P-N Junctions

DOPED SEMICONDUCTORS

THE EFFECT OF TEMPERATURE

DOPED SEMICONDUCTORS

THE EFFECT OF TEMPERATURE

x
E
E_F
E_d

T=0K

x
E
E_F
E_d

T=300K

x
E
E_F
E_d

T=1000K

P-N JUNCTIONS AT EQUILIBRIUM

P-Type Semiconductor: \(N_A\)

N-Type Semiconductor: \(N_D\)

P-N JUNCTIONS: DEPLETION REGION

P-Type Semiconductor

N-Type Semiconductor

x_p
x_n

Depletion Region: \(x_0\)

P-N JUNCTIONS: BUILT-IN FIELD

P-Type Semiconductor

N-Type Semiconductor

x_p
x_n

Depletion Region: \(x_0\)

\vec{E}

P-N JUNCTIONS: COMPUTING THE FIELD

\rho(x)
x
x_p
x_n
\rho_n = e N_D
\rho_p = -e N_A
\rho(x)= \begin{dcases} -e N_A, \, -x_{p} < x < 0 \\ e N_D, \, 0 < x < x_{n} \end{dcases}
\frac{x_p}{x_n} = \frac{N_D}{N_A} \Rightarrow N_A x_p = N_D x_n

Charge Distribution

Gauss Law

\begin{dcases} \oint \vec{E} \cdot d\vec{S} = \frac{Q}{\varepsilon_{s}\varepsilon_{0}} \\[10pt] \vec{\nabla} \cdot \vec{E} = \frac{\rho(\vec{r})}{\varepsilon_{s}{\varepsilon_{0}}} \end{dcases} \Rightarrow \frac{dE}{dx} = \frac{\rho(x)}{\varepsilon_{s}{\varepsilon_{0}}}

P-N JUNCTIONS: COMPUTING THE FIELD

\rho(x)
x
x_p
x_n

P-Type Field

\begin{dcases} \int_{-x_p}^x \frac{dE}{dx}dx = E(x) - E(-x_p) \\[10pt] \int_{-x_p}^x -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}dx = -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}(x+x_p) \\[10pt] E(-x_p) = 0 \end{dcases} \\ \Downarrow \\ E(x) = -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}(x+x_p)
\rho_n = e N_D
\rho_p = -e N_A

P-N JUNCTIONS: COMPUTING THE FIELD

N-Type Field

\begin{dcases} \int_{x}^{x_n} \frac{dE}{dx}dx = E(x_n) - E(x) \\[10pt] \int_{x}^{x_n} +\frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}dx = \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x_n-x) \\[10pt] E(x_n) = 0 \end{dcases} \\ \Downarrow \\ E(x) = \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x-x_n)
\rho(x)
x
x_p
x_n
\rho_n = e N_D
\rho_p = -e N_A

P-N JUNCTIONS: COMPUTING THE FIELD

E(x)
x
x_p
x_n

Total Field

E(x) = \begin{dcases} -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}(x+x_p) \text{, P-Type} \\[10pt] \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x-x_n) \text{, N-Type} \end{dcases}
E_{max}
E_{\max }=-\frac{e N_A x_p}{\varepsilon_{s}\varepsilon_{0}}=-\frac{e N_D x_n}{\varepsilon_{s}\varepsilon_{0}}

P-N JUNCTIONS: BUILT-IN VOLTAGE

E(x)
x
x_p
x_n
E(x) = \begin{dcases} -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}(x+x_p) \text{, P-Type} \\[10pt] \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x-x_n) \text{, N-Type} \end{dcases}
V(x)=-\int_{-x_p}^x E\left(x^{\prime}\right) d x' \\ \Downarrow \\ \begin{aligned} V(x) & = -\left(-\frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}(x+x_p) \cdot (x+x_p)\right) \\ V(x) & = \frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}(x+x_p)^2 \end{aligned}
x

P-Type

P-N JUNCTIONS: BUILT-IN VOLTAGE

E(x)
x
x_p
x_n
E(x) = \begin{dcases} -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}(x+x_p) \text{, P-Type} \\[10pt] \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x-x_n) \text{, N-Type} \end{dcases}
V(x)=-\int_{0}^x E\left(x^{\prime}\right) d x \\ \Downarrow \\ \begin{aligned} V(x) & = \frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}x_p^2 + (\frac{e N_D x_n}{\varepsilon_{s}\varepsilon_{0}} + \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x_n-x)) \cdot \frac{x}{2}\\ V(x) & = \frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}x_p^2 +\frac{e N_D}{2\varepsilon_{s}\varepsilon_{0}}(2 x_n x -x^2)\\ V(x) & = \frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}x_p^2 + \frac{eN_D}{2\varepsilon_{s}\varepsilon_{0}}x_n^2 - \frac{e N_D}{2\varepsilon_{s}\varepsilon_{0}}(x-x_n)^2\\ \end{aligned}
x

N-Type

P-N JUNCTIONS: BUILT-IN VOLTAGE

V(x) = \begin{dcases} \frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}(x+x_p)^2 \text{, P-Type} \\[10pt] \frac{e}{2\varepsilon_{s}\varepsilon_{0}}(N_A x_p^2 + N_D x_n^2) - \frac{e N_D}{2\varepsilon_{s}\varepsilon_{0}}(x-x_n)^2 \text{, N-Type} \end{dcases}\\ \Downarrow \\ V\left(x_n\right)=V_0=\frac{e}{2 \varepsilon_{s}\varepsilon_{0}}\left(N_A x_p^2+N_D x_n^2\right)
\begin{dcases} x_p+x_n=x_0 \\ x_p N_A=x_n N_D \\ \end{dcases} \Rightarrow \begin{dcases} x_p=\frac{x_0}{1+\frac{N_A}{N_D}}\\ x_n=\frac{x_0}{1+\frac{N_D}{N_A}} \end{dcases}\\ \Downarrow\\ V_0=\frac{e}{2 \varepsilon_{s}\varepsilon_{0}} x_0^2 \frac{N_A N_D}{N_A + N_D}

P-N JUNCTIONS: DEPLETION WIDTH

V_0=\frac{e}{2 \varepsilon_{s}\varepsilon_{0}} x_0^2 \frac{N_A N_D}{N_A + N_D}\\[5pt] \Downarrow \\[5pt] x_0=\sqrt{\frac{2 \varepsilon_{s}\varepsilon_{0} V_0}{e}\frac{N_A+N_D}{N_A N_D}}\\[5pt] \Downarrow \\[5pt]
x_n=\sqrt{\frac{2 \varepsilon_{s} \varepsilon_0 V_{0}}{e} \frac{N_A}{N_D\left(N_A+N_D\right)}}
x_p=\sqrt{\frac{2 \varepsilon_{s} \varepsilon_0 V_{0}}{e} \frac{N_D}{N_A\left(N_A+N_D\right)}}

BUILT IN VOLTAGE \(V_0\): FERMI LEVELS

E_{f,n}=E_v+\frac{E_g}{2}+\frac{3}{4} k_B T \ln \left(\frac{m_v^*}{m_c^*}\right)+k_B T \ln \left(\frac{N_D}{n_i}\right)=E_{f,i}+k_B T \ln \left(\frac{N_D}{n_i}\right)
E_{f,p}=E_v+\frac{E_g}{2}+\frac{3}{4} k_B T \ln \left(\frac{m_v^*}{m_c^*}\right)-k_B T \ln \left(\frac{N_A}{n_i}\right)=E_{f,i}-k_B T \ln \left(\frac{N_A}{n_i}\right)

N-Type Doping

P-Type Doping

BUILT IN VOLTAGE \(V_0\): FERMI LEVELS

x
E
E_{f,p}

P-Type

\Delta E = E_{f,n} -E_{f,p}
x
E
E_{f,n}

N-Type

BUILT IN VOLTAGE \(V_0\): FERMI LEVELS

eV_0 = E_{f,n} -E_{f,p}
E_n

P-Type

E_{f}

N-Type

x
E_p

BUILT IN VOLTAGE \(V_0\): FERMI LEVELS

\begin{dcases} E_{f,n}=E_{f,i}+k_B T \ln \left(\frac{N_D}{n_i}\right) \\[10pt] E_{f,p}=E_{f,i}-k_B T \ln \left(\frac{N_A}{n_i}\right) \end{dcases}\\[5pt] \Downarrow \\[5pt] V_{0}=\frac{k_B T}{e} \ln \left(\frac{N_A N_D}{n_i^2}\right)

BUILT IN VOLTAGE \(V_0\): FERMI LEVELS

P-N JUNCTIONS: EQUILIBRIUM

\(p_{p0} \simeq N_A\): Majority Carriers \(\Rightarrow\) Diffusion

\(n_{p0} \simeq \frac{n_i^2}{N_A}\): Minority Carriers \(\Rightarrow\) Drift

P-Type Semiconductor

N-Type Semiconductor

\vec{E}

\(n_{n0} \simeq N_D\): Majority Carriers \(\Rightarrow\) Diffusion

\(p_{n0} \simeq \frac{n_i^2}{N_D}\): Minority Carriers \(\Rightarrow\) Drift

P-N JUNCTIONS: DEPLETION WIDTH

V_0=\frac{e}{2 \varepsilon_{s}\varepsilon_{0}} x_0^2 \frac{N_A N_D}{N_A + N_D}\\[5pt] \Downarrow \\[5pt] x_0=\sqrt{\frac{2 \varepsilon_{s}\varepsilon_{0} V_0}{e}\frac{N_A+N_D}{N_A N_D}}\\[5pt] \Downarrow \\[5pt]
x_n=\sqrt{\frac{2 \varepsilon_{s} \varepsilon_0 V_{0}}{e} \frac{N_A}{N_D\left(N_A+N_D\right)}}
x_p=\sqrt{\frac{2 \varepsilon_{s} \varepsilon_0 V_{0}}{e} \frac{N_D}{N_A\left(N_A+N_D\right)}}

P-N JUNCTIONS: DEPLETION CHARGE DENSITY

\rho(x)= \begin{dcases} -e N_A, \, -x_{p} < x < 0 \\ e N_D, \, 0 < x < x_{n} \end{dcases}

P-N JUNCTIONS: DEPLETION CHARGE DENSITY

P-Type

N-Type

P-N JUNCTIONS: BUILT-IN FIELDS

E(x) = \begin{dcases} -\frac{eN_A}{\varepsilon_{s}\varepsilon_{0}}(x+x_p) \text{, P-Type} \\[10pt] \frac{e N_D}{\varepsilon_{s}\varepsilon_{0}}(x-x_n) \text{, N-Type} \end{dcases}

P-N JUNCTIONS: BUILT-IN FIELDS

P-Type

N-Type

P-N JUNCTIONS: BUILT-IN VOLTAGE

V(x) = \begin{dcases} \frac{eN_A}{2\varepsilon_{s}\varepsilon_{0}}(x+x_p)^2 \text{, P-Type} \\[10pt] \frac{e}{2\varepsilon_{s}\varepsilon_{0}}(N_A x_p^2 + N_D x_n^2) - \frac{e N_D}{2\varepsilon_{s}\varepsilon_{0}}(x-x_n)^2 \text{, N-Type} \end{dcases}\\

P-N JUNCTIONS: BUILT-IN VOLTAGE

P-Type

N-Type

P-N JUNCTIONS: HOLES POTENTIAL ENERGY

P-Type

N-Type

P-N JUNCTIONS: ELECTRONS POTENTIAL ENERGY

P-Type

N-Type

P-N JUNCTIONS: CARRIER CONCENTRATION

P-Type

N-Type

P-N JUNCTIONS: OUT OF EQUILIBRIUM

\(p_{p0} \simeq N_A\): Majority Carriers \(\Rightarrow\) Diffusion

\(n_{p0} \simeq \frac{n_i^2}{N_A}\): Minority Carriers \(\Rightarrow\) Drift

P-Type Semiconductor

N-Type Semiconductor

\vec{E}

\(n_{n0} \simeq N_D\): Majority Carriers \(\Rightarrow\) Diffusion

\(p_{n0} \simeq \frac{n_i^2}{N_D}\): Minority Carriers \(\Rightarrow\) Drift

P-N JUNCTIONS: OUT OF EQUILIBRIUM

\(p_{p0} \simeq N_A\): Majority Carriers \(\Rightarrow\) Diffusion

\(n_{p0} \simeq \frac{n_i^2}{N_A}\): Minority Carriers \(\Rightarrow\) Drift

\(n_{n0} \simeq N_D\): Majority Carriers \(\Rightarrow\) Diffusion

\(p_{n0} \simeq \frac{n_i^2}{N_D}\): Minority Carriers \(\Rightarrow\) Drift

\Downarrow \\[5pt] V_{0}=\frac{k_B T}{e} \ln \left(\frac{N_A N_D}{n_i^2}\right) = \frac{k_B T}{e} \ln \left(\frac{n_{n0}}{n_{p0}}\right) = \frac{k_B T}{e} \ln \left(\frac{p_{p0}}{p_{n0}}\right)\\[5pt] \Downarrow \\[5pt] \begin{dcases} n_{p0} = n_{n0} e^{\frac{-e V_0}{k_B T}} \\[10pt] p_{n0} = p_{p0} e^{\frac{-e V_0}{k_B T}} \\[10pt] \end{dcases}

P-N JUNCTIONS: CARRIER CONCENTRATION

P-Type

N-Type

\vec{E}

P-N JUNCTIONS: CARRIER CONCENTRATION

P-N JUNCTIONS: DIFFUSION CURRENT

P-N JUNCTIONS: DIFFUSION CURRENT

x
p(x)
x
n(x)
J_{D,p}=-e D_p \frac{d p(x)}{d x}
J_{D,n}=e D_n \frac{d n(x)}{d x}

P-N JUNCTIONS: DRIFT CURRENT

\begin{aligned} \vec{j}_{drift}=n \cdot e \cdot \vec{v}=\sigma \vec{E} \\ \end{aligned}\\ \Downarrow
\sigma=n_c e \mu_n + p_v e \mu_p \, \text{ with } \, \begin{dcases} \mu_n = \frac{e \tau_n}{m^*_c}\\[10pt] \mu_p = \frac{e \tau_p}{m^*_v} \end{dcases} \text{, } \begin{dcases} v_{d,n} = \mu_n E\\[10pt] v_{d,p} = \mu_p E \end{dcases}\\ \Downarrow
j_{drift} = j_{drift,n} + j_{drift,p} = n_c e \mu_n E + p_v e \mu_p E

CURRENT EQUILIBRIUM: EINSTEN RELATIONS

\begin{dcases} j_{D,p} + j_{drift,p} = -e D_p \frac{d p(x)}{d x} + p(x) e \mu_p E = 0 \\[10pt] j_{D,n} + j_{drift,n} = e D_n \frac{d n(x)}{d x} + n(x) e \mu_n E = 0 \\ \end{dcases}\\ \Downarrow\\ \begin{dcases} D_p=\frac{k_B T \mu_p}{e} \\[10pt] D_n=\frac{k_B T \mu_n}{e} \end{dcases}

WHAT ARE THE CONCENTRATION PROFILES \(p(x)\)AND \(n(x)\)? 

STEADY STATE INJECTION OF MINORITY CARRIERS

n_{p0}
\Delta n
\begin{dcases} p_n(x)=p_n(0) \exp \left(-\frac{x}{L_p}\right);\, \Delta p_n(x)= \Delta p_n(0) \exp \left(-\frac{x}{L_p}\right)\\ n_p(x)=n_p(0) \exp \left(-\frac{x}{L_n}\right); \, \Delta n_p(x)= \Delta n_p(0) \exp \left(-\frac{x}{L_n}\right) \end{dcases}
L_p=\sqrt{D_p \tau_p}\\ L_n=\sqrt{D_n \tau_n} \\ L_{p,n} \rightarrow \text{Diffusion length} \\ D_{p,n} \rightarrow \text{Diffusion coefficient} \\ \tau_{p,n} \rightarrow \text{Recombination time}

P-Type

P-N JUNCTIONS: FORWARD BIAS

P-N JUNCTIONS: REVERSE BIAS

P-N JUNCTIONS: OUT OF EQUILIBRIUM

EXCESS OF INJECTED CARRIERS

\begin{dcases} n_{p0} = n_{n0} e^{-\frac{e V_0}{k_B T}} \\[10pt] p_{n0} = p_{p0} e^{-\frac{e V_0}{k_B T}} \\[10pt] \end{dcases} \Rightarrow \begin{dcases} n_{p}(0) = n_{n0} e^{-\frac{ e(V_0-V_{ext})}{k_B T}} \\[10pt] p_{n}(0) = p_{p0} e^{-\frac{ e(V_0-V_{ext})}{k_B T}} \\[10pt] \end{dcases}\\[5pt] \Downarrow \\[5pt] \begin{dcases} \Delta n_{p}(0) = n_{p}(0)- n_{p0} = n_{p0} (e^{\frac{e V_{ext}}{k_B T}}-1) \\[10pt] \Delta p_{n}(0) = p_{n}(0) - p_{n0} = p_{n0} (e^{\frac{e V_{ext}}{k_B T}}-1) \\[10pt] \end{dcases}

P-N JUNCTIONS: OUT OF EQUILIBRIUM

CONCENTRATION PROFILE OF EXCESS CARRIERS

L_p=\sqrt{D_p \tau_p}\\ L_n=\sqrt{D_n \tau_n} \\ L_{p,n} \rightarrow \text{Diffusion length} \\ D_{p,n} \rightarrow \text{Diffusion coefficient} \\ \tau_{p,n} \rightarrow \text{Recombination time}

Diffusion Parameters

Einstein relation

D_p=\frac{k_B T \mu_p}{e} \\[10pt] D_n=\frac{k_B T \mu_n}{e}
\begin{dcases} p_n(x)=p_n(0) \exp \left(-\frac{x}{L_p}\right);\, \Delta p_n(x)= \Delta p_n(0) \exp \left(-\frac{x}{L_p}\right)\\ n_p(x)=n_p(0) \exp \left(-\frac{x}{L_n}\right); \, \Delta n_p(x)= \Delta n_p(0) \exp \left(-\frac{x}{L_n}\right) \end{dcases}

Concentration Profile

P-N JUNCTIONS: OUT OF EQUILIBRIUM

DIFFUSION CURRENT DENSITY: Holes

J_{D,p}=-e D_p \frac{d p_n(x)}{d x} = -e D_p \frac{d \Delta p_n(x)}{d x} \\[5pt] \Downarrow\\[5pt] \begin{array}{r} J_{D,p}=\frac{e D_p}{L_p} \Delta p_n(0)=\frac{e D_p}{L_p}\left(p_n(0)-p_{n 0}\right) \\ J_{D,p}=\frac{e D_p p_{n 0}}{L_p}\left[\exp \left(\frac{e V}{k_B T}\right)-1\right] \\ J_{D,p}=\frac{e D_p n_i^2}{L_p N_D}\left[\exp \left(\frac{e V}{k_B T}\right)-1\right] \end{array}

P-N JUNCTIONS: OUT OF EQUILIBRIUM

DIFFUSION CURRENT DENSITY: Electrons

J_{D,n}=e D_n \frac{d n_p(x)}{d x} = e D_n \frac{d \Delta n_p(x)}{d x} \\[5pt] \Downarrow\\[5pt] \begin{array}{r} J_{D,n}=\frac{e D_n}{L_n} \Delta p_n(0)=\frac{e D_n}{L_n}\left(n_p(0)-n_{p 0}\right) \\ J_{D,n}=\frac{e D_n n_{p 0}}{L_n}\left[\exp \left(\frac{e V}{k_B T}\right)-1\right] \\ J_{D,n}=\frac{e D_n n_i^2}{L_n N_A}\left[\exp \left(\frac{e V}{k_B T}\right)-1\right] \end{array}

P-N JUNCTIONS: I-V CURVE

\begin{gathered} J_{s 0}=e n_i^2\left(\frac{D_p}{L_p N_D}+\frac{D_n}{L_n N_A}\right) \\ J_D=J_{s 0}\left[\exp \left(\frac{e V}{k_B T}\right)-1\right] \end{gathered}

P-N JUNCTIONS: J-V CURVE