Chapter 8 THE FIELD EFFECT TRANSISTOR

Example 8_1 pgno: 267

In [1]:
#exa 8.1
Nd =10**16
print"Nd= ",Nd," /cmˆ3" # initializing value of donor ion concentration .
Er =3.9
print"Er= ",Er # initializing value of relative dielectric permittivity constant .
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
W=0.5*10**-4
print"W = ",W," cm" # initializing value of width of p−substrate .
L=10**-4
print"L = ",L," cm" # initializing value of length of p−substrate .
tox =400*10**-8
print"tox = ",tox," cm" # initializing value of thickness of p−substrate .
E=Eo*Er
print" total permittivity ,E=Eo∗Er=",E," F/cm"# calculation
Cox=(E*W*L)/tox
print"Oxide capacitance ,Cox=(E∗W∗L)/tox)=",Cox," F"# calculation
Co=(Cox/(W*L))
print"Capacitance per unit area ,Co=(Cox/(W∗L)))=",Co," F/cmˆ2"# calculation
Nd=  10000000000000000  /cmˆ3
Er=  3.9
Eo =  8.854e-14  F/cm
W =  5e-05  cm
L =  0.0001  cm
tox =  4e-06  cm
 total permittivity ,E=Eo∗Er= 3.45306e-13  F/cm
Oxide capacitance ,Cox=(E∗W∗L)/tox)= 4.316325e-16  F
Capacitance per unit area ,Co=(Cox/(W∗L)))= 8.63265e-08  F/cmˆ2

Example 8_2 pgno: 267

In [2]:
#exa 8.2
from math import log
from math import sqrt
Na =10**17
print"Na = ",Na," /cmˆ3" # initializing value of acceptor ion concentration .
Vt =0.0259
print"Vt = ",Vt,"V" # initializing value of thermal voltage .
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
ni=1.5*10**10
print"ni = ",ni,"/cmˆ3" #initializing value of intrinsic carrier concentration .
Er=11.9
print"Er = ",Er # initializing value of relative dielectric permittivity constant
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
Vs=Vt*log(Na/ni)
print"Vs=Vt∗log(Na/ni))=",Vs," V"#calculation
E=Eo*Er
print" total permittivity ,E=Eo∗Er=",E," F/cm" # calculation
Wd=sqrt(4*E*Vs/(e*Na))
print"maximum depletion width ,Wd(max)=Sqrt(4∗E∗Vs/(e∗Na)))=",Wd," cm"#calculation
Na =  100000000000000000  /cmˆ3
Vt =  0.0259 V
e =  1.6e-19  columns
ni =  15000000000.0 /cmˆ3
Er =  11.9
Eo =  8.854e-14  F/cm
Vs=Vt∗log(Na/ni))= 0.40695713106  V
 total permittivity ,E=Eo∗Er= 1.053626e-12  F/cm
maximum depletion width ,Wd(max)=Sqrt(4∗E∗Vs/(e∗Na)))= 1.03535092381e-05  cm

Example 8_3 pgno: 268

In [3]:
#exa 8.3
from math import sqrt
from math import log
Nd =3*10**18
print"Nd = ",Nd," /cmˆ3" # initializing value of acceptor ion concentration .
Vt =0.0259
print"Vt = ",Vt,"V" # initializing value of thermal voltage .
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
ni=1.5*10**10
print"ni = ",ni,"/cmˆ3" #initializing value of intrinsic carrier concentration .
Er=11.9
print"Er = ",Er # initializing value of relative dielectric permittivity constant
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
Vs=Vt*log(Nd/ni)
print"Vs=Vt∗log(Nd/ni))=",Vs," V"# calculation
E=Eo*Er
print" total permittivity ,E=Eo∗Er=",E," F/cm" # calculation
Wd=sqrt(4*E*Vs/(e*Nd))
print"maximum depletion width ,Wd(max)=Sqrt(4∗E∗Vs/(e∗Nd)))=",Wd," cm"#calculation
Nd =  3000000000000000000  /cmˆ3
Vt =  0.0259 V
e =  1.6e-19  columns
ni =  15000000000.0 /cmˆ3
Er =  11.9
Eo =  8.854e-14  F/cm
Vs=Vt∗log(Nd/ni))= 0.495048143245  V
 total permittivity ,E=Eo∗Er= 1.053626e-12  F/cm
maximum depletion width ,Wd(max)=Sqrt(4∗E∗Vs/(e∗Nd)))= 2.08485729922e-06  cm

Example 8_5 pgno: 269

In [4]:
#exa 8.5
from math import log
Vm =3.2
print"Vm = ",Vm," V" # initializing value of modified metal work function .
X=3.25
print"X = ",X," V" # initializing value of modified electron affinity .
Nd =2*10**16
print"Nd = ",Nd," /cmˆ3" # initializing value of donor concentration .
ni=1.5*10**10
print"ni = ",ni," V" # initializing value of intrinsic carrier concentration .
Vt=0.0259
print"Vt = ",Vt,"V" # initializing value of thermal voltage .
Eg=1.12
print"Eg = ",Eg,"V" # initializing value of energy gap .
Vfp=(Vt*log(Nd/ni))
print"Vfp=(Vt∗log(Nd/ni))=",Vfp," V" # calculation .
Vms=-(Vm+(Eg/2)+Vfp-Vm)
print"Vms=−(Vm+(Eg/2)+Vfp−Vm)=",Vms," V" # calculation .
Vm =  3.2  V
X =  3.25  V
Nd =  20000000000000000  /cmˆ3
ni =  15000000000.0  V
Vt =  0.0259 V
Eg =  1.12 V
Vfp=(Vt∗log(Nd/ni))= 0.365272689128  V
Vms=−(Vm+(Eg/2)+Vfp−Vm)= -0.925272689128  V

Example 8_7 pgno: 270

In [5]:
#exa 8.7
Nd =10**16
print"Nd = ",Nd," /cmˆ3" # initializing value of donor ion concentration .
Vms = -1.12
print"Vms = ",Vms," V" # initializing value of metal semiconductor work function difference .
Er =3.9
print"Er = ",Er # initializing value of relative dielectric permittivity constant .
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
tox =200*10**-8
print"tox = ",tox," cm" # initializing value of thickness of p−type substrate .
Qss =2.5*10**-8
print"Qss = ",Qss," columbs/cmˆ2" # initializing value of charge density on semiconductor surface .
Eox=Eo*Er
print"Total permittivity ,Eox=Eo∗Er=",Eox," F/cm"# calculation
Co=(Eox/tox)
print"Capacitance per unit area ,Co=(E/tox))=",Co," F/cmˆ2"# calculation
Vfb=(Vms-(Qss/Co))
print" Flat band voltage , Vfb=(Vms−(Qss/Co) ) )=",Vfb," V"# calculation
#the answer for Co after calculation is provided wrong in the book 
Nd =  10000000000000000  /cmˆ3
Vms =  -1.12  V
Er =  3.9
Eo =  8.854e-14  F/cm
tox =  2e-06  cm
Qss =  2.5e-08  columbs/cmˆ2
Total permittivity ,Eox=Eo∗Er= 3.45306e-13  F/cm
Capacitance per unit area ,Co=(E/tox))= 1.72653e-07  F/cmˆ2
 Flat band voltage , Vfb=(Vms−(Qss/Co) ) )= -1.26479910572  V

Example 8_9 pgno: 271

In [6]:
#exa 8.9
from math import log
Na =3*10**16
print"Na = ",Na," /cmˆ3" # initializing value of acceptor ion concentration .
Vms = -1.12
print"Vms = ",Vms,"V" # initializing value of metal semiconductor work function difference .
Er =11.9
print"Er = ",Er # initializing value of relative dielectric permittivity constant .
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
ni=1.5*10**10
print"ni = ",ni,"cmˆ−3" # initializing value of intrinsic concentration of electrons .
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
tox =300*10**-8
print"tox = ",tox," cm" # initializing value of thickness of p−type substrate .
Vfb=-1.12
print"Vfb = ",Vfb," V" # initializing value of flat band voltage .
Qss=10**11
print"Qss = ",Qss," electronic charge columns/cmˆ2" # initializing value of charge density on semiconductor surface .
Vt=0.0259
print"Vt = ",Vt," eV" # initializing value of thermal voltage .
er=3.9
print"er = ",er # initializing value of relative dielectric permittivity constant
Eox=Eo*Er
print"total permittivity ,Eox=Eo∗Er=",Eox," F/cm"# calculation
Vfp=Vt*(log(Na/(ni)))
print"Potential ,Vfp=Vt∗(log(Na/(ni))))=",Vfp," V"#calculation
Wd=sqrt((4*Eox*Vfp)/(e*Na))
print"Maximum depletion width ,Wd=sqrt ((4∗E∗Vs)/(e∗Nd)))=",Wd," cm"#calculation
QDmax=(e*Na*Wd)
print"Over all maximum depletion width ,QDmax=(e∗Na∗ Wd) )=",QDmax," columns/cmˆ2" # calculation
VT=(((QDmax -1.6*10**-8)*tox)/(er*Eo))+(2*Vfp+Vfb)
print "Threshold Voltage ,VT=(((QDmax−1.6∗10ˆ−8)∗tox)/(er∗Eo),(2∗Vfp+Vfb)=",VT," V" # calculation
Na =  30000000000000000  /cmˆ3
Vms =  -1.12 V
Er =  11.9
Eo =  8.854e-14  F/cm
ni =  15000000000.0 cmˆ−3
e =  1.6e-19  columns
tox =  3e-06  cm
Vfb =  -1.12  V
Qss =  100000000000  electronic charge columns/cmˆ2
Vt =  0.0259  eV
er =  3.9
total permittivity ,Eox=Eo∗Er= 1.053626e-12  F/cm
Potential ,Vfp=Vt∗(log(Na/(ni))))= 0.375774235428  V
Maximum depletion width ,Wd=sqrt ((4∗E∗Vs)/(e∗Nd)))= 1.81641933617e-05  cm
Over all maximum depletion width ,QDmax=(e∗Na∗ Wd) )= 8.71881281361e-08  columns/cmˆ2
Threshold Voltage ,VT=(((QDmax−1.6∗10ˆ−8)∗tox)/(er∗Eo),(2∗Vfp+Vfb)= 0.250027108378  V

Example 8_10 pgno: 271

In [7]:
#exa 8.10
L=1.25*10**-4
print"L = ",L," cm" # initializing value of length of channel .
un =600
print"un = ",un,"cmˆ2/V−s" # initializing value of mobility of n−channel MOS transistor .
Co =6.9*10**-9
print"Co = ",Co,"F/cmˆ2" # initializing value of capacitance per unit area .
VT =0.60
print"VT = ",VT," V" # initializing value of threshold Voltage .
Vgs=4
print"Vgs = ",Vgs," V" # initializing value of gate to source voltage .
W=12*10**-4
print"W = ",W,"cm" # initializing value of width of channel .
Id=((Co*un*W)/(L)*((Vgs-VT)**2/(2)))
print"Drain current ,Id=((Co∗un∗W)/(L)∗((Vgs−VT)ˆ2/(2)))=",Id," A"#calculation .
#The answer provided in the book (for Id) is wrong as the value of mobility used for solution is different than provided in the question and also value of (Vgs−Vt) is put wrong in the solution than given in the book .
#I have used the value given in the question i.e. answer differ .
L =  0.000125  cm
un =  600 cmˆ2/V−s
Co =  6.9e-09 F/cmˆ2
VT =  0.6  V
Vgs =  4  V
W =  0.0012 cm
Drain current ,Id=((Co∗un∗W)/(L)∗((Vgs−VT)ˆ2/(2)))= 0.00022972032  A

Example 8_13 pgno: 273

In [8]:
#exa 8.13
from math import sqrt
from math import log
Na =2*10**17
print"Na = ",Na," /cmˆ3" # initializing value of acceptor ion concentration .
Er =11.9
print"Er = ",Er # initializing value of relative dielectric permittivity constant .
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
ni=1.5*10**10
print"ni = ",ni,"cmˆ−3" # initializing value of intrinsic concentration of electrons .
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
tox =400*10**-8
print"tox = ",tox," cm" # initializing value of thickness of p−type substrate .
Vt=0.0259
print"Vt = ",Vt," eV" # initializing value of thermal voltage .
er=3.9
print"er = ",er  # initializing value of relative dielectric permittivity constant
Vfp=Vt*(log(Na/(ni)))
print"Potential ,Vfp=Vt∗(log(Na/(ni))))=",Vfp," V"#calculation
Wd=sqrt((4*Er*Eo*Vfp)/(e*Na))
print"Depletion width ,Wd=sqrt ((4∗Er∗Eo∗Vs)/(e∗Nd)))=",Wd," cm"# calculation
CTmin=(er*Eo/(((er/Er)*(Wd))+(tox)))
print"Minimum Capacitance,CTmin=(er∗Eo/((er/Er)∗(Wd)+(tox)))=",CTmin," F/cmˆ2"#calculation
CFB=((er*Eo)/((((er/Er)*sqrt(Vt*Er*Eo/(e*Na))))+(tox)))
print"Flat band capacitance ,CFB=((er∗Eo) /((( er/Er)∗sqrt(Vt∗Er∗Eo/(e∗Na))))+(tox))=",CFB," F/ cmˆ2"# calculation
#the value of Na (acceptor ion concentration) and tox ( thickness of p−type substrate ) is provided different in the question than used in the solution .
#I have used the value provided in the solution .( i . e Na=2∗10ˆ17 and tox =400∗10ˆ8cm)
Na =  200000000000000000  /cmˆ3
Er =  11.9
Eo =  8.854e-14  F/cm
ni =  15000000000.0 cmˆ−3
e =  1.6e-19  columns
tox =  4e-06  cm
Vt =  0.0259  eV
er =  3.9
Potential ,Vfp=Vt∗(log(Na/(ni))))= 0.424909643036  V
Depletion width ,Wd=sqrt ((4∗Er∗Eo∗Vs)/(e∗Nd)))= 7.48077408723e-06  cm
Minimum Capacitance,CTmin=(er∗Eo/((er/Er)∗(Wd)+(tox)))= 5.35218545918e-08  F/cmˆ2
Flat band capacitance ,CFB=((er∗Eo) /((( er/Er)∗sqrt(Vt∗Er∗Eo/(e∗Na))))+(tox))= 8.02543256028e-08  F/ cmˆ2

Example 8_14 pgno: 274

In [9]:
#exa 8.14
Vfb = -1.0
print"Vfb = ",Vfb," V" # initializing value of flat band voltage .
Vms = -0.9
print"Vms = ",Vms,"V" # initializing value of metal semiconductor work function difference .
tox =200*10**-8
print"tox = ",tox," cm" # initializing value of gate oxide thickness .
et =3.9
print"et = ",et # initializing value of relative permittivity .
eo =8.85*10**-14
print"eo = ",eo,"F/cm" # initializing value of free space permittivity .
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
eox=(eo*et)
print"eox=(eo∗et))=",eox," F/cmˆ2" # calculation
Cox=(eox/tox)
print"Oxide capacitance ,Cox=(eox/tox))=",Cox," F/cmˆ2"# calculation
Qss=((Vms-Vfb)*Cox)
print"charge density on semiconductor surface ,Qss=(( Vms−Vfb)∗Cox))=",Qss," C/cmˆ2" # calculation
Qss1=Qss/e
print"charge density on semiconductor surface (in terms of number of charges) ,Qss∗=Qss/e)=",Qss1," electrons/cmˆ2" #calculation
Vfb =  -1.0  V
Vms =  -0.9 V
tox =  2e-06  cm
et =  3.9
eo =  8.85e-14 F/cm
e =  1.6e-19  columns
eox=(eo∗et))= 3.4515e-13  F/cmˆ2
Oxide capacitance ,Cox=(eox/tox))= 1.72575e-07  F/cmˆ2
charge density on semiconductor surface ,Qss=(( Vms−Vfb)∗Cox))= 1.72575e-08  C/cmˆ2
charge density on semiconductor surface (in terms of number of charges) ,Qss∗=Qss/e)= 1.07859375e+11  electrons/cmˆ2

Example 8_15 pgno: 274

In [1]:
#exa 8.15
L=3e-6
print"L = ",L," meter" # initializing value of length of channel .
un =800.
print"un = ",un,"cmˆ2/V−s" # initializing value of mobility of n−channel MOS transistor .
VT=1.
print"VT = ",VT," V" # initializing value of threshold Voltage .
Vgs=0
print"Vgs = ",Vgs," V" # initializing value of gate to source voltage .
tox =500e-8
print"tox = ",tox," cm" # initializing value of gate oxide thickness .
et=3.9
print"et = ",et # initializing value of relative permittivity .
eo =8.85e-14
print"eo = ",eo,"F/cm" # initializing value of free space permittivity .
W=30e-6
print"W = ",W,"m" # initializing value of width of channel .
eox=(eo*et)
print"eox=(eo∗et))=",eox," F/cmˆ2"# calculation
Id=((eox*un*W)/(tox*L)*((Vgs-VT)**2/(2)))*(1e9)
print"Drain current ,Id=((eox∗un∗W)/(tox∗L)∗((Vgs−VT)ˆ2/(2))))=",Id," A"#calculation
L =  3e-06  meter
un =  800.0 cmˆ2/V−s
VT =  1.0  V
Vgs =  0  V
tox =  5e-06  cm
et =  3.9
eo =  8.85e-14 F/cm
W =  3e-05 m
eox=(eo∗et))= 3.4515e-13  F/cmˆ2
Drain current ,Id=((eox∗un∗W)/(tox∗L)∗((Vgs−VT)ˆ2/(2))))= 276120.0  A

Example 8_16 pgno: 274

In [11]:
#exa 8.16
L=2.5*10**-6
print"L = ",L," meter" # initializing value of length of channel .
un =800
print"un = ",un,"cmˆ2/V−s" # initializing value of mobility of n−channel MOS transistor .
VT =0.8
print"VT = ",VT," V" # initializing value of threshold Voltage .
Vgs=1
print"Vgs = ",Vgs," V" # initializing value of gate to source voltage .
tox =400*10**-8
print"tox = ",tox," cm" # initializing value of gate oxide thickness .
et=3.9
print"et = ",et # initializing value of relative permittivity .
eo =8.85*10**-14
print"eo = ",eo,"F/cm" # initializing value of free space permittivity .
eox=(eo*et)
print"eox=(eo∗et))=",eox," F/cmˆ2" # calculation
W=25*10**-6
print"W = ",W,"m" # initializing value of width of channel . .
Id=((eox*un*W)/(tox*L)*((Vgs-VT)**2/(2)))
print"Drain current ,Id=((eox∗un∗W)/(tox∗L)∗((Vgs−VT)ˆ2/(2))))=",Id," A"#calculation
L =  2.5e-06  meter
un =  800 cmˆ2/V−s
VT =  0.8  V
Vgs =  1  V
tox =  4e-06  cm
et =  3.9
eo =  8.85e-14 F/cm
eox=(eo∗et))= 3.4515e-13  F/cmˆ2
W =  2.5e-05 m
Drain current ,Id=((eox∗un∗W)/(tox∗L)∗((Vgs−VT)ˆ2/(2))))= 1.3806e-05  A

Example 8_17 pgno: 274

In [12]:
#8.17
un =525
print"un = ",un,"cmˆ2/V−s" # initializing value of mobility of n−channel MOS transistor .
VT =0.75
print"VT = ",VT," V" # initializing value of threshold Voltage .
Vgs=2
print"Vgs = ",Vgs," V" # initializing value of gate to source voltage .
tox =400*10**-8
print"tox = ",tox," cm" # initializing value of gate oxide thickness .
et=3.9
print"et = ",et # initializing value of relative permittivity .
eo =8.85*10**-14
print"eo = ",eo,"F/cm" # initializing value of free space permittivity .
eox=(eo*et)
print"eox=(eo∗et))=",eox," F/cmˆ2" # calculation
Id =6*10**-3
print"Id = ",Id,"A" # initializing value of width of channel . .
X=((Id*tox*2)/(eox*un*((Vgs-VT)**2)))
print"width to length ratio ,W/L=((Id∗tox∗2)/(eox∗un∗((Vgs−VT)ˆ2)))= ",X # calculation
un =  525 cmˆ2/V−s
VT =  0.75  V
Vgs =  2  V
tox =  4e-06  cm
et =  3.9
eo =  8.85e-14 F/cm
eox=(eo∗et))= 3.4515e-13  F/cmˆ2
Id =  0.006 A
width to length ratio ,W/L=((Id∗tox∗2)/(eox∗un∗((Vgs−VT)ˆ2)))=  169.532915296

Example 8_18 pgno: 275

In [13]:
#exa 8.18
Nd =2*10**16
print"Nd = ",Nd," /cmˆ3" # initializing value of donor ion concentration .
a=2*10**-4
print"a = ",a," cm" # initializing value of height of channel at pinch off .
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
Er =11.9
print"Er = ",Er # initializing value of relative permittivity .
Eo =8.85*10**-14
print"Eo = ",Eo,"F/cm" # initializing value of free space permittivity .
E=(Eo*Er)
print"E=(Eo∗Er))=",E," F/cmˆ2"#calculation
Vp=((e*Nd*a**2)/(2*E))
print"Pinch off Voltage ,Vp=((e∗Nd∗aˆ2)/(2∗E)))=",Vp," V"# calculation,
Nd =  20000000000000000  /cmˆ3
a =  0.0002  cm
e =  1.6e-19  columns
Er =  11.9
Eo =  8.85e-14 F/cm
E=(Eo∗Er))= 1.05315e-12  F/cmˆ2
Pinch off Voltage ,Vp=((e∗Nd∗aˆ2)/(2∗E)))= 60.7700707402  V

Example 8_20 pgno: 275

In [14]:
#exa 8.20
a=2*10**-4
print"a = ",a," cm" # initializing value of height of channel at pinch off .
Er =11.9
print"Er = ",Er  # initializing value of relative dielectric permittivity constant .
Eo=8.854*10**-14
print"Eo = ",Eo," F/cm" # initializing value of permittivity of free space .
un =1350
print"un = ",un,"cmˆ2/V−s" # initializing value of mobility of n−type silicon Mosfet.
W=8*10**-4
print"W = ",W," cm" # initializing value of width of p−substrate .
L=10*10**-4
print"L = ",L," cm" # initializing value of length of p−substrate 
e=1.6*10**-19
print"e = ",e," columns" # initializing value of charge of electrons .
Vp=4
print"Vp = ",Vp," V" # initializing value of thickness of p−substrate . 
Vgs=0
print"Vgs = ",Vgs," V" # initializing value of gate to source voltage .
E=Eo*Er
print" total permittivity ,E=Eo∗Er=",E," F/cm" # calculation
Nd=((Vp*2*E)/(e*a**2))
print"Donor ion concentration ,Nd=((Vp∗2∗E)/(e∗aˆ2)))=",Nd," /cmˆ3"# calculation
rds=(L/(W*a*e*un*Nd))
print"On Drain resistance ,rds=(L/(W∗a∗e∗un∗Nd)))=",rds," ohm"# calculation
a =  0.0002  cm
Er =  11.9
Eo =  8.854e-14  F/cm
un =  1350 cmˆ2/V−s
W =  0.0008  cm
L =  0.001  cm
e =  1.6e-19  columns
Vp =  4  V
Vgs =  0  V
 total permittivity ,E=Eo∗Er= 1.053626e-12  F/cm
Donor ion concentration ,Nd=((Vp∗2∗E)/(e∗aˆ2)))= 1.3170325e+15  /cmˆ3
On Drain resistance ,rds=(L/(W∗a∗e∗un∗Nd)))= 21969.9856953  ohm