Sse Lecture3

8/22/2019 Sse Lecture3

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ENERGY BANDS AND

CHARGE CARRIERS INSEMICONDUCTORS


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Bonding Forces and Energy

Bands in Solids For an electron in an isolated atoms, discrete energy levels

exist.

In solids, when atoms come closer at an inter atomic

distance, their wave functions overlap, resulting the

formation of energy bands.

Band formation is due to the change in potential energyterm in Schrodingers eq. as well as the boundary

conditions


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Bonding forces in Solids

The interaction of electrons in neighboring atoms of a

solid is the most important function of holding the crystal

together.

Electronic structure in the outermost orbit is responsible

for the different types of bonding and the electrical

properties of solids.


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Types of Bonding

Ionic Bonding: Most commonly the alkali halides such as NaCl are bound

through Ionic bonding

The electronic structure of Na(Z=11) is [Ne]3s1, and

Cl(Z=17) has the structure [Ne] 3s2

3p5

. In NaCl lattice, each Na atom gives its 3s electron to Cl

atom to complete its outermost orbit as 3s23p6.

result is the crystal of ions with the electronic structureof inert atoms Ne and ArNo electrons for the electricalconduction are available.

Each Na atom will become Na+ ion and each Cl atom willbecome Cl- ion, columbic attraction between the opposite ions is

responsible for the strong bonding force known as IONIC

BONDING


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Figure 31

Different types of chemical bonding in solids (a) an example of ionic

bonding in NaCl; (b) covalent bonding in the Si crystal, veiwed along a

direction (see also Figs. 18 and 19).


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METALLIC BONDING Outer electronic shell of metallic atoms is partially filled,usually with three electrons at maximum.

In alkali atoms, there is only one loosely bonded electron in

the outer orbit.

In solids, atoms at a distance of atomic dimensions,electrons cannot distinguish between their own core and the core of

neighboring atom, thus all the outermost electrons are not bound to

single atom, but the inner ionic cores are immersed in the sea of free

electrons giving rise to high electrical conductivities.


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COVALENT BONDING

Diamond lattice semiconductors mostly exhibit the

Covalent Bonding.

Each atom in Ge, Si or C diamond lattice is surrounded

by four nearest neighbors, each with four electrons in theoutermost orbit.

Each atom shares its valence electrons with its four

neighbors.

Shared electrons are bounded through quantummechanical interaction between them.

Shared electrons are indistinguishable except that they

must have opposite spins satisfying the Paulis exclusion

principle.


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Figure 19

Diamond lattice unit cell, showing the four nearest neighbor structure. (From

Electrons and Holes in Semiconductors by W. Shockley, 1950 by Litton

Educational Publishing Co., Inc.; by permission of Van Nostrand Reinhold Co., Inc.)


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Few Properties of Semiconductors

For elemental semiconductors, no free electrons arepresent in the lattice, resulting in an insulator behavior. (The

case at 0oK).

An electron can be thermally or optically excited out of a

covalent bond and become available for electrical conduction,

which is the most important feature of semiconductors.

Compound semiconductors like GaAs, a II-VI

compound, have mixed bonding.

Ionic, due to their positions in periodic table.

Covalent due to their atomic structure.


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ENERGY BANDS

Isolated atoms have discrete energy levels, which results

due to columbic potential of nucleus seen by the electrons.

Look at fig. 2-8, orbital model of Si atom and the energy

levels of various electrons in the columbic potential well of the

nucleus. When two atoms come close to each other at atomic

distance their wave functions overlap.

By solving the Schrodingers equation for such an interactingsystem we find that the composite two-electron wave function is

the linear combination of the individual atomic orbital (LCAO)

1: Wave function for isolated atom 1 and 2: Wave functionfor isolated atom 2

= a 1 + b 2 is the (LCAO)


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Figure 28

Electronic structure and energy levels in a Si atom: (a) The orbital model of a Si atom

showing the 10 core electrons (n = 1 and 2), and the 4 valence electrons (n = 3); (b)

energy levels in the coulombic potential of the nucleus are also shown schematically.


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Figure 32

Linear combinations of atomic orbitals (LCAO): The LCAO when 2 atoms are brought together

leads to 2 distinct normal modesa higher energy anti-bonding orbital, and a lower energy

bonding orbital. Note that the electron probability density is high in the region between the ion

cores (covalent bond), leading to lowering of the bonding energy level and the cohesion of the

crystal. If instead of 2 atoms, one brings together N atoms, there will be N distinct LCAO, and N

closely-spaced energy levels in a band.


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Bonding or Anti-bonding Orbital :

Sym LCAOsym = (1/2) 1 + (1/2) 2 bonding orbital

Non-sym. LCAO

asym = (1/2) 1 - (1/2) 2 anti-bonding orbital

sym Lowest energy state

asym Highest energy state

All other energy state are due to any other LCAO


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For the two electrons at inter-atomic distance from

each other, their wave functions overlap, but according to

Paulis exclusion principle, they cannot have same state orenergy level

So there is a splitting of these energy levels in such a

way that only one electron per energy level is available.

In solids, many atoms are brought close to each otherand same phenomenon is seen i.e., splitting of energy level

with such a small difference of energy that they almost form a

continum known as energy band.


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Band Types

Lower energy states form valence bands.

Higher energy levels split in the form ofconductionband.

There is a gap between valence band and conduction bandknown as band gap orforbidden band.


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Special case Si

In Si, outer orbit is 3S23P2 , allowed states are 2N in 3s

state and 6N in 3P state for N atoms.

In S shell is all states are occupied but in p shell 2N states

are occupied and 4N are available.

At 0K, lowest energy states are occupiedvalence bandis filled, conduction band empty.

Through thermal or optical excitation we can shift theelectrons from valence to conduction band. effectingon the electrical properties.


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Figure 33

Energy levels in Si as a function of inter-atomic spacing. The core levels (n = 1,2) in Si

are completely filled with electrons. At the actual atomic spacing of the crystal, the 2 N

electrons in the 3 s sub-shell and the 2 Nelectrons in the 3 p sub-shell undergo sp 3

hybridization, and all end up in the lower 4 Nstates (valence band), while the higher lying

4 Nstates (conduction band) are empty, separated by a bandgap.


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Metals, Semiconductors and

Insulators In metals conduction band is partially filled, so

allowed states are available for electrons to hopfrom one state to another state

In Insulator e.g., diamond, valence band iscompletely filled so there is no place for theelectrons to move and conduction band iscompletely empty so there is no electron to movethrough allowed states.

Semiconductors also has the same band structureas of diamond but there is a difference in band gap


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Band Gap of Diamond and Silicon

Semiconductors at 0K have same band structureas of diamond but the Energy gap between valence

and conduction band in Si is 1.1eV, whereas indiamond gap is 5eV. At room temperature these

gaps become 1eV and 10eV

Due to small band gap, energy could be supplied

to electrons through thermal or optical excitations,

so that they can jump from valence to conduction

band.


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Figure 34

Typical band structures at 0 K.


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Direct and Indirect semiconductors

In a solid and electron is assumed to travel through a

perfectly periodic lattice. The wave function is assumed to

be in the form of plane wave e.g., in x direction with

propagation constant k also called the wave vector. K alsodefines the momentum of the electron.

Space dependent wave function for the electron is

k(x) = U(kx, x) exp (ikxx)

Where the U(kx, x) modulates the wave function according to

periodicity of the lattice.


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For such calculations allowed values of energy could be

plotted as a function of k.

As the periodicity of the crystal is mostly different in

different directions so (E, k) plot in different directions is

different and we can get a complex surface for such a plot.

For example, the band structure of GaAs is such that the

maxima of valence band and minima of conduction band lie at

the same value of k. For Si, these minima and maxima lie at two different

values of k

Transition of an electron from the conduction band of

GaAs to valence band will result in emission of a photon ofenergy equal to band gap without changing the value of k. Such

transition is known as DIRECT TRANSTION


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Figure 35

Direct and indirect electron transitions in semiconductors: (a) direct transition

with accompanying photon emission; (b) indirect transition via a defect level.


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In Si, transition from conduction to valence band minima

will take place with the change of energy as well as momentum

or k.

Such transition take place through some defect in the

system and known as Indirect Transition.

For indirect transitions, the energy is given up to the

lattice in the form of heat instead of emitting it in the form of

photon. The remarkable difference in the behavior of direct andindirect semiconductors makes it possible to select a typical type

for a typical purpose, e.g., direct semiconductors are mostly

used in light emitting devices.OR we can use the material with vertical transition between the

defect states


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Variation of energy Bands with

Alloy Composition As III-V ternary and quaternary alloys are varied over

their composition ranges, their band structures change.

Fig illustrates the band structure of GaAs and AlAs and the

way in which the bands change with composition x in the

ternary compound AlxGa1-xAs.

In figure

is the minimum of direct conduction band,whereas two other minima for indirect conduction band are

assigned names X and L


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Comparison of Band in GaAs and AlAs

In GaAs is the lowest lying conduction band, so itbehaves as direct semiconductor with band gap of 1.43eV

Whereas in AlAs, direct minimum is much higher as

compared to X and L. X is the lowest lying minima and soindirect transitions with energy gap of 2.16eV will be more

probable.

Ternary alloy AlxGa1-xAs, has a completely different band

structure for different compositions, shown in figure


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Figure 3

6Variation of direct and indirect conduction bands in AIGaAs as a function of composition: (a)

the ( E,k) diagram for GaAs, showing three minima in the conduction band; (b) AIAs band

diagram; (c) positions of the three conduction band minima in AI x Ga 1- x As as x varies over

the range of compositions from GaAs ( x = 0) to AIAs ( x = 1). The smallest band gap, E g

(shown in color), follows the direct band to x = 0.38, and then follows the indirect X band.


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Variation of band gap with alloy

composition From x = 0 to x = 0.38, has lower energy and the

behavior is direct

For x > 0.38, X band has a lowest minima and there willbe indirect transitions

The light emitting diodes and lasers are generally made of

materials capable of direct band to band transition or of

indirect materials with vertical transition between defects


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Charge Carriers in

Semiconductors In conductors or metals, atoms are immersed in the sea of free

electrons, which can individually or as a group can move due

under the influence of an electric field. The valence band is

completely filled and conduction band is partially filled

providing enough states for the electrons to hop.

In semiconductors, valence band is completely filled and

conduction band is completely empty, so no conduction. We

can excite electrons thermally to conduction band so enoughstates are available for their movement as well as in valence

band some states are available for electron movement.

Impurities can play a vital role in controlling the conduction.


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Electrons and Holes

As we have that due to thermal excitations, electrons movefrom valence to conduction band by providing few electron inconduction band and few empty states in valence band.

an empty state in the valence band is known as Hole

If the hole is created by the thermal excitation then they arecalled electron-hole pair (EHP).

After excitation an electron is surrounded bu a large number ofunoccupied energy states, I.e., the eqbm. No. of EHP in pure Siat room temperature is about 1010 EHP/cm3, compared tp Si

atom density of 5 1022 EHP/cm3. The problem of charge transport in valence band can be

accounted for by simply tracking the holes.


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Figure 37

Electron-hole pairs in a

semiconductor.


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Charge transport in the valence band

In a filled band, all available energy states are occupied.

For every electron moving with a given velocity, there

must be another electron somewhere moving with the

opposite velocity.

By applying electric field, net current is zero because forevery electron j moving with velocity vj there is a

corresponding electron j` withvj.(effect shown in fig 3.8).

With N electrons/cm3 in the band we express the current

densityusing a sum over all of trhe electron velocities, andhaving chargeq on each electron. In a unit volume

J = (-q) iN vi = 0 (filled band)


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For a hole, which is created by removing the jth electron,

the net current density in the valence band involves the sum

over all the velocities minus the contribution of the removedelectron

J = (-q) iN vi - (-q) vj (jth electron missing)

The first term is zero so net current is qvj.

The contribution of current due to missing electron is as

due to a positively charged particle with velocity v.

We shall treat the empty states in valence band as positive

charge carrier with mass m.


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Figure 38

A valence band with all states filled, including states j and j9, marked for

discussion. The jth electron with wave vector k j is matched by an electron

at j9 with the opposite wave vector 9k j . There is no net current in the

band unless an electron is removed. For example, if the jth electron is

removed, the motion of the electron at j9 is no longer compensated.


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Comparison of energies in the two bands

In valence band hole energy increases oppositely to

electron energy in conduction band, as the two carriers

have opposite charge.

For the valence band energy minima is at the top of the

band and energy increases downward.

In conduction band, energy minima lies at the bottom of

band and increases upward.

Fig 3.9 shows the simplified version of band diagrams and

used for routine devices. E(K) diagram is a plot pf the total

electron energy (KE + PE) as a function of the crystal

direction dependent electron wave vector or momentum


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The bottom of the conduction band correspond to zero

electron velocity or KE and simply gives us the potential

energy at that point

For the valence band, top is only the potential energy.

In simplified diagram we plot the edges (PE) of the two

bands as a function of position in the device. The higher energies correspond to additional kinetic energy

of the electron.

Band edge correspond to the potential energies, so with

applied electric potential, potential energy edges have slopeas a function of device position.


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Figure 39

Superimposition of the ( E,k) bandstructure on the E-versus-position simplified band

diagram for a semiconductor in an electric field. Electron energies increase going up,

while hole energies increase going down. Similarly, electron and hole wavevectors point

in opposite directions and these charge carriers move opposite to each other, as shown.


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Explanation of figure 3-9:

an electronat position A sees an electric field given by the slopeof the band edge (PE).

Moving to point B, gains kinetic energy at the expense of electric

potential energy.

Electron starts through k and then gained momentum by reaching

at kb

The elctron loses KE by scattering mechanism and returns to the

botton of the band at B.The slope of the E(x) diagram represent the local electric field at

those points.


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Effective mass

If we treat the electron in the periodic lattice as free, thento consider the electrodynamics in the system we have to

consider the different mass as compared to the actual mass,

e.g., in the motion of hole, actually the mass of electron is

involved but its energy states shows it to be negative. By considering the effective mass, we can approximate

the motion of electron in periodic lattice as a free electron,

having kinetic energy as 2k2 / 2m*, where m* is the

effective mass. The effective mass of an electron in a band with a given

(E,k) relation is found to be

m* = 2 / (d2E/dk2)


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Last eq. shows that the curvature of the band determines

the electron effective mass.

consider fig. 3-6a, we cans see that the electron effectivemass in GaAs is much smaller in the direct conduction

band (strong curvature) than in L or X minima (weaker

curvature).

Curvature of conduction band minima is positive andcurvature of valence band maxima is negative

Thus the electrons near the top of the valence band have

negative mass.

Valence band electrons with negative charge and negative

mass move in an electric field in the same direction as a

positive charge hole with positive mass.


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Intrinsic Material

a perfect semiconductor crystal with no impurities orlattice defects is called an intrinsic semiconductor.

No charge carriers at 0oK. (filled VB and empty CB)

At high temperature EHP creates, which are the only

charge carriers in the Intrinsic material. The generation of EHP is due to the breaking of covalent

bonds in the crystal lattice.

the energy required to break the bond is the band gap

energy. (fig.3-11) fig. 3-11 is a qualitative representation of EHP creation.

Quantitative analysis could be done by the band theory.


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It seems that electron and hole are confined in the lattice.

In actual they are few lattice spacing from each other and

they can hop too.

Since they are created in pairs so concentration of electrons

in the CB is the same as the concentration of the holes in the

VB. Thus for intrinsic material

n = p = ni

At a particular temperature, there is a particular

concentration of EHP, So for a continuous EHP formation

there should be a continuous regeneration of EHP. These tw

rates should be equal to each other ri = gi

These rates are proportional to the equilibrium concentration

of electrons no and holes po

ri = rnopo = rni2

= gi


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Extrinsic Material

In addition to the intrinsic carriers, it is possible to createcarriers in semiconductors by purposely introducing

impurities in the crystal. This process is called doping.

When a crystal is doped such that the equilibrium

concentration no and po are different from the intrinsiccarrier concentration ni, the material is called extrinsic

material

By doping, a crystal can be altered so that it has a

predominance of either electrons or holes.

There are two types of doped semiconductors, n-type

(extra electrons) and p-type (extra holes)


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N-Type Material:

when impurities are added to a pure crystal, additional level are

created in the energy band structure, usually within the band gap. impurity from column V (P, As, Sb) introduces an energy level

just below the conduction band in Si or Ge.

This level is filled at electrons at 0K and very little thermal

energy is required to excite them to conduction band. See figure.

at about 50100K virtually all of the electrons in the impurity

level are donated to the CB.

Such impurity level is called donor level.

Such materials are called donor impurities.

In such materials no >> (ni, po), at room temperature.

This is n-type material


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P-Type Material:

Impurities are form column III (B. Al, Ga, In) introduces a

level in Ge or Si near the valence band.

These levels are empty at 0K.

At low temperature enough energy is available to excite the

electrons from the VB to the impurity level, thus enough holescreated in VB.

This impurity level accepts electrons from the VB, so known

as acceptor level.

columns III impurities are called acceptor impurities.

fig 3-12b indicates, doping with acceptor impurities.

po >> (no, pi), known as P-Type.


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Figure 312

Energy band model and chemical bond model of dopants in semiconductors: (a)

donation of electrons from donor level to conduction band; (b) acceptance of valence

band electrons by an acceptor level, and the resulting creation of holes; (c) donor and

acceptor atoms in the covalent bonding model of a Si crystal.

i di f d


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Binding energy of donors

We can calculate the energy required to excite the fifth

electron of the donor atom to CB. For example, As atom has four out of five covalent

bonding electrons and one loosely bounded electron.

We can make a picture like H atom as a loosely bounded

electron revolving around a tight bounded core. Magnitude of the energy for ground state n = 1, can begiven by Bohrs theory as

E = mq4 / 2K2. K =4o r column V donor levels lie approximately 0.01eV below

the CB in Ge, and the column III acceptor levels lie about0.01eV above the VB.

In Si, the usual donor and acceptor levels lie about 0.030.06eV from a band edge.


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Charge Carrier in compound semiconductors

In III-V compounds, column VI impurities occupying column V

sites serves as donors. Impurities from column II occupying column III sites can accept

an electron and will serve as a donor.

Impurities from columns IV have a very interesting behavior andcalled amphoteric, meaning that Si or Ge from column IV can

serve as donor or acceptor depending on weather they reside onthe column III or V sub lattice of the crystal.

By doping, the resistance of the semiconductor materialschanges drastically, in return changing its electronic properties

e.g, in Si, the ni is about 1010

cm-3

at room temperature. In doped Si with 1015 As atoms/cm3, the conduction electron

concentration changes by five orders of magnitude.

The resistivity of Si changes from 2 x 105-cm to 5 -cm withthe doping.


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Electrons and Holes in Quantum Well

In a solid there are two continuum region of energiesknown as CB and VB.

In doped semiconductors there is another type of discrete

levels arising due to the presence of the energy levels of

doping materials.

A third possibility is the formation of discrete levels for

electrons and holes due to quantum mechanical

confinement.

We can get such confinement in the multilayer compounds

semiconductors.

MBE and OMVPE are the good methods to create such

hetero junction materials.


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Fig. 3-13 shows multilayer structure of AlGaAs and GaAs,

where a this layer of GaAs is grown between the two layers of

AlGaAs, which has a larger energy gap as compared to GaAs.

we cans see that energy difference created a potential well.

Electrons in the CB and holes in the VB are confined in the

potential and now have discrete levels as compared to a

continuum.

Transition of an electron from E1 in CB to level Eh in VB will

emit a photon of larger energy as compared to a band gap of

GaAs.

Semiconductor lasers have been made in which such a quantum

well is used to raise the energy of the transition from the

infrared, typical of GaAs to the red portion of the spectrum.


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Figure 313

Energy band discontinuities for a thin layer of GaAs sandwiched between

layers of wider band gap AIGaAs. In this case, the GaAs region is so thin that

quantum states are formed in the valence and conduction bands. Electrons in

the GaAs conduction band reside on particle in a potential well states such a

E 1 shown here rather than in the usual conduction band states Holes in the

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