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Solid State Electronic Devices

Ben G. Streetman, Sanjay Kumar Banerjee

Chapter 4

Excess Carriers in Semiconductors - all with Video Answers

Educators

Chapter Questions

02:16

Problem 1

With $E_{F}$ located $0.4 \mathrm{eV}$ above the valence band in a Si sample, what charge state would you expect for most Ga atoms in the sample? What would be the predominant charge state of Zn? Au? Note: By charge state we mean neutral, singly positive, doubly negative, etc.

Suzanne W.
Suzanne W.
Numerade Educator
02:09

Problem 2

On a Si sample, incident light is at $t=0$ uniformly, which generates excess carriers for $t>0$. The generation rate for carriers is $6 \times 10^{22} / \mathrm{cm}^{3}$. Sample is doped with $2 \times 10^{17} / \mathrm{cm}^{3}$ As atoms. Determine the conductivity of the sample at $t=5 \mathrm{~ms}$. Find the separation of quasi Fermi levels at $300 \mathrm{~K}$.

Chai Santi
Chai Santi
Numerade Educator
00:50

Problem 3

Construct a semilogarithmic plot such as Fig. 4-7 for Si doped with $2 \times 10^{15}$ donors $/ \mathrm{cm}^{3}$ and having $4 \times 10^{14} \mathrm{EHP} / \mathrm{cm}^{3}$ created uniformly at $t=0$. Assume that $\tau_{n}=\tau_{p}=5 \mu \mathrm{s}$.

Chai Santi
Chai Santi
Numerade Educator
01:33

Problem 4

Calculate the recombination coefficient $\alpha_{r}$ for the low-level excitation described in Prob. 4.3. Assume that this value of $\alpha_{r}$ applies when the GaAs sample is uniformly exposed to a steady state optical generation rate $g_{\mathrm{op}}=10^{19} \mathrm{EHP} / \mathrm{cm}^{3}-\mathrm{s}$. Find the steady state excess carrier concentration $\Delta n=\Delta p$.

Chai Santi
Chai Santi
Numerade Educator
01:33

Problem 5

An intrinsic Si sample is doped with donors from one side such that $N_{d}=N_{0} \exp (-a x) .$ (a) Find an expression for the built-in electric field at equilibrium over the range for which $N_{d} \gg n_{i}$. (b) Evaluate the field when $a=1(\mu \mathrm{m})^{-1}$. (c) Sketch a band diagram such as in Fig. 4-15 and indicate the direction of the field.

Chai Santi
Chai Santi
Numerade Educator
13:03

Problem 6

A Si sample with $10^{15} / \mathrm{cm}^{3}$ donors is uniformly optically excited at room temperature such that $10^{19} / \mathrm{cm}^{3}$ EHPs are generated per second. Find the separation of the quasi-Fermi levels and the change of conductivity upon shining the light. Electron and hole lifetimes are both $10 \mu$ s. $D_{p}=12 \mathrm{~cm}^{2} / \mathrm{s}$.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
01:57

Problem 7

An n-type Si sample is doped at $10^{15} \mathrm{~cm}^{-3}$. We shine light on it to create EHPs at $10^{19} \mathrm{~cm}^{-3} / \mathrm{s}$. What is the steady state concentration of minority carriers, if the lifetime is $100 \mathrm{~ns}$ ? How long does it take for the hole concentration to drop $10 \%$, after the light is switched off? How long for the hole concentration to reach a value that is $10 \%$ higher than the thermal equilibrium value?

Chai Santi
Chai Santi
Numerade Educator
01:40

Problem 8

An $\mathrm{n}$ -type Si sample with $N_{d}=10^{15} \mathrm{~cm}^{-3}$ is steadily illuminated such that $g_{\text {op }}=10^{21} \mathrm{EHP} / \mathrm{cm}^{3}-\mathrm{s}$. If $\tau_{n}=\tau_{p}=1 \mu \mathrm{s}$ for this excitation, calculate the separation in the quasi-Fermi levels, $\left(F_{n}-F_{p}\right)$. Draw a band diagram such as Fig. 4-11.

Chai Santi
Chai Santi
Numerade Educator
01:45

Problem 9

For a $2-\mathrm{cm}$ -long doped $\mathrm{Si}$ bar $\left(N_{d}=10^{16} \mathrm{~cm}^{-3}\right)$ with a cross-sectional area $=0.05 \mathrm{~cm}^{2}$, what is the current if we apply $10 \mathrm{~V}$ across it? If we generate $10^{20}$ EHPs per second per $\mathrm{cm}^{3}$ uniformly in the bar and the lifetime $\tau_{n}=\tau_{p}=10^{-4} \mathrm{~s}$, what is the new current? Assume the low-level $\alpha_{r}$ doesn't change for high-level injection. If the voltage is then increased to $100,000 \mathrm{~V}$, what is the new current? Assume $\mu_{p}=500 \mathrm{~cm}^{2} / \mathrm{V}$ -s, but you must choose the appropriate value for electrons.

Chai Santi
Chai Santi
Numerade Educator
03:53

Problem 10

Design and sketch a photoconductor using a $5-\mu \mathrm{m}$ -thick film of $\mathrm{CdS}$, assuming that $\tau_{n}=\tau_{p}=10^{-6} \mathrm{~s}$ and $N_{d}=10^{14} \mathrm{~cm}^{-3}$. The dark resistance (with $g_{\text {op }}=0$ ) should be $10 \mathrm{M} \Omega$, and the device must fit in a square $0.5 \mathrm{~cm}$ on a side; therefore, some sort of folded or zigzag pattern is in order. With an excitation of $g_{\text {op }}=10^{21} \mathrm{EHP} / \mathrm{cm}^{3}-\mathrm{s}$, what is the resistance change?

Mrinal Rana
Mrinal Rana
Numerade Educator
04:02

Problem 11

A $80 \mathrm{~mW}$ laser beam with wavelength $\lambda=600 \mathrm{~nm}$ is focused on a Si sample of $80 \mu \mathrm{m}$ thickness. The absorption coefficient of the sample is $\alpha=8 \times 10^{3} / \mathrm{cm}$. Find the number of photons emitted per second, assuming perfect quantum efficiency. What power should be delivered to the sample as heat?

Shalini Tyagi
Shalini Tyagi
Numerade Educator
01:22

Problem 12

A material is doped such that electron concentration varies linearly across the sample, which is $0.5 \mu \mathrm{m}$ thick. Donor concentration varies from 0 (at $x=0$ ) to $10^{16} / \mathrm{cm}^{3}$ (at $\left.x=0.5 \mu \mathrm{m}\right)$. Write equations for total electron and hole concentrations as a function of distance $x .$ Determine electron and hole diffusion current densities if the diffusion coefficients are $D_{n}=30 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{sec}$ and $D_{p}=$ $12 \mathrm{~cm}^{2} / \mathrm{V}$ -sec. Find the expression for Fermi level $\left(E_{F}\right)$ as a function of $x$.

Chai Santi
Chai Santi
Numerade Educator
05:59

Problem 13

For the steady state minority hole distribution shown in Fig. $4-17$, find the expression for the hole quasi-Fermi level position $E_{i}-F_{p}(x)$ while $p(x) \gg p_{0}$ (i.e., while $F_{p}$ is below $E_{F}$ ). On a band diagram, draw the variation of $F_{p}(x) .$ Be careful-when the minority carriers are few (e.g., when $\Delta p$ is $n_{i}$ ), $F_{p}$ still has a long way to go to reach $E_{F}$.

Samuel Smith
Samuel Smith
Numerade Educator
01:22

Problem 14

We inject electrons into a p-type semiconductor 5 microns long such that the concentration varies linearly from $10^{20} \mathrm{~cm}^{-3}$ to 0 from left to right. If the mobility of the electrons is $500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, what is the current density if the electric fields are negligible?

Chai Santi
Chai Santi
Numerade Educator
01:22

Problem 15

We shine $10^{17}$ photons $/ \mathrm{cm}^{2}-\mathrm{s}$ which are all absorbed near the surface $x=0$ of a p-type semiconductor, raising the temperature of the sample to $500 \mathrm{~K}$. If the minority carrier lifetime is $200 \mathrm{~ns}$ in this material, electron mobility is $2000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, and hole mobility is $500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, calculate the electron diffusion current density 20 microns from the surface.

Chai Santi
Chai Santi
Numerade Educator
01:22

Problem 16

A long Si sample, $\mathrm{n}$ -doped at $10^{17} \mathrm{~cm}^{-3}$, with a cross-sectional area of $0.5 \mathrm{~cm}^{2}$ is optically excited by a laser such that $10^{20} / \mathrm{cm}^{3}$ EHPs are generated per second at $x=0 \mu \mathrm{m}$. They diffuse to the right. What is the total diffusion current at $x=50 \mu \mathrm{m}$ ? Electron and hole lifetimes are both $10 \mu \mathrm{s}$. $\mu_{p}=500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} ; D_{n}=36 \mathrm{~cm}^{2} / \mathrm{s}$

Chai Santi
Chai Santi
Numerade Educator
01:32

Problem 17

In an n-type semiconductor bar, there is an increase in electron concentration from left to right and an electric field pointing to the left. With a suitable sketch, indicate the directions of the electron drift and diffusion current flow and explain why. If we double the electron concentration everywhere, what happens to the diffusion current and the drift current? If we add a constant concentration of electrons everywhere, what happens to the drift and diffusion currents? Explain your answers with appropriate equations.

Suzanne W.
Suzanne W.
Numerade Educator
01:56

Problem 18

The current required to feed the hole injection at $x=0$ in Fig. $4-17$ is obtained by evaluating Eq. (4-40) at $x=0 .$ The result is $I_{p}(x=0)=q A D_{p} \Delta p / L_{p}$. Show that this current can be calculated by integrating the charge stored in the steady state hole distribution $\Delta p(x)$ and then dividing by the average hole lifetime $\tau_{p} .$ Explain why thís approach gives $I_{p}(x=0)$.

Chai Santi
Chai Santi
Numerade Educator
04:11

Problem 19

The direction of the built-in electric field can be deduced without math by sketching the result of a doping gradient on the band diagram. Starting with a flat Fermi level at equilibrium, place $E_{i}$ near or far from $E_{F}$ as the doping is varied for the two cases of a gradient in donor or acceptor doping as in Prob. $4.5 .$ Show the electric field direction in each case, based on Eq. (4-26). If a minority carrier is injected into the impurity gradient region, in what direction is it accelerated in the two cases? This is an interesting effect that we will use later in discussing bipolar transistors.

Chai Santi
Chai Santi
Numerade Educator
06:28

Problem 20

In Prob. $4.5$, the direction of the built-in electric field due to a gradient in doping was determined from Eqs. (4-23) and (4-26). In this problem, you are asked to explain qualitatively why the field must arise and find its direction. (a) Sketch a donor doping distribution as in Prob. $4.5$, and explain the field required to keep the mobile electrons from diffusing down the gradient. Repeat for acceptors and holes. (b) Sketch a microscopic region of the doping distribution, showing ionized donors and the resulting mobile electrons. Explain the origin and direction of the field as the electrons attempt to diffuse toward lower concentrations. Repeat for acceptors and holes.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
01:19

Problem 21

A p-type Si sample is used in the Haynes-Shockley experiment. The length of the sample is $2 \mathrm{~cm}$, and two probes are separated by $1.8 \mathrm{~cm}$. Voltage applied at the two ends is $5 \mathrm{~V}$. A pulse arrives at the collection point at $0.608 \mathrm{~ms}$, and the separation of the pulse is 180 sec. Calculate mobility and diffusion coefficient for minority carriers. Verify it from the Einstein relation.

Chai Santi
Chai Santi
Numerade Educator
03:10

Problem 22

A semiconductor bar of length $2 \mu \mathrm{m}$ with intrinsic carrier concentration of $10^{13} \mathrm{~cm}^{-3}$ is uniformly doped with donors at a concentration of $2 \times 10^{13} \mathrm{~cm}^{-3}$ and acceptors at a concentration of $10^{13} \mathrm{~cm}^{-3}$. If $D_{n}=26 \mathrm{~cm}^{2} / \mathrm{s}$ and $D_{p}=52 \mathrm{~cm}^{2} / \mathrm{s}$, calculate the electron and hole drift current densities for an applied voltage of $5 \mathrm{~V}$. In this semiconductor, electrons are in the ohmic regime for fields less than $10^{5} \mathrm{~V} / \mathrm{cm}$, but travel with a saturation velocity of $10^{8} \mathrm{~cm} / \mathrm{s}$ for fields above that. For holes, they are ohmic below $10^{4} \mathrm{~V} / \mathrm{cm}$, and travel with a saturation velocity of $10^{5} \mathrm{~cm} / \mathrm{s}$ above that field. What are the electron and hole diffusion current densities in the middle of the bar? (Assume $\mathrm{T}=300 \mathrm{~K}$.)

Chai Santi
Chai Santi
Numerade Educator
01:56

Problem 23

A recently discovered semiconductor has $N_{c}=10^{19} \mathrm{~cm}^{-3}, N_{v}=5 \times 10^{18} \mathrm{~cm}^{-3}$, and $E_{g}=2 \mathrm{eV}$. If it is doped with $10^{17}$ donors (fully ionized), calculate the electron, hole, and intrinsic carrier concentrations at $627^{\circ} \mathrm{C}$. Sketch the simplified band diagram, and specify the value of $E_{F}$ and $E_{i}$ with respect to the band edges. If we apply $5 \mathrm{~V}$ across a piece of this semiconductor $8 \mu \mathrm{m}$ long, what is the current? The piece is $2 \mu \mathrm{m}$ wide and $1.5 \mu \mathrm{m}$ thick. The diffusion coefficient of holes and electrons is $25 \mathrm{~cm}^{2} / \mathrm{s}$ and $75 \mathrm{~cm}^{2} / \mathrm{s}$, respectively.

Chai Santi
Chai Santi
Numerade Educator
02:09

Problem 24

A novel semiconductor sample has $L=2 \mu \mathrm{m}, W=0.5 \mu \mathrm{m}$, and thickness of $0.2 \mu \mathrm{m}$. It has an intrinsic carrier concentration of $10^{12} \mathrm{~cm}^{-3}$. If it has an ionized donor concentration of $2 \times 10^{12} \mathrm{~cm}^{-3}$, calculate the electron and hole currents for an applied bias of $10 \mathrm{~V}$ across the length of the bar, assuming ohmic behavior for electrons, but holes are traveling at saturation velocity. The electron and hole diffusion coefficients are $20 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$ and $5 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, respectively. The electron and hole saturation velocities are $10^{8} \mathrm{~cm} / \mathrm{s}$ and $10^{7} \mathrm{~cm} / \mathrm{s}$, respectively, in this semiconductor.

Chai Santi
Chai Santi
Numerade Educator
03:10

Problem 25

Sketch the simplified band diagram (with proper labeling of positions and energies) for a semiconductor bar with a band gap of $2 \mathrm{eV}$ and $N_{c}=10^{19} \mathrm{~cm}^{-3}$, $\mathrm{n}^{+}$ doped very heavily between $0 \mathrm{~cm}$ and $0.2 \mathrm{~cm}, \mathrm{n}$ -type doped region $\left(10^{17} \mathrm{~cm}^{-3}\right)$ from $0.2 \mathrm{~cm}$ to $0.7 \mathrm{~cm}$, and then very heavily $\mathrm{n}^{+}$ doped from $0.7 \mathrm{~cm}$ to $1 \mathrm{~cm}$, to which we hook up a $0.5 \mathrm{~V}$ battery (positive terminal connected to left side of bar). There is negligible voltage drop across the highly conducting, heavily doped $\mathrm{n}^{+}$ regions. (Draw a schematic of the bar, and align the band diagram under that schematic).
What is the current density if the electron diffusion coefficient is $100 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$. If the electrons were injected with negligible kinetic energy from the right side of the bar, and they traveled without scattering, what is the kinetic energy of the electrons at $x=0.1 \mathrm{~cm}, 0.6 \mathrm{~cm}$, and $0.9 \mathrm{~cm}$ ?

Chai Santi
Chai Santi
Numerade Educator