1. Zone Schemes 2. Brillouin Zones 3

Chapter 3 Energy Bands and Electrons in Crystals

1. Zone Schemes 2. Brillouin Zones 3. Translation Vectors and Reciprocal Lattice 4. Band Structures 5. Electrons in a Crystal

Only for Teaching 材料性質學 1 張宏宜 Zone Schemes

2 electron free theFrom free electron E = h k 2 2m 2/1 x ⋅= Econstk

材料性質學 2 張宏宜 Zone Schemes

sinαa from P α coscos kaa −−−−−−=+ )67.4( αa

coscos electron), (free 0for P = 0for (free electron), α coscos x cos( x +≡= nakaka π )2

n 0, 1, 2, ±±±= ...... 3, α x += naka 2π

2m 2/1 combined with α = 2 E h

π 2m2 2/1 , x nkthen =+ 2 E a h offunction periodic thusisenergy The thusisenergy periodic offunction 2π k with the periodicit y . x a

材料性質學 3 張宏宜 Zone Schemes

electron an If an electron propagates ain periodic potential discontinu observe always we always observe discontinu ities theof energies

1.cos when i.e., min.,or max. a has coskwhen coskwhen xa has a max. min.,or i.e., when xak ±= 1.cos

x π ,2 nnak 0, 1, 2, ±±±== ...... 3, π nk ⋅= x a

材料性質學 4 張宏宜 Zone Schemes

From Fig. 5.2, we can have “periodic zone scheme” (Fig. 5.3); “reduced zone scheme” (Fig. 5.4) and “extended zone scheme” (Fig. 5.5).

材料性質學 5 張宏宜 Zone Schemes

It is useful to plot free electrons in a reduced zone scheme. One considers the width of the forbidden bands to be reduced until the energy gap between the individual branches disappears completely. However, the well-known band character disappears for free electrons, and one obtains a continuous energy region as Fig. 4.1. 2 2π E h ( += nk 2 ,) n ±±±= ...... 3,2,1,0 2m x a n = ,0 n −= ,1 n = ....1

材料性質學 6 張宏宜 k vector

These E-⎪k⎪ curves relate the energy of an electron to its k- vectors, i.e., with its momentum. In Figs. 5.3, 5.4 and 5.5 the individual allowed energy regions and the disallowed energy regions, called band gaps.

The wave vector k is inversely proportional to the wavelength of the electrons. Thus, k has the unit of a reciprocal length and in therefore defined in “reciprocal space”.

Each lattice plane in real space can be represented by a vector which is normal to this plane and whose length is made proportional to the reciprocal of interplanar distance. The tips of all such vectors from sets of parallel lattice planes form the points in a reciprocal lattice. An X-ray diffraction pattern is a map of such a reciprocal lattice. 材料性質學 7 張宏宜 Brillouin Zones

→ +2π/a -2π/a ←

The E-kx curve (-π/a ~ +π/a) corresponding to the first electron band (n-band) is called the first Brillouin zone. The area between π/a ~ 2π/a and also between -π/a ~ -2π/a, which Corresponds to the m-band, is Called the 2nd Brillouin zone. The individual branches in an Extended zone scheme are 2π/a periodic.

材料性質學 8 張宏宜 2-Dimensional Reciprocal Lattice

The electron movement in 2-D has the components kx and ky , Which are parallel to the x- and y-axes in reciprocal space. Points in

The kx -ky coordinate system form a 2-dimensional reciprocal lattice.

材料性質學 9 張宏宜 2-Dimensional Reciprocal Lattice

For the first zone one construct perpendicular bisectors on

the shortest lattice vectors, G1 . The area that is enclosed by These four “Bragg planes” is the first Brillouin zone. All the zones have the same area.

材料性質學 10 張宏宜 Brillouin Zones

The Brillouin zones are useful if one wants to calculate the behavior of an electron which may travel in a specific direction in reciprocal space.

o An electron travels to45at the kx − theaxis, ofboundary (Fig.5.8) reached is zone Brilloouin the Brilloouin zone is reached (Fig.5.8) π 2 π 22 k )8.5(2 E h k 2 ==⇒−−= h crit a max 2m crit 2ma Brillouin a ofboundary the ofboundary a Brillouin zone is reached at π k −−= ),9.5( when an electron moves t theoparallet - kk axis.-or crit a x y 1 π 22 largest The largest ofenergy electrons E = h )( max 2 2ma

材料性質學 11 張宏宜 Brillouin Zones

Once the max. energy has been reached, the electron waves form standing waves (or equivalently, the electrons are reflected back into the Brillouin zone).

材料性質學 12 張宏宜 Bragg Reflection

An electron wave propagates in a lattice at an angle θ to a set of parallel lattice planes.The corresponding rays are diffracted on the lattice atoms. At a certain angle of incidence, constructive interference between rays 1’ and 2’ occurs. 2π Bragg sin2 λθ== nnarelation , n = ,....3,2,1 k π nk −−−−−−−−= )11.5( crit asinθ For perpnedicular incidence θ = o ),0( (5.11) becomes (5.9). (5.8). obtains one ,54 If θ = o ,54 one obtains (5.8).

theAt kcrit theAt transmiss ofion an electron beam through the lattice is prevented. B ragg theandincident The theandincident Bragg − reflected electron w ave form a standing wave.

材料性質學 13 張宏宜 Bragg Reflection

材料性質學 14 張宏宜 Wigner-Seitz Cell & Reciprocal Lattice

The smallest possible cell is called “primitive unit cell”.

The Wigner-Seitz cell is a special type of primitive unit cell that shows the cubic symmetry of the cubic cells.

BCC: One bisects the vectors from a given atom to its nearest neighbors and places a plane perpendicular to these vectors at the bisecting points.

材料性質學 15 張宏宜 Wigner-Seitz Cell & Reciprocal Lattice

FCC: The atoms are arranged on the corners and faces of a cube, which is equivalent to the center points of the edges and the center of the cell.

材料性質學 16 張宏宜 Wigner-Seitz Cell & Reciprocal Lattice

In Fig. 5.14(a), by combination of these ”primitive vectors” a translation vector, define

R=n1 t1+n 2 t2+n 3 t3 , In Fig. 5.14(b), the fundamental vectors

t1 , t2 , t3 are shown in a conventional unit cell of a bcc lattice.

The b1 , b2 , b3 are the reciprocal lattice vectors and a translation vector

G=2π(h1 b1 +h2 b2 +h3 b3 ) n and h are integers.

⇒⇒ b1 ·t1 =1 b1 ·t2 =0 b1 ·t3 =0

材料性質學 17 張宏宜 Wigner-Seitz Cell & Reciprocal Lattice

Kronecker − Delta symbol

tb =⋅ δ mnmn where, δ mn 1 mn ,for δ mn 0 for ≠=== mn

three vectors bn are toreciprocal the vectors tm.

b1 is perpendicu lar to 2 toand tt 3.

1 1132 tttconsttbttconstb 321 =×⋅⋅=⋅⇒×⋅= 1 1 Then, const = ×⋅ ttt 321

× tt 32 b1 =⇒ ; ×⋅ ttt 321

× tt 13 b2 = ; ×⋅ ttt 321

× tt 21 b3 = ×⋅ ttt 321 材料性質學 18 張宏宜 Wigner-Seitz Cell & Reciprocal Lattice

5.14(b)) (Fig. a"."constant lattice thehavemay crystal real The real crystal thehavemay lattice a"."constant (Fig. 5.14(b))

lattice theexpress We theexpress lattice vectors , , ttt 321 in terms theof vecto rsunit system. coordinate z y, x,in thek j, i, j, thek x,in y, z coordinate system. a a a a t )( tkji =⇒++−= ;)111( t = );111( t = )111( 1 2 1 2 2 2 3 2 kji a2 a2 a2 a2 tt −=× 111 ( jikkji ) kj )22( +=+=+−+++= kj )( 32 4 4 4 2 −111 a3 a3 a3 ttt kjkji )0()( )110( =++=++⋅++−=×⋅ 321 4 4 2 a2 + kj )( 1 1 bthen = 2 k)(j or b =+= 11)0( 1 a3 a 1 a 2 1 1 b =⇒ ;1)0(1 b = )0(11 2 a 3 a

材料性質學 19 張宏宜 Wigner-Seitz Cell & Reciprocal Lattice

The points of the reciprocal lattice of the bcc structure are identical to the lattice points in a real lattice of the fcc structure. Conversely, the reciprocal lattice points of the fcc structure and the real lattice points of the bcc structure are identical. The Winger-Seitz cell for an fcc crystal (Fig. 5.13) and the first Brillouin zone for a bcc crystal (Fig. 5.17) are identical in shape, vice versa.

A Brillouin zone can be defined as a Wigner-Seitz cell in the reciprocal lattice.

材料性質學 20 張宏宜 Band Structures

The [100], [110], and [111] direction in k-space are indicated by the letters Γ-X, Γ-K, and Γ-L, respectively. From Fig. 5.4, the reduced zone scheme is depicted for more than one direction in k-space. Similar parabolic bands can be detected in the Γ-K, and Γ-L directions. The band diagram for Al looks quite similar to the free electron bands.

Some band gaps, e.g., between the X4 ’ and X1 symmetry points, or between W3 and W2 ’. However, the Individual energy bands overlap in different directions in k-space, no band gap exist. 材料性質學 21 張宏宜 Band Structures

The band gap is clearly seen in semiconductors, e.g., Si, GaAs.

材料性質學 22 張宏宜 Fermi Energy

In a solid of one cubic centimeter at least 1022 valence electrons can be found. How are these electrons distributed among the available energy levels? Many of the electronic properties of materials, such as optical electrical, or magnetic properties, are related to the location of

the Fermi level (EF) within a band.

The Fermi energy (for metal) is often defined as the “highest energy that the electrons assume at T=0 K”. The top surface of the water contained in a vessel can be compared to the Fermi energy.

材料性質學 23 張宏宜 Fermi Distribution Function

The kinetic energy of an electron gas (free e-) is governed by Fermi-Dirac statistics, which states that the probability that a certain energy level is occupied by electrons is given by the Fermi function, F(E),

1 E)(F = -(6.1)------− EE exp( F +1) BTk = :1)(F EE is completely occupied electronsby = :0)(F EE is empty k B: Boltzmann constant; T : absolute temperature

材料性質學 24 張宏宜 Fermi Distribution

The F(E) for higher temperature (T≠0) varies around EF in a gradual manner and not by a step as for T=0. The ΔE in Fig. 6.2 at room temperature is in reality only about

1% of EF. At high energies (E>>EF) the upper end of the Fermi distribution function can be approximated by the classical (Boltzmann) distribution function. − EE E −≈ F )](exp[)(F BTk

The F(E) curve for high energies is referred to as the “Boltzmann tail”. The value of the Fermi function

F(E) at E=EF and T≠0 is ½.

材料性質學 25 張宏宜 Density of States

Assume the free electrons are confined in a square potential well. The dimensions of this potential well are thought to be identical to the dimensions of the crystal under consideration. One electron in a potential well of size a, the solution of the SchrÖdinger equation is π 22 E = h ( ++ nnn 222 ) n 2ma2 zyx

nx , ny and nz are the principal quantum numbers and a is now the length of the crystal.

A specific energy level En (energy state) can be represented by a point in quantum space (Fig.6.3).

材料性質學 26 張宏宜 Density of States

n is the radius from the origin of the coordinate system to

a point (nx , ny , nz ) where 2 2 2 2 n = (nx +ny +nz ) All points within the sphere represent quantum states with

energies smaller than En. The number of quantum states, η, with an energy equal to

or smaller than En is proportional to the volume of the space. Since the quantum numbers are positive integers, the n-values can only be defined in the positive octant of the n-space. One-eighth of the volume of the sphere with radius n gives the number of energy states, η, the energy of which is equal to or smaller than En. 1 4 π 2ma2 3 3 πηn3 =⋅= ( ) 2 E 2 8 63 π 22 材料性質學 h 27 張宏宜 Density of States

Differentiation of η with respect to the energy E provides the number of energy states per unit energy in the energy interval dE, called density of states, Z(E):

2 3 1 3 1 dη π 2ma 2 2 2mV 2 2 EZ )( == ( 22 ) E = ( 22 ) E (6.5)----- dE 4 π h 4π h = aV 3 the, electrons occupy.can energy an have that states ofnumber The ofnumber states that have an energy

or toequal or smaller th En.an A arean delement η ⋅= dEZ(E)

材料性質學 28 張宏宜 Population Density

Based on Pauli principle, each energy state can be occupied by one electron of positive spin and one of negative spin, i.e., each energy state can be occupied by two electrons. = ⋅ EFEZEN )()(2)( 2mV 3 1 1 )( (6.5) and (6.1)with (6.1)with and (6.5) EN )( = ( ) 2 E 2 22 − EE 2π h exp( F +1) BTk EN )( is thecalled (electron) population density.

T→0, E

材料性質學 29 張宏宜 Population Density

The number of electrons, N*, have an energy equal to or

smaller the energy En. dN*=Z(E)dE

T→0, E

材料性質學 30 張宏宜 Complete Density of States

In actual crystals, the density of states is modified by the energy conditions within the first Brillouin zone. Z(E) decreases with increasing E, until eventually the corners of the Brillouin zones are filled. At this point Z(E) has dropped to zero. The largest number of energy states is thus found near the center of a band.

材料性質學 31 張宏宜 Band Model

*If the highest filled s-band of a crystal is occupied by two electrons per atom, the electrons cannot drift through the crystal when an external electric field is applied. Solids in which the highest filled band is completely occupied by electrons are insulators. (Fig. 6.7(a))

*In solids with one valence electron per atom (e.g., alkali metals) the valence band is essentially half-filled. An electron drift upon application of an external field is possible; the crystal shows metallic behavior. (Fig. 6.7(b)) Bivalent metals have the upper bands partially overlap due to weak bonding forces of the valence electrons on their atomic nuclei. The valence electrons flow in the lower portion of the next higher band. They are also conductors. (Fig. 6.7(c)) 材料性質學 32 張宏宜 Band Model

*Semiconductors can accommodate 4N electrons, the valence band is completely filled with electrons. Intrinsic semiconductors have a relatively narrow forbidden energy zone. Fig. 6.7(d) A sufficiently large energy can excite electrons from the completely filled valence band into the empty conduction band.

材料性質學 33 張宏宜 Effective Mass

Experimentally determined x d fd ππωhEd 1)/2()2( dE vg ==== = physical properties of solids, t dk dk dk h dk such as optical, thermal, or dv 1 2Ed dk electrical properties, indicate g a == 2 that for some solids the mass dt h dk dt is larger while for others it is dp dk = kp ; = hhQ slightly smaller than the free dt dt electron mass. 2 2 2 The experimentally 1 Ed dp Ed md 1)v(1 Ed a =⇒ 2 2 = 2 2 = 2 2 F determined electron mass is h dk dt h dk dt h dk usually called the effective F theis force on the electron. mass, m*. F The acceleration of an From sNewton' law, a = electron in an electric field is m a. The group velocity is 2 2* Ed −1 -(6.17)--- )( mass effective the effective mass m = h 2 )( -(6.17)--- dk

材料性質學 34 張宏宜 Effective Mass

The effective mass in (6.17) is inversely proportional to the curvature of an electron band. At these places, the effective mass is substantially reduced and may be as low

as 1% of the free electron mass m0. At points in k-space for which more than one electron band is found (Γ-point in Fig.5.23) more than one effective mass needs to be defined.

In Fig.6.8(c), the effective mass of the electrons is small and positive near the center of the Brillouin zone and

eventually increases for larger values of kx. The electrons in the upper part of the given band have a negative effective mass. A negative mass means that the “particle” under consideration travels in the opposite direction to an applied electric force. 材料性質學 35 張宏宜 Effective Mass

An electron with a negative effective mass is called a “defect electron” or an “electron hole”. The hole is a positive effective mass and a positive charge instead of a negative mass and negative charge.

Solids which possess different properties in various direction (anisotropy) have a different m* in each direction. The effective mass is a tensor. An electron/hole pair is called an “exciton”. 材料性質學 36 張宏宜