Classification of Materials and Energy Band Theory Questions in English

(B) In semiconductors,the valence band and conduction band are separated by a small energy gap,typically on the order of $1 \ eV$. This allows electrons to be thermally excited from the valence band to the conduction band at room temperature,which is why semiconductors exhibit intermediate electrical conductivity compared to conductors and insulators.

(A) In conductors,the valence band and the conduction band overlap each other. This overlap allows electrons to move freely from the valence band to the conduction band even at low temperatures,which is why conductors exhibit high electrical conductivity.

(A) In conductors,the valence band and the conduction band overlap each other. This overlap allows electrons to move freely from the valence band to the conduction band even at low temperatures,which is why conductors have high electrical conductivity.

(A) In insulators,the valence band is completely filled with electrons,and the conduction band is completely empty. The energy gap (forbidden energy gap) between the valence band and the conduction band is very large (typically $> 3 \ eV$). Therefore,electrons cannot jump from the valence band to the conduction band even at room temperature,making them poor conductors of electricity. Thus,the band gap is very high and the conduction band is empty.

(A) In conductors, the valence band and the conduction band overlap each other, or the energy gap between them is effectively zero.
This overlap allows electrons to move freely from the valence band to the conduction band, which is why conductors exhibit high electrical conductivity.

(C) Metals reflect incident light due to the oscillations of free electrons under the influence of the electric field of the incident light wave.
Electrical conductivity of metals decreases with an increase in temperature because the increased random thermal motion of ions increases the scattering of free electrons,thereby increasing resistance.
Therefore,the bonding in such a solid is metallic.

(C) In a semiconductor,as the temperature $(T)$ increases,the number of charge carriers (electrons and holes) increases exponentially due to the breaking of covalent bonds.
This increase in the number of charge carriers dominates over the effect of increased lattice vibrations (which would otherwise increase resistivity).
Consequently,the resistivity $(\varrho)$ of a semiconductor decreases rapidly with an increase in temperature.
The relationship is given by $\varrho = \varrho_0 e^{E_g / 2k_BT}$,where $E_g$ is the band gap energy and $k_B$ is the Boltzmann constant.
Graph $(C)$ correctly represents this exponential decrease of resistivity with temperature.

(C) In a semiconductor,the number of charge carriers (electrons and holes) increases exponentially with an increase in temperature because more covalent bonds are broken,releasing more charge carriers.
As the number of charge carriers increases,the electrical conductivity of the semiconductor increases,which leads to a decrease in its electrical resistance.
According to Ohm's law,$I = V/R$. Since the voltage $V$ remains constant and the resistance $R$ decreases,the current $I$ in the circuit will increase.

(C) In semiconductors,the energy gap between the valence band and the conduction band is small.
At room temperature,thermal energy is sufficient for some electrons to jump from the valence band to the conduction band by overcoming the forbidden energy gap.
As a result,some electrons move to the conduction band,making it partially filled.
Simultaneously,this leaves behind vacancies (holes) in the valence band,making it partially empty.
Therefore,at room temperature,both the conduction band and the valence band are partially filled/empty.

(C) In an $n$-type semiconductor,pentavalent impurity atoms (donor atoms) are added to the intrinsic semiconductor.
These donor atoms create discrete energy levels known as donor energy levels.
These donor energy levels are located within the band gap,just below the conduction band.
Because these levels are very close to the conduction band,the electrons in these levels can easily be thermally excited into the conduction band at room temperature.
Therefore,the correct option is $(C)$.

(C) In an intrinsic semiconductor,the number of free electrons is very small compared to a conductor at room temperature.
As the temperature increases,more covalent bonds break,leading to an increase in the number of free electrons.
At absolute zero temperature $(T = 0 \ K)$,an intrinsic semiconductor behaves as a perfect insulator because there is insufficient thermal energy to break covalent bonds.
Therefore,there are no free electrons at absolute zero temperature.
Statement $C$ is incorrect because free electrons exist at temperatures above absolute zero.

(A) In insulators,the energy band gap between the valence band and the conduction band is very large (typically $> 3 \ eV$).
Because of this large energy gap,electrons cannot gain enough thermal energy to jump from the valence band to the conduction band.
Consequently,the conduction band remains empty at all temperatures,preventing the flow of electric current.

(B) In semiconductors,as the temperature increases,more charge carriers (electrons and holes) are generated due to thermal excitation. This leads to an increase in conductivity. Since conductivity is the reciprocal of resistivity,the resistivity of a semiconductor decreases with an increase in temperature. Therefore,the statement that resistivity increases with temperature is false.

(D) In a semiconductor,as the temperature increases,more covalent bonds break,leading to an increase in the number of charge carriers (electrons and holes). This increase in charge carrier density results in a decrease in electrical resistance and an increase in electrical conductivity. Therefore,the resistance decreases and the conductivity increases.

(C)
At absolute zero temperature $(T = 0 \ K)$,there is no thermal energy available to excite electrons from the valence band to the conduction band in a pure semiconductor like silicon.
Consequently,the valence band is completely filled and the conduction band is completely empty.
Due to the absence of free charge carriers,the material cannot conduct electricity.
Therefore,pure silicon behaves as an insulator at absolute zero temperature.

(A) The resistance of a conductor (like copper) is directly proportional to its temperature. As the temperature decreases,the resistance of copper decreases.
The resistance of a semiconductor (like germanium) is inversely proportional to its temperature because the number of charge carriers decreases significantly as the temperature drops. As the temperature decreases,the resistance of germanium increases.
Therefore,when both are cooled from room temperature to $40 \ K$,the resistance of copper decreases and the resistance of germanium increases.

(A) The forbidden energy gap $(E_g)$ represents the minimum energy required for an electron to jump from the valence band to the conduction band.
For Germanium $(Ge)$ at room temperature $(300 \ K)$,the forbidden energy gap is approximately $0.72 \ eV$.
For Silicon $(Si)$ at room temperature $(300 \ K)$,the forbidden energy gap is approximately $1.1 \ eV$.
Therefore,the correct value for Germanium is $0.72 \ eV$.

(C) The energy band gap $(E_g)$ is the energy difference between the conduction band and the valence band.
For Carbon (diamond),the band gap is approximately $5.4 \ eV$.
For Silicon,the band gap is approximately $1.1 \ eV$.
For Germanium,the band gap is approximately $0.7 \ eV$.
Comparing these values,we find that $(E_g)_C > (E_g)_{Si} > (E_g)_{Ge}$.
Therefore,option $(C)$ is correct.

(D) The energy gap $(E_g)$ is the energy difference between the valence band and the conduction band.
- For metals,the valence and conduction bands overlap,so $E_g = 0$.
- For semiconductors,the energy gap is small,typically $E_g < 3 \ eV$ (e.g.,Silicon is $1.1 \ eV$,Germanium is $0.7 \ eV$).
- For insulators (often referred to as non-metals in this context),the energy gap is very large,typically $E_g > 3 \ eV$.
Therefore,the correct answer is insulators or non-metals.

(B) $CdS$ (Cadmium Sulfide) is a compound semiconductor.
It is composed of elements from groups $12$ and $16$ of the periodic table.
Since it is a compound formed by inorganic elements,it is classified as an inorganic semiconductor.
Therefore,the correct option is $B$.

(D) The energy gap $(E_g)$ of a semiconductor is typically less than $3 eV$.
Copper,Iron,and Aluminium are metals (conductors) where the valence and conduction bands overlap,meaning they do not have an energy gap in the traditional sense.
Germanium is a semiconductor with an energy gap of approximately $0.7 eV$.
Since $0.7 eV < 3 eV$,Germanium is the correct answer.

(D) Copper is a conductor (metal). For metals,the resistance decreases as the temperature decreases because the lattice vibrations (collisions) decrease.
Germanium is a semiconductor. For semiconductors,the resistance increases as the temperature decreases because the number of charge carriers (electrons and holes) decreases significantly due to the reduction in thermal energy.

(B) In conductors,there is no forbidden energy gap between the conduction band and the valence band.
For materials like copper,which is a metal (conductor),the valence band and the conduction band overlap.
This overlapping allows electrons to move freely,which is why copper is a good conductor of electricity.
Silicon,germanium,and carbon (in diamond form) are semiconductors or insulators,where a forbidden energy gap exists.

(C) The forbidden energy gap $(E_g)$ of a semiconductor depends on temperature.
For Germanium $(Ge)$, the forbidden energy gap at $0 \,K$ is approximately $0.74 \,eV$.
As the temperature increases, the energy gap decreases slightly.
At room temperature $(300 \,K)$, the value is approximately $0.67 \,eV$ to $0.72 \,eV$.
Comparing the given options, $0.74 \,eV$ is the standard value for $Ge$ at $0 \,K$.

(C) The energy required to generate an electron-hole pair is equal to the forbidden energy gap $E_{g}$.
For a photon to create an electron-hole pair,its energy must be at least equal to $E_{g}$.
The relationship between energy and wavelength is given by $E = \frac{hc}{\lambda}$.
To find the maximum wavelength $\lambda_{max}$,we use the minimum energy $E_{g}$:
$\lambda_{max} = \frac{hc}{E_{g}}$
Substituting the given values:
$\lambda_{max} = \frac{12400 eV-Å}{0.72 eV}$
$\lambda_{max} = 17222.22 Å$
Rounding to the nearest integer,we get $\lambda_{max} = 17222 Å$.

(A) The resistivity of a material determines its classification as a conductor,semiconductor,or insulator.
Metals (conductors) have low resistivity,typically in the range of $10^{-2}$ to $10^{-8} \Omega-m$.
Insulators have very high resistivity,typically in the range of $10^{11}$ to $10^{19} \Omega-m$.
Semiconductors have resistivity values that lie between those of conductors and insulators,typically in the range of $10^{-3}$ to $10^6 \Omega-m$ (or $10^{-1}$ to $10^8 \Omega-cm$).
Given the options provided,the range $10^{-3}$ to $10^6 \Omega-cm$ is the standard accepted range for semiconductors.

(C) The conductivity of a semiconductor is given by $\sigma = e(n_e \mu_e + n_h \mu_h)$.
As the temperature increases,the number density of charge carriers ($n_e$ and $n_h$) increases exponentially according to the relation $n = C T^{3/2} \exp(-E_g / 2kT)$.
Although the relaxation time $(\tau)$ decreases with an increase in temperature due to increased scattering,the exponential increase in the number density of charge carriers far outweighs the decrease in mobility caused by the reduction in relaxation time.
Therefore,the net effect is an increase in the conductivity of the semiconductor.

(D) Key Idea: In nature,metals have a positive temperature coefficient of resistivity,while semiconductors have a negative temperature coefficient of resistivity.
Aluminium is a metal. As temperature increases,the collision frequency of electrons increases,which increases resistivity and decreases conductivity.
Silicon $(Si)$ is a semiconductor. As temperature increases,more charge carriers are generated,which decreases resistivity and increases conductivity.
The assertion $(A)$ states that the conductivity of aluminium increases and silicon decreases,which is the opposite of the physical reality. Therefore,$(A)$ is incorrect.
The reason $(R)$ correctly states that metals have a positive temperature coefficient of resistivity and semiconductors have a negative temperature coefficient of resistivity. Therefore,$(R)$ is correct.
Thus,$(A)$ is incorrect but $(R)$ is correct.

(A) The classification of materials based on the energy band gap $(E_g)$ is as follows:
$1$. For conductors,the energy gap is $E_g \approx 0 \ eV$.
$2$. For semiconductors,the energy gap is typically $E_g < 3 \ eV$.
$3$. For insulators,the energy gap is large,typically $E_g > 5 \ eV$.
Since the given energy gap is $5.4 \ eV$,which is greater than $5 \ eV$,the substance is an insulator.

(D) In solid-state physics,materials are classified based on their energy band gaps.
Metals have overlapping valence and conduction bands,meaning the band gap is effectively $0 eV$.
Semiconductors have a small band gap,typically around $1 eV$ to $3 eV$.
Insulators have a very large energy band gap,which is generally greater than $3 eV$,preventing electron flow from the valence band to the conduction band under normal conditions.
Therefore,insulators have the largest band gap among the given options.

(C) At absolute zero temperature $(T = 0 \ K)$,all valence electrons are tightly bound to their respective atoms in the crystal lattice.
There is no thermal energy available to excite electrons from the valence band to the conduction band.
Since there are no free charge carriers (electrons or holes) available for conduction,the intrinsic semiconductor behaves as a perfect insulator.

(C) In a semiconductor,at $0 \ K$,all electrons are in the valence band,and no electrons are present in the conduction band. Therefore,there are no free electrons at $0 \ K$. Thus,statement $(A)$ is true.
As the temperature increases,electrons gain thermal energy,which allows them to break covalent bonds and move into the conduction band. Consequently,the number of free electrons increases with temperature. Thus,statement $(C)$ is true.
Statement $(B)$ is false because free electrons are generated as temperature increases.
In a semiconductor,the number of free electrons is significantly lower than in a conductor because most electrons are bound in the lattice structure. Thus,statement $(D)$ is true.
Therefore,statements $(A), (C),$ and $(D)$ are true,while $(B)$ is false.

(C) In semiconductors,the energy gap between the valence band and the conduction band is small.
As the temperature increases,more thermal energy is available to the electrons in the valence band.
This allows more electrons to overcome the energy gap and jump into the conduction band.
Consequently,the number of free electrons in the conduction band increases,which leads to an increase in conductivity and a decrease in resistance.
Therefore,the correct statement is that the number of electrons in the conduction band increases.

(D) The resistance of conductors depends on the temperature. For a conductor like copper,the resistance decreases as the temperature decreases.
For a semiconductor like germanium,the resistance increases as the temperature decreases because the number of charge carriers (electrons and holes) decreases significantly at lower temperatures.
Therefore,when cooled from room temperature to $77 \ K$,the resistance of copper decreases and the resistance of germanium increases.

(C) Copper is a metallic conductor, and its resistance decreases as the temperature decreases because the frequency of collisions between electrons and lattice ions decreases.
Germanium is a semiconductor. In semiconductors, the number of charge carriers (electrons and holes) decreases exponentially as the temperature decreases, which leads to a significant increase in resistance.
Therefore, when both are cooled from room temperature to $80 \text{ K}$, the resistance of copper decreases, and the resistance of germanium increases.

(B) The band gap energy is given as $E_g = 0.7 \ eV$.
To create an electron-hole pair,the energy of the incident photon must be at least equal to the band gap energy,i.e.,$E = h\nu = \frac{hc}{\lambda} \geq E_g$.
For the maximum wavelength $\lambda_{max}$,the energy of the photon must be exactly equal to the band gap energy:
$\frac{hc}{\lambda_{max}} = E_g$
$\lambda_{max} = \frac{hc}{E_g}$
Substituting the given values:
$\lambda_{max} = \frac{4.136 \times 10^{-15} \ eV \cdot s \times 3 \times 10^8 \ m/s}{0.7 \ eV}$
$\lambda_{max} = \frac{12.408 \times 10^{-7}}{0.7} \ m$
$\lambda_{max} \approx 17.7257 \times 10^{-7} \ m$
Converting this to the required format:
$\lambda_{max} \approx 1772.57 \times 10^{-9} \ m \approx 1773 \times 10^{-9} \ m$.

(D) In the energy band theory of solids:
$1$. $A$ conductor has a conduction band that is partially filled with electrons,or the valence band and conduction band overlap.
$2$. An insulator has a large energy gap between the valence band and the conduction band,and the conduction band is completely empty at $0 \ K$.
$3$. $A$ semiconductor has a small energy gap between the valence band and the conduction band,and the conduction band is empty at $0 \ K$ but can be partially filled at higher temperatures.
Therefore,the statement that a solid is a conductor if its conduction band is partially filled is correct.

(D) At absolute zero temperature $(T = 0 \ K)$,all the valence electrons are tightly bound to their respective atoms in the crystal lattice.
There is no thermal energy available to excite electrons from the valence band to the conduction band.
Consequently,the conduction band remains completely empty,and the valence band is completely filled.
Due to the absence of free charge carriers,the semiconductor acts as a perfect insulator.

(C) The number of unit cells in volume $V$ is given by $\frac{V}{V_0}$.
Since each unit cell contains $N_0$ atoms,the total number of atoms in volume $V$ is $\frac{V}{V_0} \times N_0$.
The number of moles of germanium in volume $V$ is the total number of atoms divided by Avogadro's constant $N_A$,which is $\frac{V}{V_0} \times \frac{N_0}{N_A}$.
The mass $m$ is the number of moles multiplied by the molar mass $M$.
Therefore,$m = \frac{V}{V_0} \times \frac{N_0}{N_A} \times M$.

(D) The classification of materials based on energy band theory is as follows:
$1$. Conductors: The valence band and conduction band overlap,or the energy gap is negligible.
$2$. Semiconductors: The energy band gap is small,typically around $1 \ eV$ (e.g.,$Si \approx 1.1 \ eV$,$Ge \approx 0.7 \ eV$).
$3$. Insulators: The energy band gap is very large,typically greater than $3 \ eV$.
Since the given bandgap is $5.0 \ eV$,which is significantly greater than $3 \ eV$,the material is classified as an insulator.