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Chapter 4 Metal-Semiconductor contacts and semiconductor-semiconductor heterojunctions 4.1 Metal-Semiconductor contacts (Schottky junction) 4.1.1 Electrostatic analysis 1. Schottky barrier models (1) The Schottky-Mott model According to Schottky and Mott (1938) the height of the barrier between a metal and an n-type semiconductor is given by qΦb

n = q(ΦM - χs). (4.1.1) Here, Φb

n is the work function of the metal, χs represents the electron affinity of the semiconductor. Both quantities are measured with respect to the vacuum level, and Eq. 4.1.1 therefore assumes that the vacuum level is continuous across the interface. The Schottky barrier heights for n- and p-type material, i.e., the barrier Φb

n for electrons and the barrier Φbp for

holes, are expected to add up the bandgap energy Eg of the semiconductor according to

q(Φbn + Φb

p) = Eg. (4.1.2) It was realized early that most metal-semiconductor contacts do not follow Eq. 4.1.1, which predicts that the barrier height increases linearly as SΦ = dΦb/dΦM = 1, with the work function of the metal. Experimental reveal deviations from Eq. 4.1.1, i.e., SΦ < 1, a finding which is usually ascribed to interface states.

Fig. 4.1.1 Band diagram (a) of a metal and an n-type semiconductor before the contact; (b) after intimate contact; (c) intimate contact for a p-type semiconductor.

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(2) Bardeen’s model In 1947, Bardeen proposed the first model for interface states in order to explain the deviations from the Schottky-Mott theory. Bardeen took into account that the semiconductor surface without a metal may already have certain density of surface states within the bandgap. The net charge of these surface states should be zero if the state were filled up to the so-called neutrality level Φ0. In case of the Fermi level position EF

s at the semiconductor’s surface dose not coincide with Φ0, as surface charge Qss should build up. Even without being in contact with the metal, the bands of the semiconductor could therefore be bent upwards. The total charge in the surface states Qss = - q Dit

2D [ Eg - q(Vb0 + Vn + Φ0 )] (4.1.3) depends on Dit

2D, the density of surface states per unit area and unit energy, and the quantity qVn represents the energy difference between the conduction bade edge and the Fermi level in the semiconductor. In Eq. 4.1.3, we have introduced the band bending Vb0 = Vb (V =0). To keep overall charge neutrality, the charge within the surface states must be counterbalanced by the electronic charge, Qsc, in the depletion region of the semiconductor, i.e., it holds Qss+ Qsc = 0. In the limit of a large density of surface states, the expression in the bracket of Eq. 4.1.3 must go to zero. In this case the Fermi level position at the semiconductor surface , EF = q(Vb0 + Vn), corresponds then approximately to the energy location of the neutrality level Φ0: i.e., the Fermi level at the surface is pinned by the large density of surface states. For Si, Dit

2D = 1012 cm-2eV-1 would be sufficient to pin the Fermi level at the neutrality level. If a metal is brought into contact with the semiconductor, the Fermi level remains pinned as long as there is still a finite distance δ between the charges at the semiconductor and the metal surface. Within this Bardeen model, the Schottky barrier height is given as qΦb

n = Eg - qΦ0. (4.1.4)

Hence in the Bardeen limit, the Schottky barrier height is independent of the chemical nature of the metal, and depends only on the neutrality level Φ0 of the surface states on the semiconductor.

Fig. 4.1.2 Band diagram of a metal and n-type semiconductor with surface states (a) before the contact and (b) after the contact with an interfacial layer of width δ. The interface states are assumed to have a charge neutrality level Φ0.

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(3) The linear model of Cowley and Sze The above two models represent, however, two extremes: The SM-model assumes the Schottky barrier height to vary linearly with (SΦ = 1) the work function ΦM of the metal, while B-model assumes the barrier height to be independent of the metal, i.e., SΦ = 0. For most semiconductors, both models, both models do not predict the observed behavior. However, the experimental data may often be approximated by qΦb

n =c1 qΦM + c2 (4.1.5) with c1 < 1. The S-M model assumes c1 = 1 and c2 = -q(χ + Vn), whereas the B-model holds for c1 = 0 and c2 = Eg - qΦ0. The model of Cowley and Sze ( 1965) describes the dependence of the Schottky barrier height on the metal work function and the density of surface states, thus connecting the B-model with the SM-model. The theory is based on the assumptions of (i) an interfacial layer of thickness δ between the metal and the semiconductor, and (ii) the density of surface state to be independent of energy E within the bandgap. The charge Qss in these states is determined by the position of EF with respect to the Φ0. It thus yields.

Qss = - q Dit2D [ Eg – q(Φb + Φ0 )] (4.1.6)

in the interface states. The voltage drop ∆ across the interfacial layer is given by

∆ = QM δ / (εεo), (4.1.7)

where QM is the charge density at the metallic side of the interface. The Schottky barrier height, in contrast the SM-model, is represented as

qΦbn = q(ΦM - χs - ∆). (4.1.8)

Fig. 4.1.3 Experimental data for the barrier heights vs. the work function of the metal. We find a slope parameter c1 = 0.175 and the intercept c2 =-0.12 eV from the data

Depending on the voltage drop ∆ across the interfacial layer. It finally results in

qΦbn = c1 q(ΦM - χs)+ (1 - c1)( Eg - qΦ0) ≡ c1 qΦM + c2. (4.1.9)

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Where the constants c1 and c2 are defined as

cq Dit

D1 2 20

11

=+ δ ε/ ( )ε

, (4.1.10)

and c2 = (1 - c1)( Eg - qΦ0) - c q χs. (4.1.11) Bene and Walser in 1977 experimentally determined the constants c1 and c2 for various Schottky barriers on n-type Si using a least square fit of their measured data, which resulted in c1 = 0.175 and c2 = - 0.12 eV. From Eqs. 4.1.10, and 4.1.11, they obtain Dit

2D ≈ 5.17 x 1013 cm-2eV-1 and E0 = qΦ0 ≈ 0.4 eV above the valence band with assumptions of δ = 5 Å and ε = 1. Further consideration associated this work was an empirical correlation of metal work function and their Pauling electronegativity XM.. Based on Michaelson s work function data, people have found the following correlation ΦM ≈ 1.79 XM + 1.11 (V). (4.1.12) 2. The space charge region The electronic properties of Schottky diodes are not controlled by the barrier Φb

n directly, but by the space charge region of width xm and the concomitant band bending Vb with the semiconductor. Within the space charge region, the semiconductor is depleted from free carriers; the built-in electric field and the electrostatic potential for electrons are calculated from Poisson’s equation

d

xqN D

2

20

φ∂ ε

= −ε

(4.1.13)

based on the depletion approximation (ρ = qND , 0 ≤ x≤ xm). The electric field can then be solved as

Eddx

qNx xx

Dm= − = −

φε ε0

( ) , (4.1.14)

i.e., the electric field increases linearly from the edge of the space charge region up to the maximum value

E xqN

xDmmax ( )= =0

0ε ε (4.1.15)

right at the metal-semiconductor interface at x = 0. Furthermore, there is a parabolic solution for the electrostatic potential

φε ε

( ) ( )xqN

x xDm= − −

2 0

2 = − −Vx

xbm

(1 2) . (4.1.16)

Finally, the width of the space charge region is related to the band bending (if it is biased, Vb →Vb - VB according to

xqN

V VmD

b B=2 0ε ε

( − ) , (4.1.17)

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with qVb = q(Φb

n - Vn). (4.1.18) The total charge Qsc = qNDxm per unit area in the depletion layer can be expressed as Qsc = [2qNDεεo(Vb - VB)]1/2 . (4.1.19) When we take free carriers into account, Vb → Vb + kT/q. 4.1.2 DC analysis 1. A general description of the current transport processes through a Schottky barrier In this section we will discuss the current transport mechanisms of metal-semiconductor contacts. As shown in the figures, for an n-type semiconductor, the transport process for majority carriers only are

(i) Thermionic emission of electrons over the top of the barrier (a); (ii) Thermally assisted tunneling, (b1); (iii) Tunneling at the bottom of the conduction band in the quasi-neutral region (b2).

Carrier diffusion often is not a good description. Since the current transport in a Schottky diode is due to majority carriers and the mobility for most of single crystalline semiconductors used is high (electron mean free path λe >> xm), there is no concentration gradient like minority carriers in a p-n junction limited by the lifetime through the metal-semiconductor contact. Diffusion theory can only be applied to low mobility semiconductors, e.g., CuO2, α-Si, etc..

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Fig. 4.1.5 Various carrier transport processes through a Schottky barrier Process involving also minority carriers are sketched as

(i) Recombination of holes and electrons within the space charge region (c); (ii) Recombination in the quasi-neutral region (d1); (iii) Recombination at the back contact (d2) (hole injection).

2. Emission over the barrier (1) Thermionic emission For an n-type semiconductor, assuming that the electrons inside the semiconductor crystal have a Maxwell-Bolzmann distribution function f(E) = exp[(EF - E)/(kT)], (4.1.20) we have dn = N(En)f(En)dE

= 4 2 3 2

3

π ( )exp( )

* /mh

E EE E qV

kTdEn

cc n− −

− −. (4.1.21)

The kinetic energy of electrons is presented as E - Ec = mn*v2/2. (4.1.22) Hence, we obtain dE = mn*vdv, (4.1.23) and (E - Ec)1/2 = v (mn*/2)1/2. (4.1.24) Therefore, we can write

dnmh

qVkT

m vkT

v dvn n n= − − ⋅22

432

2( ) exp( ) exp( ) ( )* *

π , (4.1.25)

where v2 = vx

2 + vy2 + vz

2. The current density flown from the semiconductor to the metal can thus represented as

J qs m xE qF b− +

∞= ∫ φ

v dn

= 22 2

32 2 2

0q

mh

qVkT

vm v

kTdv v

m vkT

dv vm v

kTdvn n

xn x

v x yn y

y zn z

zx

( ) exp( ) exp( ) exp( ) exp( )* * * *

− − − −∞

−∞

∞

−∞

∞

∫ ∫ ∫ 2

= ( ) exp( ) exp( )* *4

2

2

32 0

2π qm kh

TqVkT

m vkT

n n− − n x . (4.1.26)

The velocity vox is the minimum velocity required in the x direction to surmount the barrier and is given by (mn* v0x

2) /2 = q(Vb - VB). (4.1.27)

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Hence, we find

Jqm kh

Tq V V

kTqV

kTs mn b n

− = −+⎡

⎣⎢⎤⎦⎥

( ) exp( )

exp( )*4

2

2

32π B

= A Tq V V

kTqVkT

b n B* exp( )

exp( )2 −+⎡

⎣⎢⎤⎦⎥

, (4.1.28)

where,

A* = 4 2

3

π qm kh

n*

(4.1.29)

is the effective Richardson constant for thermionic emission, neglecting the effects of optical phonon scattering and quantum mechanical reflection. For free electrons, Ae* = 120 Acm-2K-2. We can then refer the Richardson constant for semiconductors to the one for free electrons, i.e., A* = α Ae*. (4.1.30) The α values for various semiconductors are listed below

Semiconductors Ge Si GaAs p-type 0.34 0.66 0.62 n-<111> 1.11 2.2 0.068 (low field) n-<100> 1.19 2.1 1.2 (high field)

Similarly, we can derive the equation to describe the current density from the metal to the semiconductor,

J A TqkTm s

bn

− = − −* exp( )2 Φ. (4.1.31)

Hence, the total current density under forward bias, VB = VF > 0 is given by

J J J A TqkT

qVnkTs m m s

bn

F= + = −⎡

⎣⎢

⎤

⎦⎥ −⎡⎣⎢

⎤⎦⎥− − * exp( ) exp( )2 1

Φ

= J JqVnkTss

F=⎡⎣⎢

⎤⎦⎥

exp( ) 1− , (4.1.32)

where, Φb

n = Vb + Vn. For reverse bias, VB = VR < 0, JR = -Jss (4.1.33) (2) Barrier lowering due to image force (Schottky effect)

According to Eq. 4.1.35., the current density value at reverse bias is a constant. In an real device, however, the reverse current actually increases with the bias voltage. This has been attributed to a barrier lowering effect due to image force, which was first studied by Schottky.

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When an electron is at a distance x from the metal surface, a positive charge will be induced on the metal surface. The force of attraction between the electron and the induced positive charge is equivalent to the force that would exist between the electron and an equal charge located at -x. This positive charge is referred to as the image charge. The attractive force, called the image force, is presented by,

Fq

xq

xxim = − = −

2

20

2

024 2 16π ε π ε( )

, (4.1.34)

where, ε0 is the permittivity of free space. The work done by transferring an electron from infinity to the point x is given by

E x F dxq

xdx

qxx

im

x x' ( ) = − = − = −

∞ ∞

∫ ∫161

1602

0πε πε. (4.1.35)

This energy term corresponds to the potential energy of an electron at a distance x from the metal surface.

Fig. 4.1.6 Due to the induced charge effect, the potential barrier in a Schottky junction is low.

An electron at s distance x above the metal sees an induced image charge When an external electric field is applied, the total potential energy E(x) as a function of distance is given by

E x q xq

xq x

qx

q Nx xD

m( ) ' ( ) ( ) ( )= − = − − = − + −Φ Φ2

0

2

0

2

0

2

16 16 2πεε πεε εε (4.1.36)

E(x) has a maximum value at x’, which can be solved by the following equation

q

xq N

x xDm

2

02

2

0160

πεε εε'( ' )− − = . (4.1.37)

Because x’ << xm , therefore,

xN xD m

'( ) /= −

14 1 2π

. (4.1.38)

Hence,

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∆Φ = Φ(x=0) - Φ’(x=x’)

= − − + −q

xq

xq

x xm2 16 20

2

0 0

2

εε πεε εε'( 'm ) . (4.1.39)

Because of (xm - x’)2 = xm

2 - 2xmx’ + (x’)2, where 2xmx’ >> (x’), and ∆Φ(x=x’) = Φ’(x=x’) - Φ’(x=x’) ≈ 0, we thus obtain an expression to describe the lowering the Schottky barrier due to this image force effect

∆Φif ≈ −qN

x xDmεε0

'

=14

2 3

20

3

1 4q N

V VkTq

Db Bπ ε ε( )

(/

− −⎡

⎣⎢

⎤

⎦⎥) . (4.1.40)

Discussion:

When applying an electric field εx = 105 V/cm, q∆Φif = 0.12 eV, x’ = 60 Å.

When applying an electric field εx = 107 V/cm, q∆Φif = 1.2 eV, x’ = 10 Å.. Therefore, the image force effect lowers the Schottky barrier considerably at high fields. This effect also explains why Js

Schottky > Jsp-n with a much strong dependence of the reverse bias voltage.

Fig. 4.1.7 Energy diagram describes the image force effect for a metal n-type semiconductor under different biasing conditions.

2. Tunneling through the barrier In Schottky barriers on highly doped semiconductors the depletion layer becomes so narrow that electrons can tunnel through the barrier near the top, where the barrier is thin. This process is called thermionic-field-emission. The number of electrons with a given energy, E, exponentially decreases with energy as exp[-(E/(kT)]. When doping is further increased to reach a degenerate level, especially in semiconductor with small effective mass, e.g., GaAs, electrons can tunnel through the barrier near the Fermi level, and the tunneling current is dominant. Such a mechanism is called field-emission. The I-V characteristics of a Schottky junction in case of thermionic-field-emission or field-emission can be described by the following equations, based on quantum mechanical calculations of the tunneling

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transmission coefficient and the number of electrons as a function of energy and integrating over the states in the conduction band. For the thermionic-field-emission

J JqVEstf

B= exp( )0

, (4.1.41)

where

E EEkT0 00

00= coth( ) , (4.1.42)

and

Eqh N

mD

000

1 2

4=

π ε ε(

*) / (eV) (4.1.43)

The pre-exponential term was calculated by Crowell and Rideout in 1996,

[ ]

JA T E q V V

k E kTqVkT

q VEstf

bn

n B n bn

n=− −

− −−⎡

⎣⎢

⎤

⎦⎥

* ( )cosh( / )

exp( )

/π 00

1 2

00 0

Φ Φ. (4.1.44)

In GaAs the thermionic field emission occurs roughly for ND > 10

17 cm-3 at 300 K. and for ND > 1016 cm-3 at 77 K.

If field emission takes place at very high doping concentration, the width of the depletion region becomes so narrow that direct tunneling from the semiconductor to the metal may occur. This happens when E00 >> kT. We then have

J JqVEsf

B≈ exp( )00

, (4.1.45)

where

JA T

kC kTCqEsf

bn

=⎡

⎣⎢

⎤

⎦⎥

ππ

*sin( )

exp1 1 00

Φ− , (4.1.46)

and

C EV

Vbn

B

n1 00

124

= −−⎡

⎣⎢

⎤

⎦⎥−( ) ln

( )Φ. (4.1.47)

The effective resistance of the Schottky barrier in the field-emission regime is quite low. Therefore, it can be used for Ohmic contacts 3. Generation-recombination in the space charge region

Schottky contacts are usually regards as pure majority carrier devices. Nevertheless, minority carriers may cause an additional current across the interface. Here, the most important process is the generation-recombination of minority carriers in the depletion region of the junction. In this case, for a Schottky junction under forward bias, the product of the concentration pn exceeds its equilibrium value and a recombination current flows across the interface. As we have known, recombination of holes with electrons is most efficient via recombination centers with energies located close to the intrinsic Fermi level (the mid-gap states). We thus have

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J qxn qV

kTR mi

t

F=2 2τ

exp( )

= qxN N E

kTqV

kTJ

qVkTm

v c

t

g FR

F

2 2 2 20

τexp( ) exp( ) exp( )= . (4.1.48)

Under reverse bias, the current via deep traps is due to the generation of holes and electrons, and VR < 0, We thus obtain

J qxN N E

kTJG m

v c

t

gG= − = −

2 20

τexp( ) . (4.1.49)

Note that n = 2 is again present in the exponential term. Similar to the p-n junction, the so-called ideality factor can thus be used to describe the quality of the Schottky junction. 4. Minority carrier injection Minority carrier injection is usually neglected in the description of current transport in Schottky diodes under low injection condition. For long life time semiconductors like Si, minority carrier injection may play an important role preferably for electronic transport under AC conditions and under high forward bias. The influence of the back contact is more severe when the sample thickness is smaller than or comparable to the minority carrier diffusion length because under these circumstances the recombination at the back-contact outstrips bulk recombination. The minority carrier injection effect is usually characterized using the minority carrier injection ratio γ (defined as the ratio of minority current to total current). At a low injection condition, γ can be described by the following equation.

γ =+

≈ =−

JJ J

JJ

qn DN L A T q kT

p

n p

p

n

i p

D p b

2

2* exp( / )Φ. (4.1.50)

Example: For a Au/n-Si junction with ND = 1016 cm-3, the injection ratio γ is of the order of 5x10-5, which has been obtained by both measurements and calculations based on Eq. 4.1.50.

Fig.4.1.8 Energy band diagram of an epitaxial Schottky barrier incorporating the minority carrier injection effect

At sufficiently large forward bias, however, the electric field causes a significant carrier-drift current, the expression of γ should thus be modified as

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γ = qµppnε ∼ JJs

, (4.1.51)

where J is the total current density and Js is the majority carrier (electron) saturation current. As can be seen, for the high current limiting condition the injection ratio increases linearly with current density. Generally, a high injection ration would be obtained from a metal-semiconductor system with high barrier height (low Js), high resistivity (low doping) and a wide band gap (low ni). 5. Inhomogeneities in Schottky contacts

Up to this point, we have only used 1-D theory for Schottky contacts. In this case the influence of randomly distributed dopant atoms on the electric transport is neglected. This assumption seems justified as long as one regards the electrostatic properties of the diode. It has been revealed that the spatial potential fluctuations can, however, strongly alter the electronic transport when there are inhomogeneities in the range of a fraction of 10-3

or 10-4 of the total interface area. This is because that the current density J depends exponentially on the local value of the Schottky barrier height. There are several reasons for such potential Inhomogeneities at the interface of metal-semiconductor contacts

(i) The dopant atoms are randomly distributed within the semiconductor. (ii) Atomic steps and lattice defects at the interface modulated the barrier height even at single crystal epitaxial

contacts. (iii) The dependence of the Schottky barrier on the relative orientation of semiconductor and metal atoms results in

barrier height fluctuations over the area of polycrystalline Schottky contacts (iv) At polycrystalline contacts, grain boundaries in the metal may modify the Schottky barrier. (v) Interface roughness results in spatially varying effective Schottky barriers by local barrier lowering due to field

emission even for nominally homogeneous contacts. (vi) In the particular case of reactive contacts, e.g., silicide/Si, different phase of metal yield different barrier

heights. (vii) The metal atoms can diffuse into the semiconductor and there will be a redistribution of dopant atoms in the

vicinity of the interface. (viii) Contact edges are often sites of charges which locally modify the Schottky barrier height. The contact appears

therefore also inhomogeneous. 4.1.3 AC analysis 1. Differential conductance When an AC modulated signal is added to the DC bias of a Schottky junction, the small signal junction conductance for a diode area A at a give bias voltage is obtained differentiating the diode current equation 4.1.34 with respect to the voltage:

G AdJ V

dVAJ

qnkT

qVnkTd ss

B= =( )

exp( ) . (4.1.52) 2. Depletion capacitance The junction capacitance of the diodes in this case is determined only by the space charge depletion capacitance, CD, but there is no diffusion capacitance since the Schottky junction is a majority carrier device.

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C V AdQdV

q NV kT q Vd

SC D

bn

n B( )

( /= =

− − −

20

2ε ε

Φ ), (4.1.53)

where the free carrier effect has been taken into account. 3. Small signal equivalent circuit of a Schottky diode

Fig. 4.1.10 Equivalent circuit of a Schottky diode 4. Frequency limit for the high frequency performance of devices, the cut-off frequency is governed by the following equation

fR Cc

s SC= =

12

12π τ π

. (4.1.54)

As can be seen from the equivalent circuit, Schottky diodes are more suitable for the high frequency applications, since the total capacitance is much small than that of p-n junctions. In order to maximize fc for Schottky diodes, to minimize the series resistance is important. For a circular metal-semiconductor contact of area A = πr2, the area dependence of fc can be calculated by

fC t Ac

d epi sub

=+

12

1π ρ πρ

, (4.1.55)

where t is the epi-layer thickness if there is, and CD is depletion capacitance per unit area. In practice, the use of microwave detector and mixer diodes is one of most important technological applications for Schottky diodes 4.1.4 Schottky barrier measurements 1. I-V measurements The determination of the Schottky barrier height from I-V measurements is based upon the diode equation, i.e.,

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I AJqVnkTss

B= −⎡⎣⎢

⎤⎦⎥

exp( ) 1− . (4.1.56)

Φb can be extrapolated at zero bias condition. If the effective Richardson constant is not known, Φb can be obtained from an activation energy plot of ln(Jss/T2) vs. 1/T. We have

ln( ) ln( *)JT

AkT

ss b2 = −

Φ. (4.1.57)

Thus, the slope of this plot yields Φb and from the y-axis intercept one obtains the A* value. The derivation of the barrier height base on I-V characteristics, however, needs to be done very carefully, since many non-ideal effects can be involved. 2. C-V measurements The C-V measurements of a Schottky junction can be used to determined the doping profiles of an epi-layer or the substrate, similar to the case of a p-n junction. As a consequence of Eq .4.1.45, a plot of measured capacitance C-2 = Cd

-2 versus the bias voltage should yield a straight line, with an x-axis intercept at V = Vb . The Schottky barrier height can the be deduced from

Φb = Vb + Vn + kT (4.1.58)

3. Internal photoemission

Internal photoemission is the most accurate and direct method for the determination of the Schottky barrier height. Monochromatic light, incident on a metal-semiconductor contact, excites charge carriers at the Fermi level of the metal. If the photon energy hv exceeds the threshold hv0 = Φb - ∆Φif, some of these excited carriers cause a photo current in the external circuit. This current obeys the proportionality

Ip ∼ (hv - hv0)2 (4.1.59) as long as hv - hv0 > 5 kT. Thus, a plot of Ip

1/2 vs. hv should yield a straight line with an x-axis intercept at hv0 = Φb - ∆Φif.

Ch-4 15

Ch-4 16

4.2 Semiconductor-Semiconductor heterojunctions 4.2.1 Band offset models In section 2.1.3, we have introduced the concept of heterojunction band offsets. Three types of the band offset exist, which are straddling (type I), staggered (type II) or broken-gap (type III) in character. In this section, we will discuss some models in order to provide some useful insight how the band offsets are formed at the heterojunction interface. 1. Andeson’s model

For an ideal abrupt heterojunction without interface traps, Aderson (1962) proposed an energy band model, which is very similar to the Schottky-Mott model for metal-semiconductor junction. According to his model the energy bands in both semiconductor materials constituting a heterostructure are not affected each other. He then assumed that the vacuum energy level is continuous and, hence, the conduction band offset is determined by the difference of electron affinities, i.e.,

∆Ec = χ1 - χ2 (4.2.1) The valence band offset is then given by

∆Ev = ∆Eg - ∆Ec (4.2.2) or ∆Ev = ∆Eg - χ1 + χ2, (4.2.3) where, ∆Eg is the energy gap discontinuity. However, this model is in disagreement with the experimental data. 2. Tersoff’s model Tersoff postulated in 1984 that the band offset in the heterojunctions are controlled by the same mechanism as the barrier height in the Schottky barrier diodes. This mechanism is the electron tunneling from one material into the energy gap of the other semiconductor at the heterointerface, leading to the formation of the interfacial dipole layer. The conduction band offset can then be related to the difference in the Schottky barrier heights for the two semiconductor materials forming the heterojunction

∆Ec =q( Φb1 - Φb2). (4.2.4) Generally, a good correlation has been found when comparing the theoretically predicted band offset values using this model with experimental results for lattice matched heterojunctions, e.g., GaAs/AlxGa1-xAs (x=0 - 1). 3. Role of the lattice strain No mater how accurate they are, the above two models can only be used to describe the lattice matched (i.e., two semiconductors forming a heterojunction have the nearly same lattice constant in plane, e.g, GaAs/Ge and GaAs/AlGaAs) heterostructures. If the difference of the lattice constants in plane for two semiconductors is small (e.g., ≤ 5%), using modern epitaxial growth techniques materials (MBE or LTCVD) a thin layer of one material can be grown on the substrate of another material without generation of misfit dislocations. The in plane lattice constant a// of the thin layer will be constrained to match with that of the substrate (biaxial strain) while the perpendicular lattice constant a⊥ will be either extended or contracted (uniaxial strain). At the same time, the fractional volume change also leads to the hydrostatic strain. Such a growth mode is call pseudomorphic or commensurate growth. The critical thickness to keep pseudomorphic growth is limited by the lattice mismatch (in term of the lattice strain) between two semiconductors, and is also dependent on the growth temperature. Above this critical thickness limit, the lattice

Ch-4 17

stain in the heterojunction structures will be released, instead a large density of misfit dislocations will be formed at the heterojunction interface.

Fig. 4.2.1 The critical thickness of SiGe pseudomorphic growth vs. Growth temperature and Ge fraction. Insert figures show schematically (a) a strained, and (b) a relaxed heterostructure

Unstrained hydrostatic strain uniaxial strain

Fig. 4.2.2 Schematic representation of the effect of strain on a triply degenerate band The strain in the heterostructure has tow main effects on the band structure of a semiconductor: hydrostatic strain due to the volume change of the strained layer shifts the energetic position of a band, and uniaxial strain splits degenerate bands. The band offsets of a strained heterojunction do not follow the bulk band properties of the respective semiconductor. In particular, different configurations of strain (compressive or tensile) will result in different spilt shifts, eventually different band offset values. An example of strained Si/SiGe heterostructures is shown in Fig. 4.2.2 unstrained Si Strained SiGe unstrained Si strained Si Strained SiGe (Bulk) (compressive) (Bulk) (tensile) (compressive) ∆(4) ∆Ec ∆(2) ∆(2) Ec ∆(6) ∆(4) ∆(6) ∆(2) ∆(4) ∆Ec

hh ∆Ev hh ∆Ev lh lh lh Ev hh, lh hh, lh hh s.o s.o s.o s.o

Fig. 4.2.3 Strain dependence of the band offsets for various Si/SiGe heterojunctions

4.2.2 Built-in voltage and space charge region across a heterojunction interface

Ch-4 18

Depending on doping, heterojunctions can be classified as isotype (n-n or p-p) and anisotype (p-n) two groups. 1. Anisotype n-p heterojunction

When two semiconductors for an heterojunction, the Fermi level must be continuos throughout. Just like a conventional p-n junction, this requirement leads to band bending. The built-in voltage, Vb is found to be qVb = Eg1 - ∆En - ∆Ep + ∆Ec. (4.2.5)

Here, Eg1 is the energy gap of the narrow gap semiconductor, which was assumed to doped p-type. ∆En is the difference between the bottom of the conduction band in the wide gap n-type material, and ∆Ep is the difference between the Fermi level and the top of the valence band. Note that, in case of equilibrium, although charge depletion occurs at both side of the heterojunction interface, an energy spike is formed at the wide-gap semiconductor side, which is an additional electron (n-doped) or hole (p-doped) barrier for the carrier transport. Using the depletion approximation, we find how Vb is divided between the p- and n-region:

VN

N NVb

D

D Ab1

2

2 1=

+ε

ε ε, (4.2.6)

VN

N NVb

A

D Ab2

1

2 1=

+ε

ε ε. (4.2.7)

The depletion widths and capacitance can be obtained by solving Poisson’s equation for the step charge density function on either side of the interface. One boundary condition is the continuity of the electric displacement, i.e., ε1 E1 = ε2 E2, at the interface. We obtain

xN V V

qN N NdnA b B

D D A

=−

+⎡

⎣⎢

⎤

⎦⎥

2 1 2

2 1

1/2ε ε

ε ε( )

( ), (4.2.8)

xN V V

qN N NdpD b B

A D A=

−+

⎡

⎣⎢

⎤

⎦⎥

2 1 2

2 1

1 2ε εε ε

( )( )

/

, (4.2.9)

and

Cq N N

N N V VdA D

D A b B

=+ −

⎡

⎣⎢

⎤

⎦⎥

ε εε ε

1 2

2 1

1/2

2( )( ). (4.2.10)

According to Eqs, 4.2.6 and 4.2.7, we have he relative voltage supported in each semiconductor is

V VV V

NN

b B

b B

A

D

1 1

2 2

1

2

−−

=εε

, (4.2.11)

where, VB = VB1 + VB2.

Ch-4 19

Fig. 4.2.4 (a) Energy diagram for an ideal n-n isotype heterojunction. (b) and (c) Energy diagrams for ideal p-n and p-p heterojunctions, respectively.

2. Isotype heterojunction The case of an n-n isotype heterojunction is somewhat different. Since the Fermi level position is higher in the wide-gap material, depletion happens at the wider-gap semiconductor side, while charge accumulation occurs at the narrower gap semiconductor side. Under an equilibrium condition, the relation between Vb1 and Vb2 can be found from the boundary condition of continuity of electric displacement at the interface. The electric displacement at the interface (x = 0) in region 1 for an accumulation governed by Boltzmann statistics is given by

r rD E x qN

kTq

qVkT

VDb

b1 1 1 1 11

1

1 2

0 2 1= = = − −⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

ε ε( ) (exp )/

, (4.2.12)

and for a depletion in region 2 is given by

r rD E x qN VD b2 2 2 2 2 2

1 20 2= = =ε ε( ) ( ) / . (4.2.13)

By equaling above two equations, and using Vb = Vb1 + Vb2, we hence derive

V VNN

kTq

qVkT

Vb bD

D

bb= + − −

⎡

⎣⎢

⎤

⎦⎥1

1 1

2 2

111

εε

(exp ) . (4.2.14)

Ch-4 20

Discussion: (i) When Vb1 < kT/q,

VkTq

NN

Vq

kTNNb

D

Db

D

D1

2 2

1 1

1 1

2 2

1 21 2 1≈ +⎡

⎣⎢

⎤

⎦⎥

εε

εε

( / −) , (4.2.15)

Vb2 = Vb - Vb1, (4.2.16)

x x xq

VNd

b

D2 2 0

2 2

2

1 22= − = ( /)

ε. (4.2.17)

(ii) When ε1 ND1 ≈ ε2 ND 2, and Vb = Vb1 + Vb2 >> kT/q,

exp[qVb1/(kT)] ≈ qVb/(kT). (4.2.18) (iii) When ND1 >> ND 2,

according to Eq. 4.2.15, Vb1 ≈ 0. In this case, the heterojunction behaves similarly to the metal-semiconductor contact.

The above equations are derived under the equilibrium condition, in case of biases and free carriers, , Vb1 → Vb1 - VB1 -kT/q, Vb2 → Vb2 - VB2 -kT/q, and Vb → Vb - VB -kT/q, where VB = VB1 + VB2. The derived results can also be used for p - p heterojunctions by only changing ND to NA. 4.2.2 Current transport through heterojunctions The following six processes are usually considered to be responsible for the current transport through heterojunctions under deferent conditions: (i) Diffusion; (ii) Diffution-Thermionic emission; (iii) Thermionic emission; (iv) Thermionic emission- recombination; (v) Tunneling, (vi) Tunneling-recombination. 1. Anisotype p-n heterojunction Both theories of diffusion and thermionic emission can been used to describe the carrier transport through the heterojunction. Care needs to be taken when the shape on the energy barrier (e.g., a big energy spike) at heterojunction interface is complicated. We here restrict the discussion only in the situation when the energy level of the spike at the wide-gap side is lower than that of the bulk conduction band at narrow-gap side. This situation happens when band offset value is small and doping concentrations at both sides of semiconductors are not too high, i.e. relatively smaller band bendings. In this case the diffusion theory can be directly applied, similar to normal p-n homojunctions. (i) Diffusion theory As shown in Fig. 4.2.4, under equilibrium, the energy barrier for hole moving from the p- to n-side is q(Vb + ∆Ev), and for electrons moving from the p- to n-side is q(Vb - ∆Ec). Therefore, the current passing through the junction will be mainly the electron current. Similar to the relation between the minority electron concentration at p-side and the electron concentration at n-side, we have n n qV E

kTb c

1 2= − −exp( )∆ (4.2.19)

Ch-4 21

Fig. 4.2.5 Energy diagrams of an n-p heterojunction: (a) zero bias and (b) forward bias.

Under forward bias, V, the boundaries of the space charge region are x = 0 at the junction interface, -x1 at p-side, x2 at n-side, respectively. At the steady state, neglecting the carrier generation-recombination at the space charge region, the continuity equation of the minority electron injection to the p-region can be written as

Dd n x

xn x n

nn

21

21 1 0

( ) ( )∂ τ

−−

= (4.2.20)

The solution is

n x n AxL

BxLn n

1 1( ) exp( ) exp( )− = − + . (4.2.21)

A and B are determined using the boundary conditions, when x = -∞, n1(-∞) = n1, (4.2.22) x = - x1, [n x n q V V E

kTb B c

1 1 2( ) exp ( )− = − − −∆ ] , (4.2.23)

according to Eq. 4.2.19, Hence, we obtain, A = 0, (4.2.24)

[ ]B nxL

qV EkT

qVkT

n

b c B= − − −−2

11exp( ) exp( ) exp( )∆. (4.2.25)

Inserting these results into Eq. 4.2.21, we find

[ ]n x n nx x

LqV E

kTqVkT

n

b c B

1 1 211( ) exp( ) exp( ) exp( )− = − − −+−∆

. (4.2.26)

The electron current density is

Ch-4 22

[ ] [J J qD

d n x ndx

qD nLn n x x

n

n

qV EkT

qVkT

b c B≈ =−

= − −=−−1 1 2

11

( )exp( ) exp( )∆ ]− , (4.2.27)

or

[ ]J qnDn

n

qV EkT

qVkT

b c B= − −−2

1 2 1( ) exp( ) exp( )/

τ∆ − . (4.2.28)

(ii) Thermionic emission The same as the carrier thermionic process across a Schottky junction, we can apply this theory to describe the current transport through an anisotype heterojunction as well, i.e., some carriers can gain enough thermal energy to be able to go over the energy barrier into another side of the junction. The derived equation is very similar to Eq. 4.2.28 obtained using diffusion theory by only replacing the term (Dn/τn)1/2 with v2

2/2,

[J qnv qV E

kTqVkT

d c B= − −−2

22

1 2

21( ) exp( ) exp( )/ ∆ ]− . (4.2.29)

2. Isotype n-n heterojunction The conduction mechanism fo an n-n heterojunction shown in Fig. 4.2.3a is governed by thermionic emission, very similar to the Schottky junction. We can thus derive [ ]J J J A T qV

kTqV

kTqV

kTb B= + = − − −− −2 1 1 2

2 2 2* exp( ) exp( ) exp( )B1 , (4.2.30)

where A* is the effective Richardson constant Substituting Eq. 4.2.18 to 4.2.30 yields the I-V relationship:

[ ]JqA TV

kVV

b qVkT

B

b

qVkT

b= − −*

exp( )( ) exp( )1 B −1 . (4.2.31)

Ch-4 23

Appendix I: Noise 1. Definition

The term “noise” refers to spontaneous fluctuations in the current passing through, or the voltage developed

across, semiconductor materials or devices. The noise is usually repented by the noise coefficient or the noise figure.

Noise coefficient: FP PP P

isignal

inoise

osignal noise=

( /( / 0

))

(I-1)

Noise figure: NF = 10 log(F) (dB). (I-2) Since the devices are mainly used to measure small physical quantities or to amplify small signals,

spontaneous fluctuations in current or voltage set a lower limit to the quantities to be measured or the signals to be amplified. It is thus important to know the factors contributing to these limits, to use this knowledge to optimize operating conditions, and to find new methods and new technologies to reduce noise.

2. Noise properties:

Observed noise is generally classified into flicker noise (the 1/f noise), shot noise, and thermal noise. Low frequency noise (< 106 Hz, for Si ∼< 103 Hz):

1/f noise (distinguished by its peculiar spectrial distribution, which is proportional to 1/fα with α generally close to unity)

Main physical cause: the carrier generation-recombination via (i) surface states, (ii) interface states (heterojunctions), or (iii) any defect induced trap states. Solution: reduce the density of defect states, surface passivation.

(e.g., The 1/f noise-power spectrum has been correlated both qualitatively and quantitatively with the loss part of MIS gate impedance due to carrier recombination at the interface trips)

White noise (frequency independent) :

Shot noise (constitutes the major noise in most semiconductor devices, and it is independent of frequency at low or intermediate frequencies)

Main physical cause: The current flowing in semiconductor devices is not smooth and continuous, but

rather it is the sum of pulse of current by flow of carriers, each carrying one electron. Solution: reduction of series resistance. Thermal noise (occurs in any conductor and semiconductor) Main physical cause: The carrier random thermal motions. Solution: low series resistance, high current gain. RF noise:

RF noise is composed mainly by thermal noise, but also shot noise. Since the current gain is decreasing at high frequencies, it apparently shows an increase of the noise figure with the frequency.

Ch-4 24

3. Noise spectrum: A typical noise spectrum is depicted in Fig. I-1, where fl and fh are called the corner frequency for the low or RF frequency region, respectively.

NF(dB)

1/f white noise RF noise 3db

fl fh f Fig. I.1 A typical noise spectrum