Electronic physics Solid State Electrical Devices

مهدی حریری

دانشکدة فنی دانشگاه زنجان)گروه الکترونیک(

Be_teacher@yahoo.com

1385-1386نیمسال اول

In the name of God

SOLID STATE ELECTRONIC DEVICES

Chapter II

The behavior of solid state devices is directly related to atomic theory, quantum mechanics, and electron models. In this chapter we shall investigate some of the important properties of electrons, with special emphasis on two points:

(1) the electronic structure of atoms and (2) the interaction of atoms and electrons with excitation, such as the absorption and emission of light.

1

First, we shall investigate some of the experimental observations which led to the modern concept of the atom, and then we shall give a brief introduction to the theory of quantum mechanics.

2

INTRODUCTION TO PHYSICAL MODELS

In the 1920s it became necessary to develop a new theory to describe phenomena on the atomic scale.

Physicists discovered that Newtonian mechanics did not apply when objects were very small or moved very fast!

3

If things are confined to very small dimensions (nanometer-scale), then QUANTUM mechanics is necessary.

If things move very fast (close to the speed of light), then RELATIVISTIC mechanics is necessary.

4

The Photoelectric Effect An important observation by Planck

indicated that radiation from a heated sample is emitted in discrete units of energy, called quanta; the energy units were described by hv, where ν is the frequency of the radiation, and h is a quantity now called Planck's constant (h = 6.63 x 10-34 J-s).

5

Electrons are ejected from the surface of a metal when exposed to light of frequency ν in a vacuum.

6

Plot of the maximum kinetic energy of ejected electrons vs. frequency of the incoming light.

The equation of line is:

E =hv - qФ m

7

Atomic Spectra Early in the 19th century, Fraunhofer saw dark bands on the

solar spectrum. In 1885, Balmer observed hydrogen spectrum and saw colored lines

: Found empirical formula for discrete wavelengths of lines. Formula generalized by Rydberg for all one-electron atoms.

8

Atomic Spectra: Modern Physics Lab

Neon Tube

DiffractionGrating

(to separate light)

Eyepiece(to observe lines)

High VoltageSupply

(to “excite” atoms)

9

Emission-line spectra of Na, H, Ca, Hg, Ne

(1) Continuous spectrum from an incandescent light bulb. (2) Absorption-line spectrum (schematic) of sun’s most

prominent lines: H, Ca, Fe, Na. 10

Some important line in the emission spectrum of hydrogen :

Photon energy hv is then related to wavelength by :

11

The various series in the spectrum were observed to follow certain empirical forms :

Where R is a constant called the Rydberg constant :

12

Each energy can be obtained by taking sums and differences of other photon energies in the spectrum :

For example :E = E - E 42 41 21

In the Lyman series

In the Balmer series 13

The Bohr Model Classical model of the electron “orbiting” nucleus is unstable. Why

unstable? Electron experiences centripetal acceleration. Accelerated electron emits radiation. Radiation leads to energy loss. Electron eventually “crashes” into nucleus.

In 1913, Bohr proposed quantized model of the H atom to predict the observed spectrum.

14

To develop the model, Bohr made several postulates :

1- Electrons exist in certain stable, circular orbits about the nucleus. This assumption implies that the orbiting electron does not give off radiation as classical electromagnetic theory would normally require of a charge experiencing angular acceleration; otherwise, the electron would not be stable in the orbit but would spiral into the nucleus as it lost energy by radiation.

2- The electron may shift to an orbit of higher or lower energy, thereby gaining or losing energy equal to the difference in the energy levels :

15

3- The angular momentum P of the electron in an orbit is always an integral multiple of Planck's constant divided by 2π (h / 2π is often abbreviated ħ for convenience) :

ө

If we visualize the electron in a stable orbit of radius r about the proton of the hydrogen atom, we can equate the electrostatic force between the charges to the centripetal force:

16

17

18

The energy difference between orbits n and n is given by :1 2

The frequency of light given off by a transition between these two orbits is :

19

The factor in brackets is essentially the Rydberg constant R times the speed of light c.

Whereas the Bohr model accurately describes the gross features of the hydrogen spectrum, it does not include many fine points. For example, experimental evidence indicates some splitting of levels in additon to the levels predicted by the theory. Also, difficulties arise in extending the model to atoms more complicated than hydrogen. Attempts were made to modify the Bohr model for more general cases, but it soon became obvious that a more comprehensive theory was needed.

Electron orbits and transitions in the Bohr model of the hydrogen atom. Orbit spacing is not drawn to scale.

20

Orbital Radii and Energies, cont.

rn=0.0529n2 (nm)

En=-13.6/n2 (eV) Energy difference between the

levels E=13.6(1/nf2-1/ni

2)

For example, between n=1 and n=2 (as drawn in the picture)

E=13.6(1/nf2-1/ni

2)=13.6(1/12-1/22)=10.2 eV 

E=10.2 eV

Final state, nf

Initial State, ni

21

Bohr’s Correspondence Principle

Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small. Similar to having Newtonian Mechanics be

a special case of relativistic mechanics when v << c .

22

Successes of the Bohr Theory

Explained several features of the hydrogen spectrum Accounts for Balmer and other series Predicts a value for RH that agrees with the experimental

value Gives an expression for the radius of the atom Predicts energy levels of hydrogen Gives a model of what the atom looks like and how it behaves

Can be extended to “hydrogen-like” atoms Those with one electron Ze2 needs to be substituted for e2 in the Bohr equations

Z is the atomic number of the element (=number of protons)

23

Pioneers of Quantum Mechanics

HeisenbergFermiSchrodinger

24

Probability and the Uncertainty Principle

25

In any measurement of the position and momentum of a particle, the uncertainties in the two measured quantities will be related by :

Similarly, the uncertainties in an energy measurement will be related to the uncertainty in the time at which the measurement was made by :

A particular optical fiber transmits light over the range 1300-1600 nm (corresponding to a frequency range of 2.3x1014 Hz to 1.9x1014 Hz). How long (approximately) is the shortest pulse that can propagate down this

fiber?

a. 4 ns

b. 2 fs

c. 4 fs

d. 2 ns

E t Uncertainty Principle Example*

14

15

1 2 1

1/ 2

1/(2 0.4 10 )

4 10 4

t f t

t f

Hz

s fs

*This problem obviously does not require “quantum mechanics”. However, due to the Correspondence Principle, the quantum constraints on single photons also apply at the classical-pulse level . 26

Note: This means the upper limit to data transmission is ~1/(4fs) = 2.5x1014 bits/second = 250 Gb/s

One implication of the uncertainty principle is that we cannot properly speak of the position of an electron , for example, but must look for the probability of finding an electron at a certain position. Thus one of the important results of quantum mechanics is that a probability density function can be obtained for a particle in a certain environment, and this function can be used to find the expectation value of important quantities such as position, momentum, and energy.

27

Given a probability density function P(x) for a one-dimensional problem, the probability of finding the particle in a range from x to x + dx is P(x) dx. Since the particle will be somewhere, this definition implies that :

28

To find the average value of a function of x, we need only multiply the value of that function in each increment dx by the probability of finding the particle in that dx and sum over all x. Thus the average value of f(x) is :

If the probability density function is not normalized, this equation should be written :

The Schrodinger Wave Equation

29

30

31

32

33

Now the variables can be separated to obtain the time-dependent equation in one dimension

and the time-independent equation,

34

Potential Well Problem The simplest problem is the potential energy well with infinite

boundaries. Let us assume a particle is trapped in a potential well with V(x) zero except at the boundaries x = 0 and L, where it is infinitely large :

35

36

From Eqs. (2-30) and (2-31) we can solve for the total energy E„ for each value of the integer n.

37

The problem of a particle in a potential well :(a) potential energy diagram; (b) wave functions in the first three quantum states ;(c) probability density distribution for the second state.

Particle Motion in a Box: Example Consider the numerical example:

An electron in the infinite square well potential is initially (at t=0) confined to the left side of the well, and is described by the following wavefunction:

If the well width is L = 0.5 nm, determine the time to it takes for the particle to “move” to the right side of the well.

x

Lx

LLAtx

2sinsin

2)0,(

(x,t0)U= U=

0 xL

(x,t=0)U=U=

0 xL

15

16o

2 1 1

T h h 4.136 10 eV sect 4.6 10 sec

2 2 E E 2 3E 2 3 1.5eV

2 2

2 2

2 22

1 12 2

1.505

2

1.505 1.505 1.505

4 4(.5 )

ne n n

n

h eV nmE

m

eV nm eV nmE eV E E n

L nm

L/n2 n

period T = 1/f = 2t0

with f = (E2-E1)/h

38

Classically, when an object hits a potential thatit doesn’t have enough energy to pass, it will never go though that potential wall, it alwaysbounces back.

In English, if you throw a ball at a wall, it willbounce back at you.

ClassicalWave FunctionFor Finite SquareWell PotentialWhere E<V

Tunneling

39

In quantum mechanics when a particle hits apotential that it doesn’t have enough energyto pass, when inside the square well, the wavefunction dies off exponentially.

If the well is short enough, there will be a noticeableprobability of finding the particle on the other side.

QuantumWave FunctionFor Finite SquareWell PotentialWhere E<V

40

More graphs of tunneling:

An electron tunneling from atom to atom:

n(r) is the probability of finding an electron

V(r) is the potential

41

Quantum mechanical tunneling:

(a)potential barrier of height Vo and thickness W;

(b)probability density for an electron with energy E < V0, indicating a nonzero value of the wave function beyond the barrier.

42

Now looking more in depth at the case of tunneling from one metal to another. EF represents the Fermi energy. Creating a voltage drop between the two metals allows current.

TipSample

43

44

The Hydrogen Atom Finding the wave functions for the hydrogen atom requires a solution of

the Schrodinger equation in three dimensions for a coulombic potential field. Since the problem is spherically symmetric, the spherical coordinate system is used in the calculation. The term V (x, y, z) in Eq. (2-24) must be replaced by V (r, ө, Ф) , representing the Coulomb potential which the electron experiences in the vicinity of the proton. The Coulomb potential varies only with r in spherical coordinates :

When the separation of variables is made, the time-independent equation can be written as:

Schrödinger Equation in Spherical Coordinates

Assuming spherical symmetry, change to spherical coordinate system

r

ZekrU

2

)(

zyxEzyxzyxUzyxzyxm

,,,,),,(,,2 2

2

2

2

2

22

cossinrx

sinsinry

cosrz

y

r

z

x

sinr

sinsinr cossinr

2

2

2

2

2

22

zyx

2

2

22

22

2

sin

1

sinsin

11

r

rrr

rr

3D Cartesian:

Potential energy:

LaplacianOperator:

222 zyxr Radius :

222

1coszyx

zPolar :

xy1tanAzimuthal :

45

Separation of Variables

)()(),( tTxXtx

Begin with the time-dependant Schrodinger wave equation:

txEt

txitxxU

x

tx

m,

,,)(

,

2 2

22

:),( tx Complex wave functionU(x): Potential Energy

Assume: 1) 1D, free particle U(x)=0.2) a separable wave function,

Inserting the trial wave function into the Schrodigenr equation above, and dividing through by Ψ(x,t):

)(

)('

)(

)("

2

2

tT

tTi

xX

xX

m

xkiAexXxXxXm

)()()("2

2

ttiBetTtTtTi )()()()('

m

kkwith

AetTxXtx tkxki

2)(

)()(),(22

])([

46

Spherical Symmetric Solution of the Schrödinger Equation(1)

Let rRr ,,

EUrrr

rrrm

2

2

2222

2

2

sin

1sin

sin

11

2

constantsin

1sin

sin

1

2)(

1

2 2

2

222

22

2

2

d

d

rd

d

d

d

rmRrUE

dr

dRr

dr

d

rm

RrrUEdr

dRr

dr

d

m22

2

)(2

'2

sin

1sin

sin

122

2

2

m

d

d

d

d

d

d

Since the LHS of the equation is a function of r, the RHS is a function of Θ and Φ, the only possibility is that both sides equal to a constant Λ.

47

Spherical Symmetric Solution of the Schrödinger Equation(2)Consider the part first: 02

2

2

md

d

mBmA cossin

Since ,2 n ,...2,1,0 m

0sin

'sinsin

12

2

m

d

d

d

dThe part becomes:

If m=0, F= Legendre polynomials. The solution cannot be finite unless where λ= positive integer or 0.)1(' ll

cosChange variables: F

The equation is transformed into 01

'12

22

F

m

d

dF

d

d

If m≠0, F= associated Legendre polynomials with m≤ l and are the only non-singular and physically acceptable solutions.

)1(' ll

48

Spherical Symmetric Solution of the Schrödinger Equation(3)

0

2

1222

22

22

2

RE

mr

ll

r

KZem

dr

dR

rdr

Rd

The radial part of the equation is

Let r

rurR

and weru l 1r

mE2

2

Euu

mr

ll

r

KZe

dr

ud

m

2

22

2

22

2

1

2

The radial wave equation is

Let

The equation becomes 01212 02

2

wld

dwl

d

wd

E

UkZe

E

m

2

0

2

The finiteness of w at 0 demands

120 lN where l= 0, 1, 2… and N=0, 1, 2…

Define the principal quantum number 21 0 lNn

We find the energy is quantized as 22

422

2 n

meZkEn

n=1, 2, 3…

49

Quantum Numbers for the H-atom

Principal quantum number

• Angular momentum is quantized in magnitude and direction

• Z can be any direction in free space, but will be parallel to the magnetic field.

• Bound state energies are negative.

• Energy depending only on n is a result of spherical symmetry.

n=1, 2, 3 …

where E0=13.6eVEnergy2

02

n

EZEn

Orbital quantum number l=0, 1, 2, 3 …, n-1

Orbital angular momentum 1 llL

Magnetic quantum number m=-l, -l+1, …, l-1, l

Angular momentum in Z-direction mLz

50

The Ground State Wave Functions of H-atom

Ground state n=1, l=m=0

Electrons can be anywhere, but most likely to be at r=a0 for the ground state.

where Bohr radius

0

23

0100

1 aZrea

Z

Normalization condition

nmmke

a 0529.02

2

0

Radial probability density

2

0ar1 21sin0 0

2

0

222

100

ddrdrdV

022

3

0

22 44 aZrera

zrrp

22r

0ar1 2

rp

51

The First Excited States Wave Functions of H-atom

1st excited state has three degenerated states

02

23

0200200 2 aZre

a

ZrC

n=2, l=0, m=0

n=2, l=1, m=±1

cos02

0210210

aZrea

ZrC n=2, l=1, m=0

iaZr eea

ZrC

sin02

0211121

2

200 02ar

2

210 2

211 rp

0ar2 4 6

22

210 r 22

200 r

Bohr model:Z

anr 02

52

Meaning of Quantum NumbersPrinciple Quantum Number

Principal quantum number n=1, 2, 3 …

where E0=13.6eVDetermines energy 20

2

n

EZEn

S P D F G

n=5 Degenerate states

l=0 1 432

n=4

n=3

n=2

n=1

12 EEhf

23 EEhf

53

Orbital quantum number l=0, 1, 2, 3 …, n-1Determines magnitude of orbital angular momentum 1 llL

54

Meaning of Quantum NumbersOrbital Quantum Number

Meaning of Quantum NumbersMagnetic Quantum Number

Magnetic quantum number m=-l, -l+1, …, l-1, l

Determines angular momentum in Z-direction

or equivalently the direction of L

mLz

55

Example

56

Electronic configurationsfor atoms in the ground state.

57

Periodic Table

l = 1 (p)l = 0 (s)

l = 2 (d)

l = 3 (f)

1

2

3

4

5

6

7

No

ble

Ga

s

Ha

log

en

Gro

up

VI

Gro

up

IV

Gro

up

V

Gro

up

III

n

58

Probability Distributions for Hydrogen Atom

59

Directed by :

Nasser Talebi

Electronic Group - Zanjan UniversityAban - Azar 1384 (November-December 2005)

The end of slide show for “solid State Electronic Devices”.