From 9bc54d475fbd281d9a181bea0a884fcbddffd4f6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C3=98yvind=20Skaaden?= Date: Sat, 21 Nov 2020 14:04:51 +0100 Subject: [PATCH] Started new page --- _config.yml | 4 + _includes/toc.html | 5 + _layouts/lecture.html | 2 - .../tfe4146/summary/figures/HallEffect.png | Bin 0 -> 89038 bytes .../summary/figures/carrierDensity.png | Bin 0 -> 72728 bytes .../tfe4146/summary/figures/conductivity.png | Bin 0 -> 259052 bytes .../tfe4146/summary/figures/continuityEq.png | Bin 0 -> 70006 bytes courses/tfe4146/summary/figures/dos.png | Bin 0 -> 84206 bytes .../tfe4146/summary/figures/fermiDirac.png | Bin 0 -> 143911 bytes .../summary/figures/haynes-ShockleyExp.png | Bin 0 -> 55555 bytes .../summary/figures/haynes-ShockleyTheory.png | Bin 0 -> 37464 bytes courses/tfe4146/summary/figures/intrinsic.png | Bin 0 -> 64778 bytes .../tfe4146/summary/figures/particleBox.png | Bin 0 -> 36927 bytes .../summary/figures/steadyStateBar.png | Bin 0 -> 18010 bytes .../tfe4146/summary/figures/tetteBonds.png | Bin 0 -> 98643 bytes courses/tfe4146/summary/summary.md | 325 ++++++++++++++++++ 16 files changed, 334 insertions(+), 2 deletions(-) create mode 100644 _includes/toc.html create mode 100644 courses/tfe4146/summary/figures/HallEffect.png create mode 100644 courses/tfe4146/summary/figures/carrierDensity.png create mode 100644 courses/tfe4146/summary/figures/conductivity.png create mode 100644 courses/tfe4146/summary/figures/continuityEq.png create mode 100644 courses/tfe4146/summary/figures/dos.png create mode 100644 courses/tfe4146/summary/figures/fermiDirac.png create mode 100644 courses/tfe4146/summary/figures/haynes-ShockleyExp.png create mode 100644 courses/tfe4146/summary/figures/haynes-ShockleyTheory.png create mode 100644 courses/tfe4146/summary/figures/intrinsic.png create mode 100644 courses/tfe4146/summary/figures/particleBox.png create mode 100644 courses/tfe4146/summary/figures/steadyStateBar.png create mode 100644 courses/tfe4146/summary/figures/tetteBonds.png create mode 100644 courses/tfe4146/summary/summary.md diff --git a/_config.yml b/_config.yml index 847be46..af52e0a 100644 --- a/_config.yml +++ b/_config.yml @@ -4,9 +4,13 @@ description_long: Personlig nettside for Øyvind Skaaden. 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+description: Oppsummering av faget TFE4146 høsten 2020. +math: true +permalink: /:path +date: 2020-11-20 +--- + +{% include toc.html %} + +# Grunnleggende om halvledere + +## Historie + +* **1830** - Mekanisk +* **1944** - Elektromekanisk +* **1946** - Releer og radiorør +* **1948** - Transistor +* **1958** - Første IC +* **1971** - Første mikroprosessor +* **2020** - Der i er i dag med nanoelektronikk + +Moores lov forutser hvor mange transistorer det er plass til per areal. +Skal dobles hver 18-24 måneder. + +## Halvledere + +![Oversikt over halvledere, metaller og isolatorer](./figures/conductivity.png) + +## Atomer og elektroner + +### Uskarphetsrelasjonen + +$$ \Delta x \cdot \delta p_x \geq \frac{\hbar}{2} $$ + +### Paulti prinsippet + +> To like fermioner kan ikke ha den samme kvantetilstanden. + +### Schrödingers likning + + +$$ - \frac{\hbar ^2}{2m}\frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) = -\frac{\hbar}{j}\frac{\hbar ^2}{2m}\frac{\partial \Psi(x,t)}{\partial t} $$ + +#### Løsninger + +$$ \psi(x) = Ae^{\pm ikx} \quad E= \frac{\hbar^2 k^2}{2m}$$ + +Partikkel i en "boks". + +![Partikkel i en boks](figures/particleBox.png) + +![Tette bånd](figures/tetteBonds.png) + + +## Effektiv masse + +$$ m^* = \frac{\hbar ^2}{\frac{d^2 E}{d k^2}} $$ + +Ser på krumningen til energien i k-rommet. +Høy kromming er liten effektiv masse, og vica versa. + +## Intrisisk materiale + +Inneholder bare en type materiale. + +$$ n = p = n_i $$ + +Der $n$ er tettheten av frie elektroner i lednignsbånd (CB), målt i $\text{cm}^{-3}$. +$p$ er tettheten av hull i valensbåndet (VB), målt i $\text{cm}^{-3}$. +Og $n_i$ er den intrisiske elektrontettheten, målt i $\text{cm}^{-3}$. + +![Intrisisk materiale](figures/intrinsic.png) + +## Ekstrinsiske materialer + +Disse er intrinsiske materialer som er dopet med et donor eller akseptor materiale. + +I Si er det typisk As (Arsenik, donor), eller B (Bor, akseptor). + +### n-type + +$$ n_0 \gg p_0,n_i $$ + +Der $n_0$ er elektrontettheten i termisk likevekt. + +### p-type + +$$ p_0 \gg n_0,n_i $$ + +Der $n_0$ er elektrontettheten i termisk likevekt. + +## Elektron-Hull-par i intrinsiske materialer + +Elektroner og hull genereres og rekominerer kontinuerlig. + +$$ r_i = g_i $$ +$$ r_i = \alpha n_0 p_0 = \alpha n_i^2 = g_i $$ + +![Bærertetthet](figures/dos.png) + +## Bærertetthet + +Hvordan beskrive hvordan $e^-$ og $h^+$ er fordelt i CB og VB. + +$$ \delta n(E) = N(E) \cdot f(E) \cdot \delta E $$ + +Der +* $\delta n(E)$ er tettheten av $e^-$ i CB. +* $N(E)$ er mulige av elektrontilstander +* $f(E)$ er Fermi-Dirac sannsynlighetsfordelingen +* $\delta E$ er energidifferansen vi ser på. I tilfellet over, er det $E_g$. + +Finnes flere typer +* Isotropisk båndstruktur +* Anisotropisk båndstruktur + +$$ N_C(E) = 4\pi \left(\frac{2m^*}{h^2}\right)^{\frac{3}{2}}\cdot E^\frac{1}{2} $$ + + +### Fermi-Dirac + +$$ f(E) = \frac{1}{1 + \exp{\frac{E - E_F}{k_B T}}} $$ + +#### Fermi-Dirac ved forskjellige dopinger +![Fermi-Dirac ved forskjellige dopinger](figures/fermiDirac.png) + +I et intrinsisk materiale ligger fordelingen midt i båndapet. +For en n-type doping vil fordelingen bevege seg mot CB, og i p-type vil den bevege seg mot VB. + +### Frie elektroner og hull + +Ved å se på "summen" av elektrontilstander, $ N_C $ og sannsynligheten for å finne dem der. + +$$ \int_{0}^\infty f(E)N_C(E) dE $$ + +Gir oss en likning for frie elektroner i termisk likevet. + +$$ n_0 = \underbrace{2\left(\frac{2\pi m_n^* k_B T}{h^2}\right)^\frac{3}{2}}_{N_c} e^{-\frac{E_C - E_F}{k_B T}} $$ + +Som forkortet, og på samme måte for $p_0$ + +$$ n_0 = N_c e^{-\frac{E_C - E_F}{k_B T}}, \qquad N_c = 2\left(\frac{2\pi m_n^* k_B T}{h^2}\right)^\frac{3}{2} $$ + +$$ p_0 = N_v e^{-\frac{E_F - E_V}{k_B T}}, \qquad N_v = 2\left(\frac{2\pi m_p^* k_B T}{h^2}\right)^\frac{3}{2} $$ + +### Noen resultater + +$$ n_0 p_0 = N_c N_v e^{-\frac{E_g}{k_B T}} $$ +$$ n_i p_i = N_c N_v e^{-\frac{E_g}{k_B T}} $$ + +Som sammen med $n_i = p_i$, gir følgende: + +$$ n_0 p_0 = n_i^2 $$ + +Dette gir oss igjen + +$$ n_0 = n_i e^{-\frac{E_F - E_i}{k_B T}} $$ + +$$ p_0 = n_i e^{-\frac{E_i - E_F}{k_B T}} $$ + + +### Noen eksempler på bærertetthet + +![DOS](figures/carrierDensity.png) + +## Drift av ladningsbærere + +Drifter i alle retninger, ikke noen som er preferert. Litt som en biesverm. + +Dersom det påtrykkes et elektrisk felt, vil partiklene fremdeles drifte, men de vi ha en netto bevegelse i en retning. +Elektroner vil bevege seg mot feltet, og hull vil bevege seg med. + +Strømmen er beskrevet av følgende: + +$$ J_x = q(n\mu_n + p\mu_p)E_x $$ + +Der + +$$ \mu_n = \frac{q \tau}{m_n^*} \quad \text{og} \quad \mu_p = \frac{q \tau}{m_p^*} $$ + +## Hall-effekten + +Halleffekten kan brukes til å se på mobiliteten til majoritetsladningsbærerene. F.eks. hull i p-type er majoritetsladningsbærere. + +![Hall-effekten](figures/HallEffect.png) + +Vi bruker Lortenz kraften, gitt som under: + +$$ \vec{F} = q\left(\vec{E} + \vec{v}\times\vec{B}\right)$$ + +Ved påtrykt strømm, $J_x$ og magnetfelt $B$ vil det settes opp et elektrisk felt mellom kontaktene A og B. + +La halvlederen være en p-type, da vil, pga. Lorenz-kraften gjøre at hull beveger seg mot kontakten A. Og dermed lage et målbart elektrisk felt fra A til B. + +Motsatt felt for n-type med samme strøm og magnetfelt. + +Tettheten vil da være gitt som under. + +$$ E_y = \frac{J_x}{qp_0}B_z = R_H J_x B_z $$ + +$$ R_H \equiv \frac{1}{qp_0} $$ + +Som gir følgende + +$$ p_0 = \frac{I_x B_z}{q t V_{AB}} $$ + +## Diffusjon + +Natrulig prosess , som å blande melk i kaffe/te eller hvordan oksygen tas opp i kroppen. + +> Diffusjon er å utligne konsentrasjonsforskjeller over tid, ved bruk av en tilfeldig prosess. + +To viktige parameter i diffusjon: +* Spredningstiden $\tau$, gjennomsnittlig spredningsintervall +* Spredningslengden $\bar{l}$, gjennomsnittlig lengde mellom spredninger + +### Elektronfluxen gitt av diffusjon + +For elektroner: + +$$ \phi_n(x) = -D_n \frac{dn(x)}{dx}, \quad \text{der } D_n = \frac{\bar{l}^2}{2\tau} $$ + +For hull: + +$$ \phi_p(x) = -D_p \frac{dp(x)}{dx}, \quad \text{der } D_n = \frac{\bar{l}^2}{2\tau} $$ + +$D_n$ og $D_p$ kalles diffusjonskonstantene. + +### Strømmen gitt av diffusjon + +For elektroner: + +$$ J_n^\text{diff} = q D_n \frac{dn(x)}{dx} $$ + +For hull: + +$$ J_p^\text{diff} = -q D_p \frac{dp(x)}{dx} $$ + +### Strømmen gitt av diffusjon med påsatt elektrisk felt + +For elektroner: + +$$ J_n(x) = q\mu_n n(x) E(x) + q D_n \frac{dn(x)}{dx} $$ + +For hull: + +$$ J_p(x) = q\mu_p p(x) E(x) - q D_p \frac{dp(x)}{dx} $$ + +Der summen av disse gir den totale strømmen: + +$$ J(x) = J_n(x) + J_p(x) $$ + +## Einsteinrealasjonen + +> I termisk likevekt går det ingen netto strøm. Dermed må det settes opp et E-felt for å kompensere driftsstrømmen. + +$$ J_p(x) = q\mu_p p(x) E(x) - q D_p \frac{dp(x)}{dx} = 0 $$ + +Som gir: + +\begin{align\*} + E(x) &= \frac{D_p}{\mu_p}\cdot \frac{1}{p_0(x)}\cdot \frac{dp(x)}{dx} \\\\\ + E(x) &= \frac{D_p}{\mu_p}\cdot \frac{1}{k_B T}\left(\frac{dE_i(x)}{dx} - \frac{dE_F(x)}{dx} \right) +\end{align\*} + +Ved termisk likevekt er $\frac{dE_F(x)}{dx} = 0$ og $\frac{dE_i(x)}{dx} = qE(x)$. +Dermed får vi Einsteinrelasjonen: + +$$ \frac{D}{\mu} = \frac{k_B T}{q} $$ + + +## Kontinuitetslikningen + +Viser sammenheng mellom endring i hulltetthet og strømmenm gjennom et areale. + +![Kontinuitet av strømmer](figures/continuityEq.png) + +$$ \frac{\partial \delta n(x,t)}{\partial t} = \phantom{-}\frac{1}{q}\frac{\partial J_n(x,t)}{\partial x} - \frac{\delta n}{\tau_n} $$ + +$$ \frac{\partial \delta p(x,t)}{\partial t} = -\frac{1}{q}\frac{\partial J_p(x,t)}{\partial x} - \frac{\delta p}{\tau_p} $$ + + +## Steady State + +Anta det lyses, med konstant effekt, på ene enden av en bit med n-type halvleder. + +![Bit med halvleder](figures/steadyStateBar.png) + +Man kan videre anta at det er en konstant hulltetthet på enden med lyset. + +$$ \delta p(x=0) = \Delta p $$ + +Fra diffusjon kan man anta at hullene diffunderer ut over i biten. +Siden det ikke er noen tidsavhengighet i hullkonsentrajsonen vil diffusjonsliknignen bli: + +$$ \frac{d^\delta p}{dx^2} = \frac{\delta p}{D_p \tau_p} \equiv \frac{\delta p}{L_p^2} $$ + +Der $ L_p \equiv \sqrt{D_p \tau_p} $. + +Denne har en generell løsning: + +$$ \delta p = C_1 \exp{\frac{x}{L_p}} + C_2 \exp{\frac{-x}{L_p}} $$ + +Og med grensebetingelser, $\delta p(x=0) = \Delta p$ og $\delta p(x \rightarrow \infty) = 0$, gir det oss: + +$$ \delta p(x) = \Delta p \exp{\frac{-x}{L_p}} $$ + + +## Haynes-Shockley eksperimentet + +Eksperiment som gir informasjon om minoritetsladningsbærere. + +![Haynes-Shockley Teori](figures/haynes-ShockleyTheory.png) + +Vi bruker prinsippet om steady state for å finne mobiliteten til minoritetsladningsbærere og diffusjonskonstanten. + +Ved å trykke på en strøm på ene siden, vil vi få et E-felt over halvlederen. +Ved å deretter sende inn en lyspuls på ene enden, vil det kunne detekteres en utsmørt versjon av pulsen etter en tid $t_d$. + +![Haynes-Shockley Eksperiment](figures/haynes-ShockleyExp.png) + + +# PN-overganger \ No newline at end of file