Questions on semiconductor course
Q)how do we know we are dealing with low injection or high injection? so that we can use the correct approximations for low injection. equation (63) in book (page 51) in general case to find U, then the book assumed there is low injection, and did some approximations, I need to know when i can assume low injection in a problem.
A) when the injected minority carrier density is much lower than the majority carrier density, example \(n_{n}\simeq n_{no}\)
Q)what is difference between equation 66 page 52 and equation 61 page 94, since both are for low injection and equal capture corss section for holes and electron?
A)equation 61 page 94 is used when we have an applied external volatge?
Q)what is the difference between equation 64, page 51 and equation 67, page 52 since both are for low injection? it looks like equation 64 is when \(E_{t}=E_{i}\) while equation 67 is not.
Q) about problem 1 set 2, where is the 1/cosh term in the \(I_{EP}\)calculation?
Q1) why is low injection in an n-type semiconductor \(\Rightarrow \) \(n_{n}\gg p_{n}\)
Q) how do you decide to use short base diode or long base diode for finding the diffusion capacitance?
for long base \(C_{d}=\frac{AqL_{p}p_{no}}{KT}e^{\frac V{V_{T}}}\) while for short base \(C_{d}=\frac I{V_{T}}\left ( \frac{W^{2}}{2D_{p}}\right ) \)
example, the diffusion capacitance of the Emitter-Base junction is \(\frac{I_{E} }{V_{T}}\left ( \frac{W^{2}}{2D_{p}}\right ) \) where \(I_{E}=I_{En}+I_{En}\)
where \(A\) the area of the junction
\(q\) electron charge
\(L_{p}\) the diffusion length \(=\sqrt{D_{p}\tau _{p}}\)
\(D_{p}\) the diffusion constant
\(V_{T}=\frac q{KT}=0.0259\;eV\) at room temp.
\(K\) is Blotzmann constant=\(1.38066\times 10^{-23}J/K\)
\(q\) \(=1.60218\times 10^{-19}C\)
Q) how to find the electron effective mass \(m_{n}\) ?
Q)what is the transit time? \(\tau _{t}?and\;what\;is\;\frac 1\beta =\frac{\tau _{t}}{\tau _{p}}?\)
I think \(\tau _{p}=\)time required by hole to traverse the base=\(\frac{W^{2}}{2D_{p}}\)
Q)is , under LOW level injection, the recombination life time \(\tau _{r}\) the same thing as \(\tau _{p}?\) is the difference is that \(\tau _{r}\) is only when \(\sigma _{n}=\sigma _{n}=\sigma _{o}\) ?
\(\bullet \)common-base current gain \(\alpha _{o}=\frac{I_{CP}}{I_{E}}\)
\(\bullet \)emitter efficiency \(\gamma =\frac{I_{EP}}{I_{E}},\) note: if \(\frac{N_{E}}{N_{B}}\gg 1,like\;5000,then\;\gamma =1\)
\(\bullet \)base transport facotr \(\alpha _{T}=\frac{I_{CP}}{I_{EP}}\)
\(\bullet \alpha _{F}\) the forward common-base current gain (page 132)
\(\bullet \alpha _{R}\) the reverse commmon-base current gain=\(\frac{\text{number of minority carries collected at emiiter}}{\text{number of minority carries injected into base at the forward-biased collector-base junction }}\)
\(\bullet \alpha _{o}=\gamma \alpha _{T}\)
\(\bullet \)common-emitter current gain \(\beta _{o}=\frac{\Delta I_{C}}{\Delta I_{B}}=\frac{\alpha _{o}}{1-\alpha _{o}}\) if \(\gamma \) very close to 1,\(\beta _{o}=\frac{\gamma \alpha _{T}}{1-\gamma \alpha _{T}}\simeq \frac{\alpha _{T}}{1-\alpha _{T}}= \frac{2L_{p}^{2}}{W^{2}}\)
\(\bullet \) diffusion current \[ J_{n}=-q\left ( carrie\;flow\right ) =-q\left ( -D_{n}\frac{dn}{dx}\right ) \]
\(\bullet \)Einstein relation relates mobility to diffusivity\[ D_{n}=\left ( \frac{KT}q\right ) \mu _{n} \]
\(\bullet \) electron concentration in thermal equilibrium\[ n=n_{i}e^{\left ( \frac{E_{F}-E_{i}}{KT}\right ) } \]
\(\bullet \) \(\tau _{g}=\)the generation life time
\(\bullet \) \(\tau _{p}=\)the life time of the excess minority carriers
\(\bullet \)\(\tau _{c}=\) mean free time, the average time between collisions
\(\bullet \)when there is no doping (i.e. \(N_{A}=0,N_{D}=0\)), then , assuming this is an n-type semiconductore, then \[ n_{no}=p_{no}=n_{i} \] this means that\[ n_{no}p_{no}=n_{i}^{2} \]
when we have dopig added, then \(n_{n}=n_{no}+N_{D}\) and \(p_{n}=p_{no}+N_{A}\) and assuming full ionization, we have that \[ n_{n}p_{n}=n_{i}^{2} \]
and\[ n_{n}+N_{A}=p_{n}+N_{D} \]
\(\bullet \) mobility \(\mu =\frac{q\tau _{c}}{m_{n}}\)
\(\bullet m_{n}\) the electron effective mass
\(\bullet \)resistivity \(\rho \equiv \frac 1\sigma \) where \(\sigma \) is the conductivity.
\(\bullet \) capture cross section = \(\sigma _{p}\) or \(\sigma _{n}\) describes the effectiveness of the center to capture an electron and is a measure of how close the electron has come to the center to be captured. \(\sigma _{n}\) is in \(cm^{2}\) , so it is an area.
\(\bullet \) \(v_{th}\sigma _{n}=\) this may be visualized as the volume swept out per unit time by an electron with cross section \(\sigma _{n}\)
\(\bullet \)to find the built-in potional between 2 different doped materials, say \(p-n\) use this\[ V_{bi}=\frac{KT}q\ln \left ( \frac{N_{A}N_{D}}{n_{i}^{2}}\right ) \]
\(\bullet \) to find what is the width of the depletion region :
\[ W=\sqrt{\frac{2\epsilon _{s}}q}\sqrt{\left ( \frac 1{N_{D}}+\frac 1{N_{A}}\right ) \left ( V_{bi}-V_{external}\right ) } \]
\(\epsilon _{s}=1.0409\times 10^{-12}\)
\(\frac{2\epsilon _{s}}q=\frac{2\times 1.0409\times 10^{-12}}{1.60218\times 10^{-19}}=12993546\;\;\;\;\;\;\;\;\;\sqrt{\frac{2\epsilon _{s}}q}=3604.6\)
note that when forward bias, \(V_{external}\) is positive, and when revesrsed biased it is negative.
example, for \(W_{EB}=\sqrt{\frac{2\epsilon _{s}}q}\sqrt{\left ( \frac 1{N_{D}}+\frac 1{N_{A}}\right ) \left ( V_{bi}-V_{EB}\right ) }\) and \(W_{CB}=\sqrt{ \frac{2\epsilon _{s}}q}\sqrt{\left ( \frac 1{N_{D}}+\frac 1{N_{A}}\right ) \left ( V_{bi}+V_{CB}\right ) }\)
to find the extent of depletion region into only one side or the other use, example, given a base-collector juntion with \(W\) width, we want to find how far this \(W\) goes into the base, use:\[ W_{base^{\prime }part}=\frac{N_{C}}{N_{C}+N_{B}}W \]
the length into the collector is\[ W_{collecter^{\prime }part}=\frac{N_{B}}{N_{C}+N_{B}}W \]
\(\bullet \;\)\(n_{i}=1.45\times 10^{10}cm^{-2}\) where \(n_{i}\) is the intrinsic carrie density. it is the number of electrons per unit volume in the conduction band and equal to number of holes in the valence band. intrinsic semiconductor is one that contains relatively small amounts of impurities compared to the thermaly generated electrons and holes.
\(\bullet \) Fermi distribution function \(F(E)=\frac 1{1+e^{\frac{\left ( E-E_{F}\right ) }{KT}}}\) where \(k\) is Blotzmann constant, \(T\) is abs. temp. in Kelvin and \(E_{F}\) is the fermi level energy. the fermi level is the energy at which the probability of occupation by electron is exactly .5
\(\bullet \) about mobility, the KE energy of electron is given by\[ \frac 12m_{n}v_{th}^{2}=\frac 32KT \]
where \(m_{n}\) is the electron effective mass=?, and \(v_{th}\) is the electron average thermal velocity.
\(\bullet \)conductivity=\(\left ( qn\mu _{n}+qp\mu _{p}\right ) \) where \(\mu \) is the carrier mobility.
when we have \(\Delta n=\Delta p\ll N_{D}\) we call this low level injection
this means no recombination centers exist.
when the light is on, at steady state, \[ G_{L}=U=\frac{p_{n}-p_{no}}{\tau _{p}}\;\;\;\;\;\;\;\left ( 1\right ) \]
when the light switched of, the excess carries die away at rate untill we get back to \(p_{no}\)
\[ p_{n}\left ( t\right ) =p_{no}+\tau _{p}G_{L}e^{-\frac t{\tau _{p}}} \]
also we have
\[ G=G_{L}+G_{th} \]
befor we reach steady state, i.e just after the light shine is on,\[ \frac{dp_{n}}{dt}=G-L=G_{L}+G_{th} \]
in steady state
\[ G_{L}=R-G_{th}=U \]
this means there are recomination centers \(N_{t}\)
under steady state, non-equilibrium (i.e. source that causes generation exist and is ON), we have\[ U=R_{a}-R_{b}=\frac{v_{th}\sigma _{n}\sigma _{p}N_{t}\left ( p_{n}n_{n}-n_{i}^{2}\right ) }{\sigma _{p}\left ( p_{n}+n_{i}e^{\left ( \frac{E_{i}-E_{t}}{KT}\right ) }\right ) +\sigma _{n}\left ( n_{n}+n_{i}e^{\left ( \frac{E_{t}-Ei}{KT}\right ) }\right ) }\;\;\;\;\;\;\;\;\left ( 2\right ) \]
where\[ R_{a}=\left ( \sigma _{n}v_{th}\right ) nN_{t}\left ( 1-F\left ( E_{t}\right ) \right ) \]
\[ R_{b}=e_{n}N_{t}F\left ( E_{t}\right ) \]
at thermal equilibrium, \(R_{a}=R_{b}\)
where \(e_{n}\) is called the emission probability.\[ e_{n}=v_{th}\sigma _{n}n_{i}e^{\left ( \frac{E_{t}-E_{i}}{KT}\right ) } \]
this shows that as \(E_{t}\) goes away from \(E_{i}\) and closer to the conduction band, there is more probability of process \(R_{b}\) i.e. electron emmision from the center up to the conduction band.
see page 50 in book.
when we have low injection, so that \(n_{n}\gg p_{n}\) , and \(E_{t}\) close to the center of the gap, \(U\) simplifies to
\[ U\simeq v_{th}\sigma _{p}N_{t}\left ( p_{n}-p_{no}\right ) \]
compare this equation to equation (1) in the direct recombination, we see that \[ \tau _{p}=\text{ life time of holes in n-type }=\frac 1{v_{th}\sigma _{p}N_{t}} \]
this is valid only under LOW LEVEL INJECTION and \(E_{t}\) close to center of gap.
back to the general case, now assume that \(\sigma _{n}=\sigma _{p}=\sigma _{o}\) then \(U\) equation (2) becomes\[ \fbox{U=$\frac{v_th\sigma _oN_t\left ( p_nn_n-n_i^2\right ) }{p_n+n_n+2n_i\cosh \left ( \frac{E_t-E_i}{KT}\right ) }$}\;\;\;\;\;\;\;(3) \]
now, if we have LOW INJECTION, \(U\) simplied to
\[ \fbox{U=v$_th$$\sigma $$_o$N$_t$$\frac{p_nn_no}{1+\left ( \frac{2n_i}{n_no+p_no}\right ) \cosh \left ( \frac{E_t-E_i}{KT}\right ) }$=$\frac{p_n-p_no}{\tau _r}$} \]
where \(\tau _{r}\) is the recombination life time
1) when we have injection of excess carries, \(pn>n_{i}^{2}\) this is called then RECOMBINATION, as in FORWARD biased juntions.
2)when we have \(pn<n_{i}^{2}\) this is then called GENERATION. this is as in REVERSE BIASED junction
3)to find \(G\) let \(p_{n}<n_{i}\) and \(n_{n}<n_{i}\) in equation (3) we get\[ \frame{G=-U=$\frac{v_th\sigma _oN_tn_i}{2\cosh \left ( \frac{E_t-E_i}{KT}\right ) }$=$\frac{n_i}{\tau _g}$} \]
where \(\tau _{g}\) is the generation life time. the above ASSUMES \(\sigma _{n}=\sigma _{p}=\sigma _{o}\) only.
\[ \frac{\tau _{g}}{\tau _{p}}=2\cosh \left ( \frac{E_{t}-E_{i}}{KT}\right ) \]
4)when a \(p^{+}-n\) junction is forward biased, to find total current through it, we add the juntion current to the recombination current.
\[\begin{array} [c]{c}J=J_{p}\left ( x_{n}\right ) +J_{n}\left ( -x_{p}\right ) \\ =J_{s}\left ( e^{\frac V{V_{T}}}-1\right ) \end{array} \]
where\[ J_{s}=\text{saturation current density=}\frac{qD_{p}p_{no}}{L_{p}}+\frac{qD_{n}n_{po}}{L_{n}} \]
the recombination current\[ J_{rec}=\int _{0}^{w}qU\;dx\simeq \frac{qWn_{i}}{2\tau _{r}}e^{\frac V{2V_{T}}} \]
where\[ \tau _{r}=\frac 1{\sigma _{o}v_{th}N_{t}} \]
so, total forward current in forward biased p-n junction under High Injection is\[ J_{F}=J_{s}\left ( e^{\frac V{V_{T}}}-1\right ) +J_{rec} \]
when the emitter is much more doped than teh base, ie. \(p_{no}\gg n_{po}\)and \(V\gg \frac{3KT}q\) then we approixmate \[ J_{F}=q\sqrt{\frac{D_{p}}{\tau _{p}}} \frac{n_{i}^{2}}{N_{D}}e^{\frac V{V_{T}}}+\frac{qWn_{i}}{2\tau _{r}}e^{\frac V{V_{T}}}\sim e^{\frac V{\eta V_{T}}} \]
where \(\eta \) is called the ideality factor
5) in either generation or recombination, the most effective centers are those located near \(E_{t}\)
\(\bullet \) diffusion current \(=-qD_{p}\frac{dp}{dx}\)
\(\bullet \)drift current \(=q\mu _{p}p\mathit{E}\)