013         Lecture 7, Slide 4
     Metal-Semiconductor Contacts
There are 2 kinds of metal-semiconductor contacts:
• rectifying
   “Schottky diode”




• non-rectifying
   “ohmic contact”




EE130/230M Spring 2013   Lecture 7, Slide 5
  Ideal M-S Contact: FM > FS, n-type


Band diagram instantly
after contact formation:




                                                     qVbi = FBn– (Ec – EF)FB
Equilibrium band diagram:                n

Schottky Barrier Height:
      F Bn  F M  
                                                 W

 EE130/230M Spring 2013     Lecture 7, Slide 6
  Ideal M-S Contact: FM < FS, n-type


Band diagram instantly
after contact formation:




Equilibrium band diagram:




 EE130/230M Spring 2013     Lecture 7, Slide 7
  Ideal M-S Contact: FM < FS, p-type
                                             p-type
                                             semiconductor

Band diagram instantly
after contact formation:




Equilibrium band diagram:
Schottky Barrier Height:    FBp
  F Bp    EG  F M                                qVbi = FBp– (EF – Ev)FB
                                                 W

 EE130/230M Spring 2013     Lecture 7, Slide 8
  Ideal M-S Contact: FM > FS, p-type
                                             p-type
                                             semiconductor
Band diagram instantly
after contact formation:




Equilibrium band diagram:



 EE130/230M Spring 2013     Lecture 7, Slide 9
    Effect of Interface States on FBn
                                               • Ideal M-S contact:
                                                  FBn = FM – 

                                               • Real M-S contacts:
                                                 A high density of
                                                 allowed energy states in
   FM                                            the band gap at the M-S
                                                 interface “pins” EF to be
         FBn
                                                 within the range 0.4 eV
                                                 to 0.9 eV below Ec




EE130/230M Spring 2013   Lecture 7, Slide 10
Schottky Barrier Heights: Metal on Si

            Metal         Er     Ti       Ni          W      Mo     Pt
            FM (eV)      3.12   4.3       4.7         4.6    4.6    5.6
            FBn (eV)     0.44   0.5      0.61         0.67   0.68   0.73
            FBp (eV)     0.68   0.61     0.51         0.45   0.42   0.39

 • FBn tends to increase with increasing metal work function




EE130/230M Spring 2013          Lecture 7, Slide 11
Schottky Barrier Heights: Silicide on Si

                     Silicide ErSi1.7 TiSi2   CoSi2      NiSi   WSi2   PtSi
                    FM (eV) 3.78 4.18       4.6 4.65 4.7    5
                    FBn (eV) 0.3      0.6 0.64 0.65 0.65 0.84
                    FBp (eV) 0.8      0.52 0.48 0.47 0.47 0.28

 Silicide-Si interfaces are more stable than metal-silicon
 interfaces and hence are much more prevalent in ICs.
 After metal is deposited on Si, a thermal annealing
 step is applied to form a silicide-Si contact. The term
 metal-silicon contact includes silicide-Si contacts.


 EE130/230M Spring 2013                 Lecture 7, Slide 12
        The Depletion Approximation
The semiconductor is depleted of mobile carriers to a depth W



 In the depleted region (0  x  W ):
                     = q (ND – NA)

  Beyond the depleted region (x > W ):
                    =0



 EE130/230M Spring 2013               Lecture 7, Slide 13
                          Electrostatics
• Poisson’s equation:               qN D
                                     
                                 x  s s




• The solution is:        x      qN D
                                       s
                                              W  x 



      V x     ( x)dx



EE130/230M Spring 2013                Lecture 7, Slide 14
                    Depletion Width, W
                                    qN D
                          V x           W  x 2
                                   2K S 0


   At x = 0, V = -Vbi

                              2 sVbi
                          W
                               qN D


   • W decreases with increasing ND



EE130/230M Spring 2013                  Lecture 7, Slide 15
Summary: Schottky Diode (n-type Si)
      metal       FM > FS     n-type Si



                                                         Eo

               Si
 FM


        FBn                 qVbi = FBn – (Ec – EFS)FB
                                                         Ec
                                                         EF

                                                              Depletion width

                                                                 2 sVbi
                                                         Ev   W
                     W
                                                                  qN D
EE130/230M Spring 2013             Lecture 7, Slide 16
Summary: Schottky Diode (p-type Si)
      metal        FM < FS   p-type Si
                                                       Eo


                                       Si


                                                       Ec
 FM




                                                       EF
                                                       Ev   Depletion width
         FBp                 qVbi = FBp– (EF – Ev)FB           2 sVbi
                                                            W
                     W
                                                                qN A
EE130/230M Spring 2013           Lecture 7, Slide 17

								
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