FET

文章目录

• How FET got its Name
• JFET (Junction FET)
• I-V Characteristics
• Questions
• Pinch-off voltage
• Channel Current
• Gain (Transconductance)
• MESFET (Metal Semiconductor FET)
• Metal-Semiconductor (MS) Contact
• Schottky Diode
• Heavily Doped (Degenerately Doped) Semiconductor
• Schottky Diode Calculations
• MOSFET (Metal-Oxide FET)
• Composition
• Terminal Naming
• Functionality (NMOS)
• MOS Energy Band Diagram
• MOS Calculations
• $\Phi_S$ΦS​: Surface Potential
• $\Phi_F$ΦF​
• Depletion Width $W$W
• Threshold Voltage $V_T$VT​
• Cap-Voltage Characteristics
• Supplement
• JFET pinch-off
• Why doesn't current goto 0
• How does current flow after pinch-off
• Gradual Channel Approx
• JFET Transconductance
• JFET Transconductance

• How FET got its Name
• JFET (Junction FET)
  • I-V Characteristics
    • Questions
  • Pinch-off voltage
  • Channel Current
  • Gain (Transconductance)
• MESFET (Metal Semiconductor FET)
  • Metal-Semiconductor (MS) Contact
  • Schottky Diode
  • Heavily Doped (Degenerately Doped) Semiconductor
  • Schottky Diode Calculations
• MOSFET (Metal-Oxide FET)
  • Composition
  • Terminal Naming
  • Functionality (NMOS)
  • MOS Energy Band Diagram
  • MOS Calculations
    • KaTeX parse error: Undefined control sequence: \PhiS at position 1: \̲P̲h̲i̲S̲: Surface Potential
    • KaTeX parse error: Undefined control sequence: \PhiF at position 1: \̲P̲h̲i̲F̲
    • Depletion Width $W$W
    • Threshold Voltage $VT$VT
    • Cap-Voltage Characteristics
  • Supplement
  • JFET pinch-off
    • Why doesn’t current goto 0
    • How does current flow after pinch-off
  • Gradual Channel Approx
  • JFET Transconductance

How FET got its Name

A voltage applied to the metallic plate modalated the conductance of the underlying semiconductor, which in turn modulated the carrent flowing between ohmic contacts A and B. This phenomenon, where the conductivity of a semiconductor is modulated by an electric field applied normal to the surface of the semiconductor, has been named the field effect.

JFET (Junction FET)

Suppose we connect S to ground, reverse bias at S is $V_G$VG​.

I-V Characteristics

1. $V_G$VG​ = 0. When VD is small, ID is small. Linear I-V. No change in depletion width across channel.
   
   
2. $V_G$VG​ increases. Channel pinches-off.
   
   
   

Questions

1. Should/Should not $I_D$ID​ = 0 beyond pinch-off?
   No. Carriers can also pass depletion region, but see a much higher resisitance.
2. Why does $V_D$VD​ > $V_D^{Sat}$VDSat​ have no effect on ID?
   Increasing $V_D$VD​ will also increase the length of pinched-off region. These two effects cancel out.

Pinch-off voltage

$V_p$Vp​ = Reverse bias between n-channel and $p^+$p+ gate at the drain end $(x=0)$(x=0).
$h(x)$h(x) = Channel half-width at any $x$x
$a$a = half width of channel
Assumptions:

1. Channel with at $x=0$x=0 decreases uniformly as the reverse bias increases to pinch-off.
2. $V_{bi}$Vbi​ neglected.
3. $p^+-n$p+−n gate junction.

$V_p=\frac{qa^2N_D}{2\varepsilon}$Vp​=2εqa2ND​​
$a$a: half thickness

Channel Current

$L$L: length
$Z$Z: depth
$2a$2a: thickness

$I_D=G_oV_P\left[\frac{V_D}{V_P} + \frac23\left(-\frac{V_G}{V_P}\right)^{3/2} - \frac23\left(\frac{V_D-V_G}{V_P}\right)^{3/2}\right]$ID​=Go​VP​[VP​VD​​+32​(−VP​VG​​)3/2−32​(VP​VD​−VG​​)3/2]
$G_o=\frac{2aZ}{\rho L}$Go​=ρL2aZ​
$I_D(sat)=G_oV_P\left[\frac{V_D}{V_P} + \frac23\left(-\frac{V_G}{V_P}\right)^{3/2} - \frac23\right]$ID​(sat)=Go​VP​[VP​VD​​+32​(−VP​VG​​)3/2−32​]

Gain (Transconductance)

$g_m(sat)=\frac{\partial I_D(sat)}{\partial V_G}=G_o\left[1-\left(-\frac{V_G}{V_P}\right)^{1/2}\right]$gm​(sat)=∂VG​∂ID​(sat)​=Go​[1−(−VP​VG​​)1/2]

MESFET (Metal Semiconductor FET)

Metal-Semiconductor (MS) Contact

Energy Band Diagram of Metal and Semiconductor

• $\Phi_M$ΦM​: Metal Work Function (the one in photoelectric effect)
• $\Phi_S$ΦS​: Semiconductor Work Function
• $\chi$χ: Electron Affinity. $\chi=(E_0-E_C)|_{surface}$χ=(E0​−EC​)∣surface​

$\Phi_S=\chi+(E_c-E_F)_{FB}$ΦS​=χ+(Ec​−EF​)FB​

• $(E_c-E_F)_{FB}$(Ec​−EF​)FB​: Energy difference between $E_c$Ec​ and $E_F$EF​ at flat band (i.e.) zero bias condition.

$\Phi_B=\Phi_M-\chi$ΦB​=ΦM​−χ

• $\Phi_B$ΦB​: surface potential-energy barrier encountered by electrons with $E=E_F$E=EF​ in the metal.

Schottky Diode

• $\Phi_M>\Phi_S$ΦM​>ΦS​: Applying $V_A>0$VA​>0 lowers $E_{FM}$EFM​ below $E_{FS}$EFS​, reduces the barrier seen by electrons in the semiconductor.

• $\Phi_M<\Phi_S$ΦM​<ΦS​: Non-rectifying, Ohmic.

Heavily Doped (Degenerately Doped) Semiconductor

When the barrier is thin enough, the carriers can tunnel through.
Upper: forward bias. Below: rev bias
Additional component of current

Schottky Diode Calculations

• Built-in Voltage

$V_{bi}=\frac1q\left[\Phi_B-(E_c-E_F)_{FB}\right]$Vbi​=q1​[ΦB​−(Ec​−EF​)FB​]

• $\rho$ρ
  • Metal: delta function (charge only on surface)
  • Semiconductor: $\rho=qN_D$ρ=qND​
• $E$E:

$E(x)=-\frac{qN_D}{\varepsilon_{Si}}(W-x)\ldots 0\le x\le W$E(x)=−εSi​qND​​(W−x)…0≤x≤W

• $V$V:

$V(x)=-\frac{qN_D}{2\varepsilon_{Si}}(W-x)^2\ldots 0\le x \le W$V(x)=−2εSi​qND​​(W−x)2…0≤x≤W

• Depletion Width

$W=\sqrt{\frac{2\varepsilon_{Si}}{qN_D}(V_{bi}-V_A)}$W=qND​2εSi​​(Vbi​−VA​)​

• $\Phi(x)$Φ(x)

$\Phi(x)=\frac{qN_Dx^2}{2\epsilon_S}$Φ(x)=2ϵS​qND​x2​

• Current Density

$J=J_S(e^{qV_a/kT}-1)$J=JS​(eqVa​/kT−1)
$J_S=A^*T^2\exp\left(-\frac{q\Phi_B}{kT}\right)$JS​=A∗T2exp(−kTqΦB​​)
$A^*$A∗ is the Effective Richardson Constant

MOSFET (Metal-Oxide FET)

Composition

• MOS Capacitor
• Two pn juncitons

Terminal Naming

• Carriers enter the structure through Source (S)
• Leave through the Drain (D)
• Subject to the control of the Gate (G)

Functionality (NMOS)

• When $V_G\le V_T$VG​≤VT​, i.e. $V_G$VG​ is in accumulation or depletion biased, the gated region contains mostly holes and few electrons, an open circuit is formed.
• When $V_G>V_T$VG​>VT​, i.e. $V_G$VG​ is inversion biased, an inversion layer (channel) containing mobile electrons is formed.
• As $V_D$VD​ increases, the channel finally pinches-off, the current saturates.

MOSFET Pinches-off

Channel Length Modulation of Short-Channel Device

MOS Energy Band Diagram

Band Diagram for NMOS

• Accumulation ($V_G<0$VG​<0) holes accumulate on the semiconductor side of the gate
• Depletion ($0<V_G<V_T$0<VG​<VT​) holes repelled away, leaving Ionized acceptors atoms
• Inversion ($V_G>V_T$VG​>VT​) electron density increase
  • Initially, $n<n_i$n<ni​.
  • $n=n_i$n=ni​ when $E_i=E_F$Ei​=EF​
• When $V_G=V_T$VG​=VT​, $n=N_A$n=NA​, the semiconductor seems no longer to be depleted. Instead, it now behave similar to n-type. The channel has formed.

MOS Calculations

$\Phi_S$ΦS​: Surface Potential

$\Phi_S=\frac1q[E_i(bulk)-E_i(surface)]$ΦS​=q1​[Ei​(bulk)−Ei​(surface)]

$\Phi_F$ΦF​

$\Phi_F=\frac1q[E_i(bulk)-E_F]$ΦF​=q1​[Ei​(bulk)−EF​]
In p-type, $N_A\gg N_D$NA​≫ND​, $p_{bulk}=n_i\exp([E_i(bulk)-E_F]/kT)= N_A$pbulk​=ni​exp([Ei​(bulk)−EF​]/kT)=NA​
$\Phi_F=\frac{kT}{q}\ln\left(\frac{N_A}{n_i}\right)$ΦF​=qkT​ln(ni​NA​​)
In n-type, $N_D\gg N_A$ND​≫NA​, $n_{bulk}=n_i\exp([E_F-E_i(bulk)])=N_D$nbulk​=ni​exp([EF​−Ei​(bulk)])=ND​
$\Phi_F=-\frac{kT}{q}\ln\left(\frac{N_D}{n_i}\right)$ΦF​=−qkT​ln(ni​ND​​)
When $V_G=V_T$VG​=VT​,
$\Phi_S=2\Phi_F$ΦS​=2ΦF​

Depletion Width $W$W

Valid before strong inversion:

$W=\sqrt{\frac{2\varepsilon_{Si}}{qN_A}\Phi_S}$W=qNA​2εSi​​ΦS​​
At strong inversion:

$W_m=2\sqrt{\frac{\epsilon kT}{q^2N_A}\ln\left(\frac{N_A}{n_i}\right)}$Wm​=2q2NA​ϵkT​ln(ni​NA​​)​
When $V_G=V_T$VG​=VT​, $\Phi_S=2\Phi_F$ΦS​=2ΦF​, the depletion width
$W_T=\sqrt{\frac{4\varepsilon_{Si}}{qN_A}\Phi_F}$WT​=qNA​4εSi​​ΦF​​

Threshold Voltage $V_T$VT​

For N(-channel)MOS (P-bulk)

$V_T=2\Phi_F+\frac{\epsilon_{OXIDE}x_O}{\epsilon_{Si}}\sqrt{\frac{4qN_A}{\epsilon_{Si}}\Phi_F}$VT​=2ΦF​+ϵSi​ϵOXIDE​xO​​ϵSi​4qNA​​ΦF​​
For PMOS (N-bulk)
$V_T=2\Phi_F-\frac{\epsilon_{OXIDE}x_O}{\epsilon_{Si}}\sqrt{\frac{4qN_D}{\epsilon_{Si}}(-\Phi_F)}$VT​=2ΦF​−ϵSi​ϵOXIDE​xO​​ϵSi​4qND​​(−ΦF​)​
$x_O$xO​ is the thickness of the OXIDE

Cap-Voltage Characteristics

PMOS (n-bulk)

NMOS (p-bulk) a: Low Freq, b&c: Hi Freq

Supplement

JFET pinch-off

Widening everywhere as $V_{SD}$VSD​ grows

Why doesn’t current goto 0

If current is 0, the pinch-off will disappear. To maintain pinch-off, a non-zero current must be present.

How does current flow after pinch-off

Gradual Channel Approx

Formula for depetion layer width remainis same at every point and edge of depletion layer is not a multi-valued function at any point

JFET Transconductance

$g_m=\frac{\partial I_D(sat)}{\partial V_G}$gm​=∂VG​∂ID​(sat)​
Proportional to $\sqrt{V_G}$VG​​
e at every point and edge of depletion layer is not a multi-valued function at any point

JFET Transconductance

$g_m=\frac{\partial I_D(sat)}{\partial V_G}$gm​=∂VG​∂ID​(sat)​
Proportional to $\sqrt{V_G}$VG​​