MOSFET Depletion Region Electric Field and Charge Density Forms
which is Poisson's equation.
Substitute the depletion region charge density
at the surface
far from the surface
Subtracting:
| Depletion Derivation Line | Inversion Derivation Line |
![{rho}= q(p-p_0)=q{p_0}[e^{{-phi}/{phi_t}}-1]](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_984_4dd2a3cac57cfdcb0e83693a1c63b663.png "{rho}= q(p-p_0)=q{p_0}[e^{{-phi}/{phi_t}}-1]")
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![{rho}= -q(n-n_0)=-q{n_0}[e^{{phi}/{phi_t}}-1]=-q{{{n_i}^2}/{p_0}^2}{p_0}[e^{{phi}/{phi_t}}-1]](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_972_cb393267fffdae6ba027968c2919a1e0.png "{rho}= -q(n-n_0)=-q{n_0}[e^{{phi}/{phi_t}}-1]=-q{{{n_i}^2}/{p_0}^2}{p_0}[e^{{phi}/{phi_t}}-1]") 
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![{{d^2{phi}}/dx^2} = {{-qp_0}/{varepsilon_s}} {[e^{{-phi}/{phi_t}}-1]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_980_0d32fc1dc1d2e5f0f3627f1964b61cea.png "{{d^2{phi}}/dx^2} = {{-qp_0}/{varepsilon_s}} {[e^{{-phi}/{phi_t}}-1]}")
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![{{d^2{phi}}/dx^2} = {{qp_0}/{varepsilon_s}}{e^{{-2{phi_F}}/{phi_t}}}{[e^{{phi}/{phi_t}}-1]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_980_1da1e4d8020d7ad656ee9eb54068f93a.png "{{d^2{phi}}/dx^2} = {{qp_0}/{varepsilon_s}}{e^{{-2{phi_F}}/{phi_t}}}{[e^{{phi}/{phi_t}}-1]}")
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![int{0}{phi_s}{2{{d^2{phi}}/dx^2}{d{phi}/{dx}}} = {{-2qp_0}/{varepsilon_s}} int{0}{phi_s}{{[e^{{-phi}/{phi_t}}-1]} {{d{phi}}/{dx}}}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_970_cf689f6808b304be6fef6b6d3142e24c.png "int{0}{phi_s}{2{{d^2{phi}}/dx^2}{d{phi}/{dx}}} = {{-2qp_0}/{varepsilon_s}} int{0}{phi_s}{{[e^{{-phi}/{phi_t}}-1]} {{d{phi}}/{dx}}}")
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![int{0}{phi_s}{2{{d^2{phi}}/dx^2}{d{phi}/{dx}}} = {{-2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} int{0}{phi_s}{{[e^{{phi}/{phi_t}}-1]} {{d{phi}}/{dx}}}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_970_47cc76f98d33f2c0e60d1a1f4a502862.png "int{0}{phi_s}{2{{d^2{phi}}/dx^2}{d{phi}/{dx}}} = {{-2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} int{0}{phi_s}{{[e^{{phi}/{phi_t}}-1]} {{d{phi}}/{dx}}}")
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![({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}} {[{phi_t}e^{{-phi}/{phi_t}}+{phi}]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_982_b94ca0b8b61fd804c5d93182533275c4.png "({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}} {[{phi_t}e^{{-phi}/{phi_t}}+{phi}]}")
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![({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} {[{phi_t}e^{{phi}/{phi_t}}+{phi}]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_982_94ceb345a83febe14b4bea6fbbd2a0f0.png "({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} {[{phi_t}e^{{phi}/{phi_t}}+{phi}]}") right hand side evaluated 0 to 
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![({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}} {[e^{{-phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_982_cad6e7942674fb8ffe7df7f4bdb4c8f0.png "({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}} {[e^{{-phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]}")
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![({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}}{[e^{{phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_982_c93af984bf525a86e68c687ee01d3fa1.png "({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}}{[e^{{phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]}")
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![({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{[{phi_t}e^{{-phi_s}/{phi_t}}+{phi_s}-{phi_t}]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_982_a8eeaf2fb83f8e9cbf19cf11590626eb.png "({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{[{phi_t}e^{{-phi_s}/{phi_t}}+{phi_s}-{phi_t}]}")
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![({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}}{[{phi_t}e^{{phi_s}/{phi_t}}+{phi_s}-{phi_t}]}](http://www.amarketplaceofideas.com/wp-content/uploads/wpmathpub/math-img/math_982_7efc87786cd0c076c7f14618b580965c.png "({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}}{[{phi_t}e^{{phi_s}/{phi_t}}+{phi_s}-{phi_t}]}")
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Identity Table
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