Nanostructures – density of states

Faculty of Physics UW [email protected] Local density of states

Gęstość stanów (ogólnie) można zdefiniować jako:

푁 퐸 = ෍ 훿 퐸 − 휀푛 푛 Jak widać po scałkowaniu:

퐸2 퐸2 퐸2 න 푁 퐸 푑퐸 = න ෍ 훿 퐸 − 휀푛 푑퐸 = ෍ න 훿 퐸 − 휀푛 푑퐸 퐸1 퐸1 푛 푛 퐸1 Przykładowo:

1 1 1 2푚 푁1퐷 퐸 = ෍ 훿 퐸 − 휀 푘 = න 훿 푘 − 푘′ 2 푑푘 = 2π 퐸′ 푘 휋 퐸 푘 1 1 푚 푁2퐷 퐸 = ෍ 훿 퐸 − 휀 푘 = න 훿 푘 − 푘′ 2휋푘 푑푘 = 2π 2 퐸′ 푘 휋ℏ2 푘 3/2 1 1 1 2푚 푁3퐷 퐸 = ෍ 훿 퐸 − 휀 푘 = න 훿 푘 − 푘′ 4휋푘2 푑푘 = 퐸 2π 3 퐸′ 푘 2휋2 ℏ2 푘

2017-06-05 2 Electrons statistics in crystals

The case of a semiconductor, in which both the electron gas and hole gas are far from the degeneracy: the probability of filling of the electronic states: 퐸 퐸 휉 − 퐺 − 푒 + 퐸퐺 푓 ≈ 푒 2푘퐵푇 푘퐵푇 푘퐵푇 퐸 = + 퐸 퐸푒 푒 2 푒 퐸푐 and of holes 푓ℎ = 1 − 푓푒 퐸 퐸 휉 − 퐺 − ℎ − 2푘 푇 푘 푇 푘 푇 푓ℎ ≈ 푒 퐵 퐵 퐵

∞ 휋 න 푥푒−푥 푑푥 = 2 0 퐸푣 퐸퐺 Thus: 퐸 = − − 퐸ℎ 퐸ℎ ∗ 3/2 퐸 휉 − 퐸 −휉 2 푚푒푘퐵푇 − 퐺 푐 푛 휉 = 2 푒 2푘퐵푇 ⋅ 푒푘퐵푇 = 푁 푇 푒 푘퐵푇 2휋ℏ2 푐 ∗ 3/2 퐸 휉 − 휉−퐸 푚ℎ푘퐵푇 − 퐺 − 푣 푝 휉 = 2 푒 2푘퐵푇 ⋅ 푒 푘퐵푇 = 푁 푇 푒 푘퐵푇 2휋ℏ2 푣

2017-06-05 3 Electrons statistics in crystals What is the concentration of carriers for T>0? In the thermodynamic equilibrium for an intrinsic semiconductors (półprzewodniki samoistne), the concentration of electrons in the conduction band is equal to the concnetration of holes in the valence band (because they appear only as a result of excitation from the valence band).

푛 = 푝 = 푛푖 (an intrinsic case) ln(n)

3 3 퐸𝑔 퐸𝑔 2 푘퐵푇 ∗ − − 푛 ⋅ 푝 = 푛 = 4 푚∗푚 2 푒 푘퐵푇 = 푁 푁 푒 푘퐵푇 푖 2휋ℏ2 푒 ℎ 푐 푣 3 2 3 퐸𝑔 퐸𝑔 푘퐵푇 ∗ − − 푛 = 푝 = 푛 = 2 푚∗ 푚 4 푒 2푘퐵푇 = 푁 푁 푒 2푘퐵푇 푖 2휋ℏ2 푒 ℎ 푐 푣

퐸𝑔 − 푒 2푘퐵푇

1/T 2017-06-05 4 Intrinsic carrier concentration What is the concentration of carriers for T>0? In the thermodynamic equilibrium for an intrinsic semiconductors (półprzewodniki samoistne), the concentration of electrons in the conduction band is equal to the concnetration of holes in the valence band (because they appear only as a result of excitation from the valence band). 3 3 퐸𝑔 퐸𝑔 2 푘퐵푇 ∗ ∗ − − 푛 ⋅ 푝 = 푛 = 4 푚 푚 2 푒 푘퐵푇 = 푁 푁 푒 푘퐵푇 band 푖 2휋ℏ2 푒 ℎ 푐 푣

(general formula)

mpty e Eg 푛 = 푝 = 푛 푖 (an intrinsic3case) 2 3 퐸𝑔 퐸𝑔 푘퐵푇 ∗ − − 푛 = 푝 = 푛 = 2 푚∗ 푚 4 푒 2푘퐵푇 = 푁 푁 푒 2푘퐵푇 푖 2휋ℏ2 푒 ℎ 푐 푣

2휉−퐸𝑔 ∗ T 푁푐 1 3 푚ℎ 푘퐵푇 = 푒 ⇒ 휉 = 퐸푐 + 퐸푣 + 푘퐵푇 ln ∗ 푁푣 2 4 푚푒

in our notation the middle band

of the band is 0 filled

2017-06-05 5 Dopants, impurities and defects

Hydrogen-like model – ionization of the dopant

n-type p-type

conduction band conduction band donor level

acceptor level Electron

vallence band vallence band

x x 2017-06-05 6 Dopants, impurities and defects

Doping The carrier concentration in extrinsic semiconductor (niesamoistny) Consider a semiconductor, in which:

NA – concentration of acceptors conduction band ED ND – concentration of donors Eg/2 pA – concentration of neutral acceptors nD – concentration of neutral donors

energy donor level nc – concentration of electrons in conduction E 0 F band pv – concentration of holes in valence band

Electron acceptor level E A From the charge neutrality of the crystal: -Eg/2 vallence band nc +(NA - pA)= pv + (ND - nD) nc + nD = (ND - NA)+ pv + pA

x 2017-06-05 7 The occupation of impurity levels

Occupation of impurity / defect levels in the thermodynamic equilibrium

„Occupation” of localized or band states means the exchange of particles (electrons) between the reservoir and considered subsystem (microstate).

The (subsystem exchange particles and energy with the environment)

Thermodynamic probability (unnormalized) of finding subsystem in a state 푗,in which there are 푛푗 particles (electrons) and which subsystem energy is 퐸푗 (the total energy of all 푛푗 particles): 1 −훽 퐸푗−푛푗휉 푃푗 = 푒 , 훽 = 푘퐵푇 휉- .

Statistical sum:

−훽 퐸푗−푛푗휉 푍 = ෍ 푃푗 = ෍ 푒 푗 푗

2017-06-05 8 The occupation of impurity levels

Statistical sum :

−훽 퐸푗−푛푗휉 푍 = ෍ 푃푗 = ෍ 푒 푗 푗 Statistical means: −훽 퐸푗−푛푗휉 σ푗 퐴푗 ⋅ 푒 퐴 = −훽 퐸푗−푛푗휉 σ푗 푒 Examples: free electron occupying (or not) the quantum state of the 푘-vector and spin: 2 possible states of the subsystem (microstate): 푛0 = 0; 퐸0 = 0 푛1 = 1; 퐸1 = 퐸 (the occupation only for one spin state) the average number of particles of the subsystem: 0 ⋅ 푒0 + 1 ⋅ 푒−훽 퐸−휉 1 푛 = = = 푓 퐸, 푇 1 + 푒−훽 퐸−휉 푒훽 퐸−휉 + 1 (Fermi-Dirac distribution)

2017-06-05 9 The occupation of impurity levels

Examples : free electron occupying (or not) the quantum state of the 푘-vector and ANY spin: 4 possible states of the subsystem (microstate):

푛0 = 0; 퐸0 = 0 푛1 = 1; 퐸1 = 퐸 (spin ↑) 푛2 = 1; 퐸2 = 퐸 (spin ↓) 푛3 = 2; 퐸3 = 2퐸 (spin ↑↓) the average number of particles of the subsystem: 0 ⋅ 푒0 + 1 ⋅ 푒−훽 퐸−휉 + 1 ⋅ 푒−훽 퐸−휉 + 2 ⋅ 푒−훽 2퐸−2휉 푒−훽 퐸−휉 1 + 푒−훽 퐸−휉 푛 = = 2 = 1 + 푒−훽 퐸−휉 + 푒−훽 퐸−휉 + 푒−훽 2퐸−2휉 푒−훽 퐸−휉 + 1 2 = 2푓 퐸, 푇

(Fermi-Dirac distribution x2)

2017-06-05 10 The occupation of impurity levels

The ratio of the probability of finding dopant / defect of 푛 + 1 electrons and of 푛 electrons: σ 푒−훽 퐸푗− 푛+1 휉 푝푛+1 푁푛+1/푁푡표푡푎푙 푗:푛푗=푛+1 푔푛+1 = = = ⋅ 푒−훽 퐸푛+1−퐸푛 −휉 푝 푁 /푁 σ 푒−훽 퐸푘−푛휉 푔 푛 푛 푡표푡푎푙 푘;푛푘=푛 푛

σ푛 푁푛 = 푁 – impurity (dopants) concentration

퐸푛+1 i 퐸푛 – the lowest of all subsystem 퐸푗 with 푛 + 1 and 푛 electrons respectively

Successive impurity energy levels are filled with the increase of the .

퐸푛+1/푛 – so-called energy level of the impurity/ defect „numbered” by charge states 푛 + 1 and 푛

푔푛+1, 푔푛 – so-called degeneration of states of subsystem of 푛 + 1 and 푛 electrons

6/5/2017 11 The occupation of impurity levels

푔푛+1, 푔푛 – so-called degeneration of states of subsystem of 푛 + 1 and 푛 electrons.

The degeneracy 푔푛+1 and 푔푛 takes into account the possibility of the existence of different subsystem states corresponding to a same number of particles (including the excited states):

−훽휀푛,푖 푔푛 = 훼푛,0 + ෍ 훼푛,푖푒 푖=1,2,…

훼푛,0 i 훼푛,푖 they are respectively: the degeneracies of the 푛-electronic ground state and its excited states of energies higher than the ground state by 휀푛,푖 (excitation energies)

Such defined degeneration 푔푛 generally depends on temperature

6/5/2017 12 The occupation of impurity levels

Example – donor (we omit the excited states) charge state (+) implemented in 1 way : 푔+ = 1 charge state(0) implemented in 2 ways (spin ↑ or ↓): 푔0 = 2 0/+ the energy of the donor state Ε = 퐸퐷 Doping concentration of donors 푁퐷

푝푛+1 푝0 푔0 −훽 퐸퐷−휉 = = ⋅ 푒 , 푝+ + 푝0 = 1 푝푛 푝+ 푔+

The occupation probability of the donor state: 1 1 푝0 = 푛 = = 푔+ 훽 퐸 −휉 1 1 + ⋅ 푒 퐷 1 + ⋅ 푒훽 퐸퐷−휉 푔0 2

0 The concentration of occupied donor states (neutral donors are 푁퐷) 푁 푁0 = 퐷 퐷 1 1 + ⋅ 푒훽 퐸퐷−휉 TUTAJ 2017.04.10 2

2017-06-05 13 Dopants, impurities and defects

Doping The carrier concentration in extrinsic semiconductor (niesamoistny) Consider a semiconductor, in which:

NA – concentration of acceptors conduction band ED ND – concentration of donors Eg/2 pA – concentration of neutral acceptors nD – concentration of neutral donors

energy donor level nc – concentration of electrons in conduction E 0 F band pv – concentration of holes in valence band

Electron acceptor level E A From the charge neutrality of the crystal: -Eg/2 vallence band nc +(NA - pA)= pv + (ND - nD) nc + nD = (ND - NA)+ pv + pA

x 2017-06-05 14 The occupation of impurity levels If both the donors and acceptors are shallow, and the electron and hole gas are not 0 0 degenerated, then : 퐸퐷 − 휉 ≫ 푘퐵푇, 휉 − 퐸퐴 ≫ 푘퐵푇, 푁퐷 ≪ 푁퐷, 푁퐴 ≪ 푁퐴 (that is, virtually all impurities are ionized) .

1 푛 = Δ푛 2 + 4푛2 + Δ푛 Δ푛 = 푛 − 푝 ≈ 푁 − 푁 2 푖 퐷 퐴 ⇒ 푛 ⋅ 푝 = 푛2 1 푖 푝 = Δ푛 2 + 4푛2 − Δ푛 2 푖

If Δ푛 > 0 (푛-type semiconductor – for 푝-type is the same) and Δ푛 ≫ 푛푖, (at 푇=300K: 푛푖(Ge) < 13 -3 11 -3 10 -3 10 cm , 푛푖(Si) < 10 cm , 푛푖(GaAs) < 10 cm ):

푛 ≈ 푁퐷 − 푁퐴

2 퐸𝑔 푛푖 푁퐶 푇 ⋅ 푁푉 푇 − 푝 ≈ = ⋅ 푒 푘퐵푇 푁퐷 − 푁퐴 푁퐷 − 푁퐴 majority carrier concentration is determined by the effective concentration of impurities, the concentration of minority carriers may be very small (e.g. Si).

2017-06-05 15 The occupation of impurity levels Compensation compensated semiconductors - containing both donors and acceptors

At appropriately high temperatures majority carrier concentration given by the effective concentration of dopants |푁퐷 − 푁퐴| the concentration of scattering centers (charges): 푁퐷 + 푁퐴 compensation ratio - the ratio of impurity concentration: minority to majority. It is a measure of the total number of ionized impurities : 푁 푘 = 퐴 – for 푛-type 푁퐷 푁 푘 = 퐷 – for 푝-type 푁퐴 in highly compensated semiconductors (푘 ≈ 1) strong electrostatic potential fluctuations are present – originating from the impurities, the disorder localization, the percolation effects etc.

2017-06-05 16 The occupation of impurity levels Equation of Charge Neutrality low temperature, thermal ionization of impurities

Consider a semiconductor, in which:

NA ≈ 0 – concentration of acceptors conduction band ED ND – concentration of donors Eg/2 EF pA – concentration of neutral acceptors n – concentration of neutral donors

energy donor level D nc – concentration of electrons in conduction 0 band

pv ≈ 0 – concentration of holes in valence

band Electron

-Eg/2 vallence band From the charge neutrality of the crystal: 0 푛 = 푁퐷 − 푁퐷

x 2017-06-05 17 The occupation of impurity levels Now, a large part of donors will be neutral (the energies calculated from the bottom of the conduction band) 푁퐷 푁퐷 푁 − 푁0 = 푁 − = ⋯ = ⋅ 푒훽 퐸퐷−휉 퐷 퐷 퐷 1 훽 퐸퐷−휉 2 1 + 2 ⋅ 푒 to calculate the occupation of the conduction band we use the Boltzmann distribution : ∗ 3/2 휉−퐸 휉 푚푒푘퐵푇 푐 푛 = 2 푒 푘퐵푇 = 푁 푇 ⋅ 푒푘퐵푇 2휋ℏ2 퐶 0 Having 푛 = 푁퐷 − 푁퐷 we got: 퐸 푘 푇 푁 휉 = 퐷 + 퐵 ln 퐷 2 2 2푁퐶 푇 퐸 푁퐶 푇 ⋅ 푁퐷 퐷 푛 푇 = ⋅ 푒2푘퐵푇 2

2017-06-05 18 Dopants, impurities and defects Equation of Charge Neutrality

conduction band ED 퐸𝑔 E /2 − g 푒 2푘퐵푇

energy donor level E 0 F 퐸 − 퐷 2푘 푇 Electron 퐵 acceptor level EA 푒

-Eg/2 vallence band

x 2017-06-05 19 Dopants, impurities and defects Equation of Charge Neutrality

conduction band ED 퐸𝑔 E /2 − g 푒 2푘퐵푇

energy donor level E 0 F 퐸 − 퐷 2푘 푇 Electron 퐵 acceptor level EA 푒

-Eg/2 vallence band

x 2017-06-05 20 The occupation of impurity levels For compensated semiconductors the thermal activation energy at low temperature is 퐸 퐸 , not 퐷 퐷 2

If there is a lot of impurities, also the wave functions of electrons bound to them overlap - ionization energies decrease, formed the impurity band.

When impurity concentration is of the order of: 1 ∗ 푎퐵 ⋅ 푁퐷 3 ≈ 0.26 there is a nonmetal - metal phase transition (Mott transition)

2017-06-05 21 The occupation of impurity levels

1 ∗ 푎퐵 ⋅ 푁퐷 3 ≈ 0.26

P.P. Edwards, M.J. Sienko, J. Am. Chem. Soc. 103, 2967 (1981)

2017-06-05 22 Dopants, impurities and defects Equation of Charge Neutrality

conduction band ED 퐸𝑔 E /2 − g 푒 2푘퐵푇

energy donor level E 0 F 퐸 − 퐷 2푘 푇 Electron 퐵 acceptor level EA 푒

-Eg/2 vallence band

x 2017-06-05 23 Quasi-Fermi level (imref) What happens if there is no equilibrium? The problem is much more complicated, requires statistical analysis, but one can introduce a very useful concept of "quasi Fermi level" close to the equilibrium.

2 푛푝 > 푛푖

퐸푐 − 퐸퐹 퐸퐹 − 퐸푣 푛 = 푁푐 exp − 푝 = 푁푣 exp −

푘퐵푇 푘퐵푇 conduction Eg

푛 푛 퐸퐹 = 퐸푐 + 푘퐵푇 ln 푁푐

T

2퐸퐹−퐸𝑔 ∗ 푁푐 1 3 푚ℎ 푘퐵푇 = 푒 ⇒ 퐸퐹 = 퐸푐 + 퐸푣 + 푘퐵푇 ln ∗

푁푣 2 4 푚푒 valence

2017-06-05 24 Semiconductor heterostructures

Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application

2017-06-05 25 Bandgap engineering

Valence band offset

134

–

303 303 (2001) 123

–

B 302 B

Physica Walukiewicz

2017-06-05 26 The construction of energy band diagrams Złącze metal-metal Suppose, that 휙2 − 휙1 ≈ 1 푒푉 Estimate the number of electrons that pass from one metal to another to create equilibrium potential difference. Assume that the distance between the metals is 5 × 10−10푚.

Δ휙 푉 Electric field: 퐸 = = 2 × 109 푑 푚 The surface charge: 휎 = 휀0퐸 휎 The concentration: 푛2퐷 = = 1.12 × 1013푐푚−2 푒

The concnetration in metal 푛3퐷 = 5 × 1022푐푚−3 푛2퐷 = 1.5 × 1015푐푚−2 Within the width of 1 lattice parameter ~1% of charge

Electrical properties of materials Solymar, Walsh (6.11) Pg. 143

2017-06-05 27 The construction of energy band diagrams Złącze metal-metal Suppose, that 휙2 − 휙1 ≈ 1 푒푉 Estimate the number of electrons that pass from one metal to another to create equilibrium potential difference. Assume that the distance between the metals is 5 × 10−10푚.

Δ휙 푉 Electric field: 퐸 = = 2 × 109 푑 푚 The surface charge: 휎 = 휀0퐸 휎 The concentration: 푛2퐷 = = 1.12 × 1013푐푚−2 푒

The concnetration in metal 푛3퐷 = 5 × 1022푐푚−3 푛2퐷 = 1.5 × 1015푐푚−2 Within the width of 1 lattice parameter ~1% of charge

Electrical properties of materials Solymar, Walsh (6.11) Pg. 143

2017-06-05 28 The doping of semiconductors Net charge densities From the Maxwell equations:

훻퐷 = 휌푠 - net charge density 푥푝 + 푥 퐸 = −훻휙 = −훻푈 − 푛

2 훻퐷 = 휀0휀 훻퐸 = −휀0휀 훻 휙 ≝ −휖Δ푈 = 휌푠 푁퐴 푁퐷

Charge conservation 푒푁퐴푥푝 = 푒푁퐷푥푛 = 푄 Poisson equation: 푑2푈 1 1 = − 휌 = 푒푁 푑푥2 휖 푠 휖 퐴

Thus the electric field in the range 푥푝, 0 : 푑푈 1 1 퐸 = − = 푒푁 푥 + 퐶 = 푒푁 푥 − 푥 푑푥 휖 퐴 휖 퐴 푝

Similarly for 0, 푥푛 : 푑푈 1 1 퐸 = − = 푒푁 푥 + 퐶 = 푒푁 푥 − 푥 푑푥 휖 퐷 휖 퐷 푛

2017-06-05 29 The doping of semiconductors Net charge densities From the Maxwell equations: 푝 푛 훻퐷 = 휌푠 - net charge density 푥푝 + 푥 퐸 = −훻휙 = −훻푈 − 푛

2 훻퐷 = 휀0휀 훻퐸 = −휀0휀 훻 휙 ≝ −휖Δ푈 = 휌푠 푁퐴 푁퐷 퐸 Charge conservation 푒푁퐴푥푝 = 푒푁퐷푥푛 = 푄 Poisson equation: 푥푝 푥푛 푑2푈 1 1 2 = − 휌푠 = 푒푁퐴 1 푑푥 휖 휖 퐸 = − 푄 푚푎푥 휖 Thus the electric field in the range 푥푝, 0 : 푑푈 1 1 퐸 = − = 푒푁 푥 + 퐶 = 푒푁 푥 − 푥 푑푥 휖 퐴 휖 퐴 푝

Similarly for 0, 푥푛 : 푑푈 1 1 퐸 = − = 푒푁 푥 + 퐶 = 푒푁 푥 − 푥 푑푥 휖 퐷 휖 퐷 푛

2017-06-05 30 The doping of semiconductors

Thus the electric field in the range 푥푝, 0 : Net charge densities 푑푈 1 1 푝 푛 퐸 = − = 푒푁 푥 + 퐶 = 푒푁 푥 − 푥 퐴 푑푥 휖 퐴 휖 퐴 푝 푥푝 + 푥푛 Similarly for 0, 푥푛 : − 푑푈 1 1 퐸 = − = 푒푁 푥 + 퐶 = 푒푁 푥 − 푥 퐷 푑푥 휖 퐷 휖 퐷 푛 푁퐴 푁퐷

0 퐸 푈 = − න 퐸퐴푑푥 푥 < 0 푥푝 푥푛 푥푝

푥푛 1 퐸 = − 푄 푈 = − න 퐸퐷푑푥 푥 > 0 푚푎푥 휖 0푝 푒 푈 = 푁 푥2 + 푁 푥2 푈 0 2휀 퐴 푃 퐷 푛 The built-in volatage in the 푝푛-junction 푒 푒 2 푈 = 푈 푥 − 푈 푥 = 푁 푥2 + 푁 푥2 푁퐴푥푃 0 푛 푝 2휀 퐴 푃 퐷 푛 2휀

푥푝 푥푛

2017-06-05 31 The doping of semiconductors

Charge conservation Depletion regions Net charge densities 푝 푛 푒푁퐴푥푝 = 푒푁퐷푥푛 = 푄 푥푝 + The total width of the depletion region 푤 − 푥푛 2휀푈0 푁퐴 푁퐷 푤 = 푥푛 − 푥푝 = + 푒 푁퐴 + 푁퐷 푁퐷 푁퐴 푁퐴 푁퐷 퐸 If, say, 푁퐴 ≫ 푁퐷 (푝-type doping) then: 푥푝 푥푛

2휀푈0 푤 = i 푥푛 > 푥푝 푒푁퐷 1 퐸 = − 푄 if the 푝-region is more highly doped, practically all of the 푚푎푥 휖 potential drop is in the 푛-region. The less donors are the 푒 wider this region is. 2 2 푈 푈0 = 푁퐴푥푃 + 푁퐷푥푛 (for 푁퐴 ≪ 푁퐷 is vice-versa!) 2휀

15 -3 E.g. 푁퐷 = 10 cm for typical 푈0 = 0.3 V 푒 푁 푥2 We have 푤 ≈ 180 nm. If the change from acceptor 2휀 퐴 푃 impurities to donor impurities is gradual, then 푤 ≈ 1 휇m 푥푝 푥푛

2017-06-05 32 The doping of semiconductors

Charge conservation Depletion regions Net charge densities 푝 푛 푒푁퐴푥푝 = 푒푁퐷푥푛 = 푄 푥푝 + The total width of the depletion region 푤 − 푥푛 2휀푈0 푁퐴 푁퐷 푤 = 푥푛 − 푥푝 = + 푒 푁퐴 + 푁퐷 푁퐷 푁퐴 푁퐴 푁퐷 퐸 If, say, 푁퐴 ≫ 푁퐷 (푝-type doping) then: 푥푝 푥푛

2휀푈0 푤 = i 푥푛 > 푥푝 푒푁퐷 1 퐸 = − 푄 if the 푝-region is more highly doped, practically all of the 푚푎푥 휖 potential drop is in the 푛-region. The less donors are the 푒 wider this region is. 2 2 푈 푈0 = 푁퐴푥푃 + 푁퐷푥푛 (for 푁퐴 ≪ 푁퐷 is vice-versa!) 2휀

15 -3 E.g. 푁퐷 = 10 cm for typical 푈0 = 0.3 V 푒 푁 푥2 We have 푤 ≈ 180 nm. If the change from acceptor ! 2휀 퐴 푃 impurities to donor impurities is gradual, then 푤 ≈ 1 휇m 푥푝 푥푛

2017-06-05 33 The construction of energy band diagrams

The metal-semiconductor junction ( 휙푀 > 휙푆) Work function 휙푆, electron affinity 휙퐵

Anderson's rule

n-type

Shottky barrier

Electrical properties of materials Solymar, Walsh (9.16) Pg. 257

2017-06-05 34 The construction of energy band diagrams

The metal-semiconductor junction ( 휙푀 > 휙푆) Shottky barrier

typ n

2017-06-05 35 The construction of energy band diagrams

The metal-semiconductor junction ( 휙푀 > 휙푆)

n-type

Theoretically there should be no Shotky barrier

Electrical properties of materials Solymar, Walsh

2017-06-05 36 The construction of energy band diagrams

The metal-semiconductor junction ( 휙푀 > 휙푆)

n-type isolator

forward bias + - napięcie przewodzenia

- +

reverse bias napięcie zaporowe Electrical properties of materials Solymar, Walsh

2017-06-05 37 The construction of energy band diagrams The surface of the semiconductor is usually charged

n-type

For the p-type the opposite is true - the bands bend downwardly the diagram!

Electrical properties of materials Solymar, Walsh

2017-06-05 38 The construction of energy band diagrams

The metal-semiconductor junction ( 휙푀 > 휙푆)

n-type

https://en.wikipedia.org/wiki/Metal%E2%80%93semiconductor_junction

2017-06-05 39 WKB approximation

푥 2 푉 푥 = 푉 1 − 푏 푑

1 1 푥 2 푘 푥 = 2푚푉 푥 = 2푚푉 1 − 푛 ℏ ℏ 푏 푑

2017-06-05 40 Heterojunction

Charge conservation 푒푁퐴푥푝 = 푒푁퐷푥푛 = 푄 TUTAJ 20151126

푒푈0 (푒 < 0)

퐸𝑔 푒 푈 = 푁 푥2 + 푁 푥2 푈 0 2휀 퐴 푃 퐷 푛

푒 2 푁퐴푥푃 푥푝 푥푛 2휀

푥푝 푥푛

2017-06-05 41 Heterojunction

Heterozłącze (heterojunction) forward bias napięcie przewodzenia

reverse bias napięcie zaporowe

2017-06-05 42