Electrical properties — conductors, semiconductors, insulators; doping (n-type, p-type)

2.4.17 · Chemistry › States of Matter (Quantitative)

Overview

Solids ki electrical conductivity mobile charge carriers (electrons ya holes) ki availability se aati hai aur ise band theory se explain kiya jaata hai. Materials ko unke band gap aur electron occupancy ke basis par conductors, semiconductors, ya insulators mein classify kiya jaata hai. Doping humein controlled impurities introduce karke semiconductor properties ko engineer karne deti hai.

Core Concepts

[!intuition] Materials Alag Tarah Se Conduct Kyun Karte Hain

Solid mein electrons ko ek multi-story building mein logon ki tarah socho. Conductor mein, ek open floor hai (conduction band) jahan log freely move kar sakte hain. Insulator mein, sablog ground floor par phanse hue hain aur upar ke floors ka darwaza locked hai (large band gap). Semiconductor ek aisa darwaza hai jise thodi si mehnat se khola ja sakta hai (small band gap) — room temperature par, kuch log upar ja ke wahan move kar sakte hain.

KEY insight yeh hai: conductivity is baat par depend karti hai ki electrons empty energy states access kar sakein ya nahin jahan wo freely move kar sakein.

[!definition] Band Theory Framework

Jab atoms mil kar solid banate hain, unke atomic orbitals overlap aur split hokar molecular orbitals banate hain. ~10²³ atoms ke saath, yeh split hokar continuous energy bands ban jaate hain:

1. Valence Band (VB): 0 K par sabse upar occupied energy band, electrons se bhara hua
2. Conduction Band (CB): Sabse neeche unoccupied band jahan electrons freely move kar sakte hain
3. Band Gap (Eₘ): VB aur CB ke beech ka forbidden energy region

$E_g=E_{CB}-E_{VB}$

Bands kyun banate hain: Jab N atoms interact karte hain, har atomic orbital N closely-spaced molecular orbitals mein split ho jaata hai. Bade N ke liye, yeh levels itne close hote hain ki ek continuum — band bana dete hain.

Yeh conductivity ko HOW explain karta hai:

• Completely filled bands mein electrons conduct nahin kar sakte (electric field ke under move karne ke liye koi empty states nahin)
• Electrons ko conduct karne ke liye energy AUR available empty states dono chahiye
• Band gap decide karta hai ki electrons conduction band tak kitni aasaani se pahunch sakte hain

[!formula] Band Gap ke Basis Par Classification

Intrinsic semiconductors ke liye conductivity-temperature relationship ka derivation:

Conduction band mein thermally excited electrons ki sankhya Boltzmann statistics follow karti hai:

$n\propto e^{-E_g/2k_{B}T}$

2 ka factor kyun? Ek mobile electron banana ke liye use VB se CB tak promote karna padta hai, peeche ek hole chhodta hua. Electron-hole pair density of states calculation mein band gap energy ko barabar share karta hai.

Conductivity $\sigma =ne\mu$ jahan $e$ electron charge hai aur $\mu$ mobility hai.

Isliye: $\sigma ={\sigma }_0{e}^{-E_g/2k_{B}T}$

Iska matlab:

• Zyada temperature → zyada electrons Eₘ overcome kar sakte hain → zyada conductivity
• Yeh metals se ulta hai (jahan conductivity temperature ke saath ghatti hai kyunki badhti lattice vibrations electrons ko scatter karti hain)

Logarithm lete hue: $\ln \sigma =\ln {\sigma }_0-\frac{E_g}{2k_{B}T}$

Iska use kaise karein: ln(σ) ko 1/T ke against plot karo; slope se $-E_g/2k_B$ milti hai, jisse band gap ka experimental determination hota hai.

Doping: Semiconductor Properties Ko Engineer Karna

[!definition] Extrinsic Semiconductors

Doping semiconductor ki electrical properties modify karne ke liye impurity atoms ka controlled addition hai. 10⁶ host atoms mein sirf 1 dopant atom bhi conductivity ko dramatically change kar sakta hai.

Doping kyun kaam karti hai: Dopant atoms ki host se alag valency hoti hai, jo extra electrons ya holes introduce karte hain jinhe mobile hone ke liye band-gap energy nahin chahiye.

n-Type Semiconductors (Electron-Rich)

KYA: Group 14 semiconductors (Si, Ge) mein pentavalent (Group 15) atoms jaise P, As, Sb se doping

HOW kaam karta hai — atomic structure se derivation:

1. Silicon ke 4 valence electrons hain (3s² 3p²)
2. Phosphorus ke 5 valence electrons hain (3s² 3p³)
3. Jab P, Si ko lattice mein replace karta hai:
   • 4 electrons neighboring Si atoms ke saath covalent bonds banate hain
   • 5th electron P nucleus se loosely bound hota hai
4. Yeh ek donor level (Eᴅ) banata hai jo conduction band ke thoda neeche hota hai (~0.045 eV P ke liye Si mein)

Energy consideration: $E_{\text{ionization}}=E_{CB}-E_D\approx 0.045\text{ eV}\ll E_g$

Room temperature par, $k_{B}T\approx 0.026$ eV, isliye thermal energy aasaani se donors ko ionize karti hai: $\text{P (neutral)}\to {\text{P}}^{+}+e^{-}$

Result:

• Majority carriers: electrons (CB mein)
• Minority carriers: holes (VB mein)
• Fermi level conduction band ki taraf shift hoti hai

[!example] n-Type Calculation

Problem: Pure Si at 300 K par intrinsic carrier concentration $n_i=1.5\times 10^{10}$ cm⁻³ hai. Ise P ke saath 1 ppm (1 part per million by atom) par dope kiya jaata hai. Electron aur hole concentrations calculate karo.

Solution:

Step 1: Dopant concentration calculate karo

• Si atomic density: $N_{Si}=5\times 10^{22}$ atoms/cm³
• Dopant concentration: $N_D=10^{-6}\times 5\times 10^{22}=5\times 10^{16}$ cm⁻³

Yeh step kyun? Hume dopant atoms ki actual number density chahiye jo electrons donate karenge.

Step 2: 300 K par complete ionization assume karo

• Saare donor atoms electrons donate karte hain: $n\approx N_D=5\times 10^{16}$ cm⁻³

Kyun valid hai? $E_D=0.045$ eV $\ll k_{B}T=0.026$ eV, isliye >99% donors ionized hain.

Step 3: Mass action law use karke hole concentration calculate karo $n\cdot p=n_i^2$

Yeh law of mass action hai — doped semiconductors mein bhi, electron aur hole concentrations ka product intrinsic value ke barabar hota hai.

$p=\frac{n_i^2}{n}=\frac{(1.5\times 10^{10}{)}^2}{5\times 10^{16}}=4.5\times 10^3{\text{ cm}}^{-3}$

Yeh kyun kaam karta hai? Electron-hole recombination ki rate $n\times p$ par depend karti hai. Equilibrium mein, generation rate (constant) = recombination rate, product ko fix karti hai.

Conclusion:

• Electrons: $5\times 10^{16}$ cm⁻³ (~10⁶ ke factor se badha)
• Holes: $4.5\times 10^3$ cm⁻³ (~10⁶ ke factor se ghata)
• Majority carrier: electrons (n-type confirmed)

p-Type Semiconductors (Hole-Rich)

KYA: Group 14 semiconductors mein trivalent (Group 13) atoms jaise B, Al, Ga, In se doping

HOW kaam karta hai — atomic structure se derivation:

1. Silicon ke 4 valence electrons hain
2. Boron ke 3 valence electrons hain (2s² 2p¹)
3. Jab B, Si ko replace karta hai:
   • 3 electrons neighbors ke saath covalent bonds banate hain
   • 4th bond position electron-deficient hai — ek hole
4. B atom valence band se ek electron accept kar sakta hai, acceptor level (Eₐ) banata hai jo valence band ke thoda upar hoti hai (~0.045 eV B ke liye Si mein)

Energy consideration: $E_{\text{ionization}}=E_A-E_{VB}\approx 0.045\text{ eV}$

Room temperature par: $\text{B (neutral)}+e^{-}\to {\text{B}}^{-}$

VB ka ek electron B site par hole fill karta hai, VB mein ek mobile hole create karta hai.

Result:

• Majority carriers: holes (VB mein)
• Minority carriers: electrons (CB mein)
• Fermi level valence band ki taraf shift hoti hai

[!example] n-Type aur p-Type Ki Tulna

Problem: Explain karo ki n-type (Si:P) aur p-type (Si:B) dono similar conductivity enhancements kyun rakhte hain jabki alag carriers se conduct karte hain.

Solution:

n-type ke liye: ${\sigma }_n=ne{\mu }_e$ p-type ke liye: ${\sigma }_p=pe{\mu }_h$

Step 1: Similar doping levels (~10¹⁶ cm⁻³) par, carrier concentrations comparable hain:

• n-type: $n\approx N_D$
• p-type: $p\approx N_A$

Kyun? Dono dopants ki low ionization energies hain, room temperature par complete ionization ensure karti hain.

Step 2: Mobility electrons aur holes ke beech alag hoti hai

• Si mein 300 K par: ${\mu }_e\approx 1350$ cm²/(V·s), ${\mu }_h\approx 450$ cm²/(V·s)

Kyun? CB mein electrons ki effective mass, band curvature differences ke karan VB mein holes se kam hoti hai. Kam mass → aasaan acceleration → zyada mobility.

Step 3: Conductivity ratio calculate karo $\frac{{\sigma }_n}{{\sigma }_p}=\frac{n{\mu }_e}{p{\mu }_h}\approx \frac{N_D\cdot 1350}{N_A\cdot 450}=3\times \frac{N_D}{N_A}$

Conclusion: Equal doping concentrations ke liye, n-type p-type se ~3× zyada conductive hai. Dono intrinsic Si ke comparison mein conductivity ko 10⁶ times enhance karte hain, isliye carrier-type difference, doping se kam critical hai.

[!example] Doped Semiconductors Ki Temperature Dependence

Problem: Ek doped semiconductor ke liye conductivity temperature ke saath kaise vary karti hai, yeh sketch karo aur explain karo.

Solution:

Conductivity mein teen regions hain:

Region 1: Low Temperature (Freeze-out, T < 100 K) $\sigma \propto e^{-E_D/k_{B}T}\text{ (ya p-type ke liye }{E}_A\text{)}$

Kyun? Dopants fully ionized NAHIN hain. Thermal energy dopants/acceptor sites se carriers free karne ke liye insufficient hai.

Conductivity temperature ke saath rapidly badhti hai jab zyada dopants ionize hote hain.

Region 2: Extrinsic Range (100 K < T < 400 K) $\sigma \approx n_{D}e\mu (T)\text{ ya }{p}_{A}e\mu (T)$

Kyun? Saare dopants ionized hain (n ≈ $N_D$, constant), lekin mobility temperature ke saath badhte phonon scattering ke karan ghatti hai: $\mu \propto T^{-3/2}$

Conductivity temperature ke saath slowly ghatti hai.

Region 3: Intrinsic Range (T > 400 K) $\sigma \propto e^{-E_g/2k_{B}T}$

Kyun? Thermal energy electrons ko band gap ke across excite karne ke liye sufficient hai. Intrinsic carriers ($\propto e^{-E_g/2k_{B}T}$) dopant carriers se zyada ho jaate hain.

Conductivity temperature ke saath exponentially badhti hai, mobility decrease ko overwhelm karti hui.

Key transition: Crossover temperature $T_c$ par: $n_i(T_c)=N_D$ $\Rightarrow T_c\approx \frac{E_g}{2k_B\ln (N_c/N_D)}$

Si ke liye 10¹⁶ cm⁻³ doping par, $T_c\approx 550$ K.

[!mistake] Doping Ke Baare Mein Common Misconceptions

Mistake 1: "Doping semiconductor mein charge add karti hai"

Kyun sahi lagta hai: N-type mein electrons ya p-type mein holes add karna, charge add karne jaisa lagta hai.

Sacchhai: Doped semiconductors electrically neutral rehte hain.

• N-type mein: har mobile electron ek positive donor ion (P⁺) se balanced hota hai
• P-type mein: har hole ek negative acceptor ion (B⁻) se balanced hota hai

Fix: Doping mobile charge carriers add karti hai, net charge nahin. Charge neutrality maintain hoti hai: $n+N_A^{-}=p+N_D^{+}$

Mistake 2: "Zyada doping matlab hamesha zyada conductivity"

Kyun sahi lagta hai: Zyada dopants → zyada carriers → zyada conductivity, sahi hai na?

Sacchhai: Excessive doping (>10²⁰ cm⁻³) conductivity ko ghata deti hai kyunki:

1. Impurity scattering: dopant ions carriers ko scatter karte hain, mobility kam karti hai
2. Band-gap narrowing: heavy doping crystal potential ko disturb karta hai
3. Bohot high concentrations par carrier-carrier scattering

Fix: Ek optimal doping concentration (~10¹⁸-10¹⁹ cm⁻³ Si ke liye) hoti hai jo carrier concentration aur mobility ke beech balance banati hai. Iske baad, mobility loss dominate karta hai.

Mistake 3: "Semiconductors metals ki tarah conduct karte hain kyunki unke paas mobile electrons hain"

Kyun sahi lagta hai: Dono ke paas mobile charge carriers hain jo electric fields ko respond karte hain.

Sacchhai: Conductivity mechanisms fundamentally alag hain:

• Metals: partially filled conduction band; electrons hamesha mobile hain; $\sigma$ T ke saath ghatti hai (phonon scattering)
• Semiconductors: carriers ko thermally/optically ek gap ke across excite karna padta hai; intrinsic range mein $\sigma$ T ke saath badhti hai

Fix: $\frac{d\sigma }{dT}<0\text{ (metal)}\text{vs.}\frac{d\sigma }{dT}>0\text{ (intrinsic semiconductor)}$

Yeh temperature dependence dono ko distinguish karne ka defining test hai.

Practical Applications

[!example] p-n Junction Formation

Problem: Jab p-type aur n-type regions saath aate hain, interface par kya hota hai?

Solution — Step-by-step derivation:

Step 1: Pehle alag alag

• p-side: high hole concentration, low electron concentration
• n-side: high electron concentration, low hole concentration

Step 2: Contact establish hota hai

• Electrons diffuse karte hain n → p (high se low concentration)
• Holes diffuse karti hain p → n (high se low concentration)

Kyun? Concentration gradient diffusion drive karta hai (Fick's law).

Step 3: Depletion region form hoti hai

• Junction ke paas, mobile carriers recombine karte hain: electrons holes fill karte hain
• Unke peeche, immobile ions exposed hote hain: n-side par N_D⁺, p-side par N_A⁻
• Yeh ek region banata hai jo mobile carriers se depleted hota hai (~0.1-1 μm wide)

Step 4: Built-in electric field Exposed ions n → p ki taraf point karta ek electric field create karte hain: $E=\frac{\rho }{ϵ}=\frac{e(N_D-N_A)}{ϵ}$

Yeh crucial kyun hai: Yeh field aage diffusion oppose karta hai, equilibrium establish karta hai.

Step 5: Equilibrium potential Built-in potential barrier: $V_{bi}=\frac{k_{B}T}{e}\ln \left(\frac{N_A{N}_D}{n_i^2}\right)$

Derivation: Equilibrium mein, electrochemical potential (Fermi level) junction ke across constant hoti hai. Electrostatic potential mein difference, carrier concentrations mein difference ko compensate karta hai.

Si ke liye $N_A=N_D=10^{16}$ cm⁻³ at 300 K: $V_{bi}=0.026\ln \left(\frac{10^{32}}{(1.5\times 10^{10}{)}^2}\right)=0.026\times 50.7=0.72\text{ V}$

Application: Yeh diodes, solar cells, LEDs, aur saare semiconductor electronics ka basis hai.

[!recall]- Feynman Explanation (Ek 12-saal ke bacche ko explain karo)

Socho tumhare paas logon se bhara ek bada apartment building hai. Conductor mein (jaise copper wire), top floor half-khali hai, isliye log wahan freely ghoom sakte hain — yeh bijli flow karne jaisa hai.

Insulator mein (jaise rubber), sablog ground floor par phanse hain, aur upper floors ka darwaza bahut bhaari aur locked hai. Koi floors ke beech move nahin kar sakta, isliye bijli flow nahin hoti.

Semiconductor (jaise silicon computer chips mein) special hai: upar ke floor ka darwaza locked hai par bahut bhaari nahin. Agar thoda garam karo (jaise room temperature), kuch log push karke upar ja sakte hain. Jitna garmi badhegi, utne zyada log upar ja ke ghoom sakte hain.

Ab yeh clever trick hai — doping: Socho tum kuch alag guests invite karte ho. Agar extra energy wale log invite karo (jaise silicon mein phosphorus atoms), wo apni chabi upar ke floor ke liye laate hain! Ab tumhare paas zyada log hain jo ghoom sakte hain — yeh n-type hai.

Ya tum aisa log invite kar sakte ho jo ground floor walon se chabi udharna chahte hain (jaise boron atoms). Jab wo chabi lete hain, to ground floor par ek khali jagah reh jaati hai jahan koi aur move kar sakta hai — yeh moving "holes" create karne jaisa hai p-type mein.

Magic kya hai? Yeh control karke ki tum kise invite karte ho (doping), tum exactly control kar sakte ho ki kitni bijli flow hogi, aur aisi transistors bana sakte ho jo tumhare phone mein billion baar per second on aur off ho sakti hain!

[!mnemonic] n-Type vs p-Type Yaad Karne Ka Tarika

"Phosphorus Provides Plenty of Negative" → n-type

• Phosphorus (Group 15, 5 valence e⁻)
• negative carriers (electrons) provide karta hai
• n = negative, notice the extra electron

"Boron Borrows, Becomes Positive" → p-type

• Boron (Group 13, 3 valence e⁻)
• Electrons borrow karta hai → holes create karta hai (positive carriers)
• p = positive holes, partial electron deficiency

Band Gap Energy Scale: "In Conductors, Gaps Close; Insulators, Gaps Grow"

• Conductors: 0 eV (overlapping)
• Semiconductors: 0.1-3 eV (Small Gap)
• Insulators: >3 eV (Giant Gap)

Connections

• Band Theory of Solids — electrical properties samajhne ki foundation
• Metalic Bonding — metals ke overlapping bands kyun hote hain
• Covalent Network Solids — explain karta hai ki diamond ka band gap itna bada kyun hai
• Crystal Defects — vacancies aur interstitials conductivity ko kaise affect karte hain
• p-n Junction Diodes — doping ka practical application
• Fermi-Dirac Distribution — electrons energy bands mein kaise populate karte hain
• Transistors and Integrated Circuits — engineering applications
• Photovoltaic Cells — light ko electricity mein convert karne ke liye semiconductors ka use
• Thermistors — temperature-dependent conductivity ka fayda uthana
• Hall Effect — carrier type aur concentration ka experimental determination

#flashcards/chemistry

Band gap kya hai aur yeh conductivity ko kaise explain karta hai? :: Band gap ($E_g$) valence band aur conduction band ke beech forbidden energy region hai. Materials tab conduct karte hain jab electrons conduction band access kar sakein jahan empty states movement allow karte hain. Chhota $E_g$ matlab electrons ki aasaan thermal excitation → zyada conductivity.

n-Type doping kya hai aur yeh kyun kaam karti hai? :: Group 14 semiconductors mein pentavalent impurities (P, As, Sb) add karna. Dopant ke 5 valence electrons hote hain: 4 bonds banate hain, 5th loosely bound hota hai. Yeh conduction band ke thoda neeche donor level create karta hai (~0.045 eV). Room temperature par, thermal energy aasaani se donors ionize kar deti hai, free electrons provide karti hai.

Agar Si ko $10^{16}$ P atoms/cm³ se dope kiya jaaye aur $n_i=1.5\times 10^{10}$ cm⁻³ ho, to n aur p nikalo :: Saare donors ionize hote hain: $n\approx N_D=10^{16}$ cm⁻³. $np=n_i^2$ use karke: $p=n_i^2/n=(1.5\times 10^{10}{)}^2/10^{16}=2.25\times 10^4$ cm⁻³. Electrons majority carriers hain (n-type).

p-n junction par kya hota hai aur built-in potential derive karo :: Electrons n→p diffuse karte hain, holes p→n diffuse karti hain concentration gradients ki wajah se. Junction ke paas, yeh recombine karte hain, ions exposed karte hain (n-side par N_D⁺, p-side par N_A⁻) depletion region banate hue. Yeh field aage diffusion oppose karta hai. Equilibrium mein Fermi level flat hoti hai, built-in potential deta hai: $V_{bi}=(k_{B}T/e)\ln (N_A{N}_D/n_i^2)$.

Concept Map