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Semiconductor Engineering 1

반 도 체 공 학노 용 한

Http://mdl.skku.ac.kr

Fundamentals of Semiconductor DevicesBetty Lise Anderson • Richard L. Anderson

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Invention of the first transistor : Disruptive Technology

The discovery of the point contact transistor in 1947: W. Shockley, J. Bardeen and W. Brattain

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Semiconductor Devices

1. Fundamentals of Semiconductor Materials : Part I

2. pn Junction Diode : Part II

3. MOSFET : Part III

Evaluation : 10%(출석) + 10%(과제) + 40%(중간고사) + 40%(기말고사)

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Goal : We will discuss the operation principles of semiconductor devices : e.g., pn junction diode, MOSFET, and BJT (very briefly). I-V characteristics of semiconductor devices.

Then, we need to understand the physical properties of semiconductor materials in quantum mechanical point of view. The core concept is “energy band”.

Once we understand the concept of energy band, the concepts of (1) charged carriers (i.e., electron and hole), (2) their distribution in semiconductor materials in view of electron energy, and (3) their actions (i.e., drift, diffusion, and ehp G-R processes) will be introduced.

Finally, based on the carrier actions, we will derive the continuity equations which could be used for I-V analysis.

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One of the most important semiconductor devices is MOSFET.

Mark Bohr, “Intel’s 90 nm Process Starting High Volume Manufacturing,” Intel Developer Forum, September 16, 2003

( ) tgdsatdd

dtGoxn

D VVVVVVVVLCZI ≥≤≤⎥

⎦

⎤⎢⎣

⎡−−=∴ ,0;

2

2µ

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MOSFET Applications in LOGIC and MEMORY:

WL

BL

FG

V

I

ox

inj

C

dtIV ∫=∆

Nondestructive readoutNo-refreshNonvolatile

FLASHINVERTERVDD

VSS

VOVin

pMOS

nMOSQ

V

WL

BL

Capacitor

∫= dv)v(CQ

Destructive readoutRefreshVolatile

DRAM

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CMOS: Complementary MOS

From Burns and Bond, “Principles of Electronic Circuits”, PWS

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CMOS Inverter Circuit

There is no dc path for current from the +VDD to ground !!!

Equivalent Circuit Vin = VDD (left) and Vin = 0 (right)

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Mechanisms of Current Conduction and Continuity Equation

Lattice VibrationIonized Impurity

Drift due to the applied electric field !

Diffusion due to the concentration difference !

Electron-Hole generation and recombination !

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Continuity Equation

processesother

GRthermalP

processesother

GRthermalN

tp

tpJ

qtp

tn

tnJ

qtn

∂∂

+∂∂

+⋅∇−=∂∂

∂∂

+∂∂

+⋅∇=∂∂

−

−

1

1

Minority Carrier Diffusion Equations

Lp

nnP

n

Ln

ppN

p

Gpx

pDtp

Gn

xn

Dtn

+∆

−∂∆∂

=∂∆∂

+∆

−∂∆∂

=∂∆∂

τ

τ

2

2

2

2

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We need to understand the concepts of electron, hole, and their densities via energy band in semiconductor materials !

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Si ATOM

ψB

ψ B

ψ A

ψ AψhybCONDUCTION BAND

VALENCE BAND

Energy gap, Eg

(a) (b) (c) (d)

3p

3s

Si CRYSTAL

ψhyb

Fig. 4.17: (a) Formation of energy bands in the Si crystal first involves hybridizationof 3s and 3p orbitals to four identical ψhyb orbitals which make 109.5° with each

other as shown in (b). (c) ψhyb orbitals on two neighboring Si atoms can overlap to

form ψB or ψA. The first is a bonding orbital (full) and the second is an antibondiong

orbital (empty). In the crystal ψB overlap to give the valence band (full) and ψAoverlap to give the conduction band (empty).

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Energy Band

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PART 1 Materials

Chapter 1 Electron Energy and States in Semiconductors: A brief review for both quantum mechanical concepts and crystallography including crystal structure, plane and direction

Chapter 2 Homogeneous Semiconductors: A review for energy band concept, semiconductor classification(intrinsic vs. extrinsic), hole concept, and electron/hole distribution in semiconductor by introducing density-of-states functions andFermi functions

Chapter 3 Current Flow in Homogeneous Semiconductors: Introduction of both carrier actions (drift, diffusion, and ehp G-R) which has been known to cause device current and continuity equations that could be used for device analysis

Chapter 4 Nonhomogeneous Semiconductors: Introduction of the consequence of nonuniform doping

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Chapter 1 Electron Energy and States in Semiconductors

The point of this chapter is to understand some fundamental physics of how electrons behave in matter.

Models of an Atom :[1] Thompson Model[2] Rutherford Model[3] Bohr Model[4] Wilson-Sommerfeld Model

Figure 1.1 : (a) The Thompson model of an atom, in which the

positive charge is uniformly distributed in a sphere and the electrons are considered to be negative point charges embedded in it;

(b) the Bohr model, in which the positive charge is concentrated in a small nucleus and the electrons orbit in circles;

(c) the Wilson-Sommerfeld model, which allows for elliptical orbits.

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1.3.1 The Bohr Model for the Hydrogen Atom (p.5)

Hydrogen atom based on the Bohr model :

[1] Potential energy

[2] The Bohr radius of the nth state is

[3] The total energy En is

E E qrp vac

o

= −2

24πε (1.6)

Figure 1.2 Potential energy diagram for an electron in the vicinity of a single positive point charge. The electron isconsidered to be a point charge.

The vacuum level Evac is definedas the potential energy at r = ∝

r nmqn

o=4 2 2

2πε η

(1.12)

E E E E mqnn Kn Pn vac

o

= + = −4

2 2 22 4( )πε η(1.16)

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EXAMPLE 1.1 (p.8)

Find the energies and radii for the first four orbits in the hydrogen atom.

Figure 1.3Allowed energies in the hydrogen atom. Higher energies occur increasingly close to each other, approaching the vacuum level.

Figure 1.4Radii of the first four atomic orbits of the hydrogen atom, according to the Bohr model.

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1.3.2 Application to Molecules: Covalent Bonding (p.11)

From isolated atoms to molecules

“Isolated” means nuclei are sufficiently far apart so that they do not influence each other.

When the nuclei are allowed to approach each other, an electron would be influenced by both nuclei according to Coulomb’s law.

Figure 1.6a Energy band diagram for two noninteracting hydrogen nuclei.

Figure 1.6b As the nuclei are brought together, the upper energy levels merge and electrons in those levels are shared between the atoms.

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Figure 1.6c The nuclei are sufficiently close together that all energy levels are shared. Since the lowest level is usually the only occupied level for hydrogen, if it is occupied by two electrons H2 molecule is stable.

1s

Electron shell

Covalent bond

H-atom H-atom

H-H Molecule1

1

2

2

12

1s

Fig. 1.4: Formation of a covalent bond between two H atoms leadsto the H2 molecule. Electrons spend majority of their time betweenthe two nuclei which results in a net attraction between the electronsand the two nuclei which is the origin of the covalent bond .From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Read the paragraph in your textbook page 12: Ek = E1 - Ep so that the velocity of electrons is smallest in between the nuclei.

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1.3.4 Covalent Bonding in Crystalline Solids (p.14)

Consider Si crystal

The electrons occupying the outer subshells are called valence electrons, which determine the valency of the atoms.

Valence Electrons : “stronly involve in chemical reactions: Covalent Bonding”

Core electrons : “remain essentially unperturbed during the chemical reactions on normal atom-atom interaction.”

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Si crystalline in 3-dimension and the bonding model in 2-dimension

valence electrons :contribute to the formation of VB

http://jas2.eng.buffalo.edu/applets/education/solid/unitCell/home.html

Figure 1.7a Two-dimensional bonding representation of a crystalline solid.

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“The four vacant states in the third shell of atoms in Si form a band in crystalline Si called the conduction band.”

Forbidden Band

Figure 1.7b Potential energy for an electron in that crystal along a row of atoms (solid line) and between rows (dashed line). In this representation the electron is considered a point charge.

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Electron Affinity, Ionization Potential, and Band Gap (p. 16)

Ionization energy : the minimum energy required to excite an electron from the top of the valence band in the crystal to the vacuum level

Electron affinity : the energy difference between the vacuum level and the vacantstate of lowest energy (i.e., Ec)

Energy gap : the minimum energy required to excite an electron from the valence band to the conduction band

See Table 1.2 in p. 17, and discuss the properties of variousmaterials.

Figure 1.8 Definitions of vacuum energy Evac, electron affinity c, ionization energy g and the energy gap Eg.

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Occupancy of energy band as a function of temperature (p. 18)

If Eapplied ≥ Eg, then a few electrons are able to excited into the conduction band. For example, ni (electrons/cm3) = 1.08 x 1010 /cm3 for Si @ room temperature: For further discussion, see the Table listed in the backside of textbook cover.

This value seems to be very high ! However, consider this: In Si, 5×1022 atoms/cm3

4 valence electrons/atom→ 2×1023 bonds(or equivalently valence electrons)/cm3. Therefore, at room temperature, only 1 electrons out of 2 x 1013 electrons/cm3 can be excited !

Even more, the average time an electron spends in the CB is on the order of 10-10 to 10-3 s, depending on the quality of material.

Figure 1.9 At room temperature, electrons are being excited up to the conduction band and relaxing back to the valence band. At any given moment there is some number of electrons in the conduction band. Hole

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입자

빈공간: 반입자

현상계(양의 에너지)

진공(음의 에너지)

BREAK: Matter vs. Antimatter

Hole theory proposed by Paul Dirac in 1928, and the discovery of positron by Carl Anderson in 1932.

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Holes indeed contribute to the formation of current in the semiconductor devices !

Figure 1.10 Movement of many electrons is treated as the movement of one positively charged “hole.”

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Discussions on

[1] Polycrystalline and Amorphous Materials (p. 19)[2] Distinctive Features in Energy Band of Various Materials (p. 20)

Figure 1.11 Energy band diagrams for (a) an insulator, (b) a semiconductor, and (c) a metal. The energies in the shaded regions are in general occupied.

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1.4 Wave-Particle Duality (p. 20) : Review of Physical Electronics

1. Classical waves have energies that are quantized. Each quantum of energy can be considered a particle. For electromagnetic radiation (e.g., light), these particles are called photons. For acoustic waves (e.g., sound), the particles are called phonons.The energy of these particles is

2. Classical particles can be considered to be waves possessing energy and wavelength.

3. The electric field of an electromagnetic wave traveling in the x direction can be expressed by a simple sinusoidal function of amplitude A,wavelength λ, and wave vector (wave number in 1-dimension problem):

E h h= = =ν

ππν ω

22 η (1.24)

( )ρ

ηE x t A x t A Kx t A Kx E t

K

( , ) cos cos cos= −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎤

⎦⎥= − = −⎛

⎝⎜⎞⎠⎟

=

2

2

πλ

ν ω

πλ

(1.25)

(1.26)

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4. Likewise, matter can also be described using waves using a wave function: Matter Wave

(1.27) : wave function

(1.28) : wavelength of the matter wave

(1.29) : general form of a wave function of the matter wave in a vacuum (A=constant)

[ ]

Ψ

Ψ

( , ) sin

( / )

x t A Kx E t

K

(x,t)=Ae j Kx E t

= −⎛⎝⎜

⎞⎠⎟

=

−

η

η

λ π2

1.5 The Wave Function in 3-dimension with time variation (p. 22)

Ψ Ψ= ( , , , )x y z t

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Probability and the Wave Function

“A basic connection between the properties of the wave function and the behavior of the associated particle is the probability density P(x, t).”

where Ψ*(x, t) is the complex conjugate of Ψ*(x, t).

( ) ( )P x t x t x t( , ) , ,*= Ψ Ψ

In summary,

• IΨ(x, y, z, t)I2 is the probability of finding the electron per unit volume at x, y, z, at time t.

• IΨ(x, y, z, t)I2dxdydz is the probability of finding the electron in a small elemental volume dxdydz at x, y, z at time t.

• If we are just considering one dimension, then the wave function is Ψ(x, t), and IΨ(x, t)I2dx is the probability of finding the electron between x and (x+dx) at time t.

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1.6 The Electron Wave Function

Remind the discussion we had in the course of “Physical Electronics”. Time-dependent wave function can be derived by the multiplication of a space-dependent part (i.e., time-independent wave function) by a time-dependent part.

Now, ψ(x) can be derived by the time-independent Schrödinger’s equation.

( )Ψ(x,t)= x j E tψ − ( / )η(1.34) : time-dependent wave function

−+ =

η2 2

22md x

dxE x x E xP

ψ ψ ψ( ) ( ) ( ) ( ) (1.35)

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Figure 1.12 The free electron model for an electron; (a) the physical picture; (b) the potential is assumed constant everywhere inside the crystal.

1.6.1 The Free Electron in One Dimension : constant potential energy and no collision in a solid (p. 23)

The total energy of an electron in this case is E = Ep + Ek.Now, the Schrodinger’s equation is

Since both Eo and the total energy E are constant, the eq. (1.36) becomes

−+ =

η2 2

22md x

dxE x E xo

o

ψ ψ ψ( ) ( ) ( ) (1.36)

d xdx

m E E xoo

2

2 22 0ψ ψ( ) ( ) ( )+ − =η

(1.37)

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The solution to Equation (1.37) is

Here A, B, C, and D are constants to be determined from the boundary conditions, and

As discussed in eq. (1.34), the total wave function for the free electron is

The electron could be going either way.

(1.38)ψψ

( )( ) sin( ) cos( )x Ae Bex C Kx D Kx

jKx jKx= += +

−

(1.39)

K m E E m Eo o o K=−

=2 2

2 2( )η η

(1.40) : Ek = kinetic energy

Ψ( , ) [ ( / ) ] [ ( / ) ]x t Ae Bej Kx E t j Kx E t= +− − +η η (1.41)

Traveling wave in the -x directionTraveling wave in the +x direction

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The phase velocity is the velocity of a point of constant phase of the wave :

The phase velocity is not unique because the total energy E is dependent on the choice of potential energy reference.

The group velocity is the velocity associated with the center of mass of the particle :

ν pxt

EK

= =η

νgdxdt

dEdK

= =1η

(1.43)

http://www.ee.mu.oz.au/staff/summer/applets/group_velocity.html

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The energy E of the electron is

E E Kmo

o

= +η2 2

2 (1.44)

Kinetic Energy

∂∂

2

2

2EK mo

=η

(1.47)

The curvature (the second derivative) of the E-K locus is inversely proportional to the mass.

Figure 1.13 The E-K diagram for the free electron.

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1.6.2 The De Broglie Relationship (p. 25)

Presented in 1924, and awarded Nobel prize in 1929

“Just as a photon has a light wave associated with it that governs its motion, so a material particle (e.g., an electron) has an associated matter wave (or pilot wave) that governs its motion.” ----- A Grand Symmetry of Nature

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Filament

50kVTwo slits

Electrons

Fluorescent screen

VacuumElectron diffraction fringes on thescreen

Fig 3.12: Young's double slit experiment with electrons involves anelectron gun and two slits in a cathode ray tune (CRT) (hence invacuum). Electrons from the filament are accelerated by a 50 kVanode voltage to produce a beam which is made to pass through theslits. The electrons then produce a visible pattern when they strike afluoresecent screen (e.g. a TV screen) and the resulting visual patternis photographed (pattern from C. Jönsson, D. Brandt, S. Hirschi,Am. J. Physics, 42, Fig. 8, p. 9, 1974.

λ =hp

This equation holds only when the potential energy is constant over the path of the electron.

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1.6.4 The Quasi-free Electron Model (p. 27)

Now, consider the more realistic model. For a periodic function of Ep(x), Ep(x) = Ep(x ± na). Then the Schrodinger’s equation becomes

Note that we do not know the exact form of the potential energy Ep(x), so that we cannot derive the exact solution of Eq. (1.55). ⇒ Use the Bloch theorem.

d xdx

m E E x xop

2

2 22 0ψ ψ( ) [ ( )] ( )+ − =η

(1.55)

L = na

Figure 1.14 The electron potential energy in a crystal is a periodic function.

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Bloch Theorem (p. 28)

For an electron in a periodic potential [i.e., EP(x) is periodic function], the time-independent wave function is

where UK(x) is some function (or called as the unit cell wave function) that is also periodic in x with the periodicity of the crystal.

Based on the discussion in p.23 and on Eq. (1.34), the time-dependent Bloch wave function is

ψ ψ( ) ( )x a x e jKa+ =ψ ( ) ( )x U x eKjKx= or(1.56)

[ ]Ψ( , ) ( ) ( / )x t U x eKj Kx E t= − η (1.57)

A unit amplitude plane wave modulated by some periodic function UK(x) with period a.

U x U x naK K( ) ( )= + (1.58)

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E – K Relationship : Rough Interpretation

We know that the crystalline forces acting on the electron are independent of the direction of propagation (sign of K): Therefore, energy in K-space is

E(K) = E(-K)

This suggest that E(K) has an extremum (either a relative maximum or minimum) at K=0.

After some mathematical derivation shown in p. 29, we are able to show

That is, the E(K) relation is periodic in K, with period 2π/a.

E K E K na

( ) = +⎛⎝⎜

⎞⎠⎟

2π(1.66)

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Figure 1.15a One possible E versus K diagram for the periodic potential. E versus K.

Summary:1. E(K) is periodic in K space, with period 2π/a.2. Equivalent extrema in E exist at K = 0, ±2π/a, ±4π/a …..3. Equivalent extrema exist at K = ±π/a, ±3π/a …..4. The slope of the E-K curve is zero at K = 0, ±π/a, ±2π/a, ±3π/a, ±4π/a …..5. The group velocity is periodic in K space with the same periodicity as the E-

K curve.

νgdxdt

dEdK

= =1η

(1.43)

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Figure 1.15b One possible E versus K diagram for the periodic potential. The group velocity vg versus K.

Reduced Zone (or First Brillouin Zone)

Figure 1.16 The reduced, or first Brillouin, zone.

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Energy Band Diagram : E - K vs. E - x

Figure 1.17 (a) The E-K diagram; (b) the corresponding energy band (E-x) diagram.

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More accurate calculation of E-K via Kronig-Penny model shows

-π/a +π/a

No BraggDiffraction

Bragg diffraction occurs at the BZ boundary, resulting in standing wave and energy gap where no waves are allowed to be traveled.

http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm

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Reference : Band Gap of Si

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1.6.5 Reflection and Tunneling (p. 32)

Figure 1.18 An electron wave is extended in space. (a) When the wave reflects from the potential barrier, the electron wave function extends a short distance into the forbidden region. Thus some fraction of the electron charge is found to the right of the barrier. (b) If the barrier is very thin, the electron wave function Y may extend all the way through it. Since the probability density Y*Y is not zero on the far side of the barrier, there is some (small) chance that the electron will cross through the barrier and emerge on the other side.

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게르트 비니히 하인리히로러(Gerd Binnig) (Heinrich Rohrer)

www.zurich.ibm.com/st/nanoscience/index.html

Scanning Tunneling Microscope (STM)

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x

V(x)

Metal

ψ(x) Second MetalVacuum

Vo

(b)x

V(x)

Metal

ψ(x)Vacuum

Vo

E < Vo

(a)

Materialsurface

Probe ScanItunnel

x

Itunnel

Image of surface (schematic sketch)

(c)Fig. 3.17: (a) The wavefunction decays exponentially as we move away from thesurface because the PE outside the metal is Vo and the energy of the electron, E < Vo..(b) If we bring a second metal close to the first metal, then the wavefunction canpenetrate into the second metal. The electron can tunnel from the first metal to thesecond. (c) The principle of the Scanning Tunneling Microscope. The tunneling currentdepends on exp(-αa) where a is the distance of the probe from the surface of thematerial and α is a constant.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca www.iap.tuwien.ac.at/www/Surface/STM_Gallery/stm_animated.gif

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[Thermomechanical storage and AFM] Source: www.zurich.ibm.com/st/nanoscience/thermomech.html

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1.7 Optical Emission and Absorption

Light energy must be absorbed or emitted in integer multiples of hν.

E h pht= =ν ωη (1.70)

Figure 1.19 (a) A photon of energy 2.06 eV is incident on a material of energy gap 2.5 eV. The photon cannot be absorbed. (b) The band gap is small enough that allowed states separated by 2.06 eV exist, thus the photon can be absorbed. The photon’s energy is given to the electron. (c) In emission, the electron goes to a lower energy state, releasing the extra energy in the form of a photon.

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Example 1.5 Optical Communication

Solve this example by yourself.

Figure 1.20 (a) A communication fiber optic link contains a light source, a fiber, and a photodetector. (b) Typical absorption spectrum for optical fiber.

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1.8 Crystal Structures, Planes, and Directions

Crystallography is of great interest to people who fabricate semiconductor devices: This knowledge is especially important in MOSFET technology.

Crystalline Amorphous Poly-crystalline

Crystals are regular structures in which the atoms are arranged in a pattern that repeats throughout the material.

Electrical and physical properties strongly depend on the atomic arrangement in the material.

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Figure 1.21 Cubic crystals: (a) simple cubic; (b) face-centered cubic, an atom in the center of every face, and (c) body-centered cubic.

Figure 1.22 (a) The diamond structure consists of two interpenetrating FCC lattices. The second FCC cube is offset by one-quarter of the longest diagonal. The dashed lines indicate the part of the second FCC lattice that is outside the unit diamond cell. (b) A zinc blende material has the same structure, but two types of atoms. The black atoms are one type (for example, gallium) and the colored atoms are the other (arsenic).http://jas2.eng.buffalo.edu/applets/educa

tion/solid/unitCell/home.html

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Figure 1.23 The three most important crystallographic planes (in parentheses) and the corresponding crystallographic directions (square brackets).

The device performance including the mobility is dependent upon the plane and direction discussed here !