PHYSICAL PROCESSES at the METAL/SEMICONDUCTOR INTERFACE

Stefan ANTOHE

Introduction

The transport theory of semiconductors based on the band theory of solids which was first

formulated in 1931 by Wilson, was later applied to the metal/semiconductor contacts. In 1938

Schottky suggested that the potential barrier could arise from stable space charges in the

semiconductor without the presence of a chemical layer. The model arising from this

consideration is known as the Schottky barrier. At the same time Mott devised an appropriate

theoretical model for swept-out metal/semiconductor contacts that is known as the Mott barrier.

Because of their importance in direct current and microwave applications and as tools in

the analysis of some fundamental physical parameters, metal/semiconductor contacts have been

extensively studied.

1. Metal/Semiconductor Contacts (M/S)

The M/S contact is an important part of the electronic and optoelectronic devices, either

as an active element or as a charge carrier collector. The M/S interface processes and the

diffusion potential value at the M/S contact depend on two factors:

a) The difference between the metal work function M and the semiconductor work

function S;

b) The presence of the surface states in the semiconductor

These two factors act simultaneously, but for simplicity limiting cases involving one

factor or the other are analysed.

1.1. Metal / n- type Semiconductor contact, M > S.

The energy band diagram before and after contact is shown in figure 1, where ICn is the

semiconductor ionization energy; M is the metal work function; Sn is the semiconductor work

function; Sn is the electron affinity of the semiconductor (measured from the bottom of the

conduction band to the vacuum level); Eg is the energy gap and n is the semiconductor chemical

potential.

The semiconductor work function being lower than the metal work function, the electrons

flow from the semiconductor towards the metal generating a negative charge inside the metal and

a positive one in the semiconductor. As a result, an electric field appears at the interface M/S,

oriented from semiconductor towards metal. At equilibrium this field opposes to the diffusion of

the electrons into the metal. At the M/S interface, the energy bands bend up due to the depletion

of electrons in the semiconductor. The electrons flowing towards the metal must cross over a

barrier height bn. This “built-in barrier” related to the barrier potential is:

(1)

On the other side the barrier height for the electrons flowing from the metal into the

semiconductor is:

(2)

The metal-semiconductor electron barrier is higher than the semiconductor-metal one. Because

the charge carrier concentration in semiconductor is lower than the carrier concentration in metal,

the depletion layer w0 extends only inside of the semiconductor.

2

n

SnSn

ICnEg

M

EFM

Metal (M) Semiconductor n (Sn)

(a)

Fig. 1. Energy band diagram of M/Sn contact, before(a) and after contact (b)

w0

qVbn n

Sn

Sn

Ei

ICn

M

MS

Eg

SnM

(b)

At forward bias (+ on metal, - on semiconductor) the energy bands lifts by qV amount,

Fig. 2 (a). The electrons flowing from the semiconductor into the metal must surmount a lower

barrier q(Vbn – V) than the electrons crossing backwards (MS barrier is independent with respect

to the applied voltage). The high current through the structure is generated by the majority charge

carriers (in this case the electrons), crossing from the semiconductor into the metal.

At reverse bias the barrier for the electrons flowing from the semiconductor towards the

metal increases to q(Vbn + V), so the current through the structure is low, practically equal to the

current flowing from the metal into semiconductor. Fig.2. (b) . The M/Sn contact acts as a

blocking contact (rectifying contact).

1.2. Metal / n- type Semiconductor contact, M < S

In this situation the energy bands bend down because an increased number of electrons

cross from metal towards the semiconductor than backwards, Fig. 3. As a results an accumulation

electron layer appears at the M/S interface, the contact resistance is very low and do not change

when the bias changes, the contact behaves like an ohmic contact, being a very good electron

(majority charge carriers) injector into semiconductor.

3

qV

Eext

q(Vbn+V)qVbnMS

Ei

Sn

+M -

(b)

Eext

qV

q(Vbn-V)

qVbnMS

Ei

Sn

-M+

(a)

Fig. 2. Energy band diagrams for M/Sn contact under forward (a) and reverse bias (b)

At reverse bias (“+” on metal and “-”on semiconductor) the electronic current from the

semiconductor into metal is high because the electrons have no barrier to escalate, while the

backward hole current is very low due to the high barrier holes which must be surmounted when

they try to flow from metal towards the semiconductor.

At forward bias the electronic current is also high (the electrons must escalate a low

barrier when crossing from metal towards the semiconductor). This type of contact has no

rectifying properties, does not change the device impedance and has no influence on the

equilibrium charge carrier concentration.

1.3. Metal / p- type Semiconductor contact, M > Sp

In this case the electrons from the semiconductor spread into metal, the energy bands

bend up and an electron depleted layer appears, to the M/S interface, Fig. 4.

At forward bias (“-” on p semiconductor and “+” on metal) the holes are easily injected

from metal into the semiconductor, the main current being a hole current (holes are the majority

carriers). The metal cannot inject electrons due to the high barrier which they must surmount

MSp = M – Sp. Although the electrons crossing backwards must escalate a relatively low

barrier bp = M – Sp = qVbp, they are in small number, being the minority charge carriers for

the p-type semiconductor. This type of contact is an ohmic contact (it has no rectifying properties

and does not allow the minority electron injection).

4

Fig. 3. Band diagram for M/Sn contact when M < Sn

Ei

Eg

EV

EF

EC

M

SM

Sn

S

M

EV

EC

EF

Eg

M

1.4. Metal / p- type Semiconductor contact, M < Sp

In this situation the energy bands bend down, the holes flow from the semiconductor

towards the metal and therefore a holes depletion layer appears. An internal electric field from M

to Sp is created Fig. 5. The holes still crossing from Sp into M must escalate the bp = Sp – M =

qVbp barrier and the holes flowing backwards, from M into Sp have to surmount the: MSp = IG –

M = Eg + p – M = Eg – (M – p) barrier, while the barrier for metal towards semiconductor

crossing electrons is (M – p). The sum of these barriers is Eg.

(3)

The equation (3) shows that the barrier viewed by holes from metal towards the

semiconductor is higher than that viewed by them from semiconductor towards the metal.

At forward bias (“+” on semiconductor and “-” on metal), the barrier for the holes

crossing from Sp into M decreases to q(Vbp – V), so the current through the structure is a majority

hole current.

At reverse bias the barrier for the holes crossing from Sp into M increases to q(Vbp + V),

while the barrier for the holes flowing backwards is not changed. The hole current will be quite

low, actually negligible, as well as the minority electronic current. This is a blocking contact

(rectifying contact).

5

Fig. 4. Band diagram for M/Sp contact when M > Sp

IC

Spp

S

M

EV

EC

EF

Eg

M

qVbp

Eg

EV

EF

ECMSp

SM

++

Ei

Fig. 5. M/Sp Contact, when M < Sp

S

IC

Spp

MEC

Eg

MEV

EF

p

qVbp

p

Sp

Ei

Eg

EV

EF

EC

MSp

SM

++

2. Depletion layer theory

The charge carrier concentration inside the semiconductor is low with respect to the

metal, therefore the depletion layer extends over a w0 distance only in the semiconductor. Inside

this layer the diffusion potential follows the Poisson equation. Its solution gives the electric field

and potential dependence on distance, over the depletion layer. The Poisson one-dimensional

equation is:

(4)

Inside the neutral region (x > w0), = 0, E = -dV/dx = 0 and V = 0, and inside the depletion

region (x < w0), = qND. The integration of equation (4), once and twice between the x = 0 and x

= w0 limits, leads to the following relations for the electric field and the potential over the

depletion layer, respectively:

(5)

(6)

6

w0

w0

Em

Ex

Vbn

Vx

Fig. 6. Electric field and potential dependence on distance along the depletion layer.

The plots, for equations (5) and (6) are shown in Fig. 6. The maximum field strength and

maximum potential value occur at x = 0.

The depletion region thickness w0 is related to the equilibrium barrier potential (Vbn = V(x) at

x=0) as it follows:

(7)

The space charge inside the depletion layer is:

(8)

As a result the depletion layer equilibrium capacitance will be:

(9)

At reverse or forward biasing, the Vbn value for potential is replaced by Vbn ± V, so relations (7)

and (9) change according to:

(10)

The expression for the depletion layer capacitance, suggests the possible method for the

evaluation of the doping profile and the diffusion potential using the capacitance – voltage

characteristics (C-V). Equation (10) can be also written in the form:

(11)

7

Plotting 1/Cb2 = f(V) at reverse bias, a straight line is obtained and ND can be determined from the

graph slope (if ND is constant throughout the depletion region). From the graph intersection with

the voltage axis Vbn and then w0 could be also determined. At equilibrium, through the depletion

region there are a drift current and a diffusion current flowing in opposite directions, therefore

there is no total current through the structure. At contact biasing, the flowing current depends on

the applied voltage.

3. Models of Charge Carrier Transport Theory at the Metal/Semiconductor Contact

The charge carrier transport in Metal/Semiconductor barriers is mainly due to majority charge

carriers, in contrast to p-n junctions, where the minority charge carriers are responsible. To

establish the I-V characteristics there are three different models:

1) The simple isothermal thermionic emission theory by Bethe;

2) The simple isothermal diffusion theory by Schottky;

3) The mixed thermionic emission and diffusion theory by Crowell and Sze.

3.1. The simple isothermal thermionic emission theory by Bethe

Bethe devised this theory using the following assumptions:

a) The barrier height qbn is much larger than kT;

b) Carrier collisions within the depletion region are neglected;

c) The effect of image force is also neglected.

Because of these assumptions the shape of the barrier profile has no influence on the

electric current flow. At equilibrium the current flowing from the semiconductor to the metal is

equal to the current flowing backwards (JSM = JMS), so there is no current through the structure

(the Fermi level is the same both in the semiconductor and in the metal)

(12)

Here, A* is the Richardson constant for the semiconductor: A* = 4qm*k2/h3

8

For a n-type semiconductor, when M/Sn structure is forward biased, the Fermi level of the

semiconductor rises an amount equal to qV, Fig. 2, the barrier for the electrons crossing from Sn

into M becomes q(Vbn – V) and the current through the structure is J = JSM – JMS. The barrier for

the electrons crossing from M into Sn (which is called Schottky barrier) is not changed, while the

current from the semiconductor towards the metal becomes:

(13)

The total current crossing the structure will be:

(14)

Using the relations (15), (16) representing the average value of thermal velocity of

electrons and their concentration at the surface of the semiconductor:

(15)

(16)

the saturation current density (after replacing the Richardson constant) becomes:

(17)

This relation shows the presence of a majority carrier current.

Bethe’s theory is applicable for semiconductors having high carrier concentrations and

narrow energy gap (Ge, InSb, GaAs, Si, etc.).

At forward bias (V > 0, |V| > kT/q) the current grows exponentially, and at reverse bias (V

< 0, |V| > kT/q) the current is low and practically constant, being the saturation current for a

Schottky diode.

9

Schottky Effect

When electrons are thermionic emitted from metal into vacuum, they may form a space

charge not so far from the metal surface, thus an image interaction force will appear (given by

Coulomb law) between the emitted electrons and their images in the metal. The potential energy

of the electron in the image force field is:

(18)

The potential energy of the electron in the vicinity of the metal surface will be:

(19)

When x → ∞, W → W0 (electron energy in vacuum), therefore the work function becomes:

(20)

When an external electric field is applied, due to the image force, the effective Schottky barrier is

lowered by (Schottky effect, fig.7). The potential energy of the electron both in the external

field and in the image force field is given by:

(21)

The applied electric field rejects the electron from the metal, but the image force attracts the

electron to the metal, therefore at a certain distance xmax these effects will compensate each other,

the resulting force will be zero and the total potential energy will be maximum Wmax. The location

of the maximum is determined as follows:

(22)

(23)

In the presence of the external electric field the effective work function is reduced to:

10

(24)

As a result of the external electric field, the work function is lowered by the amount:

(25)

Regarding to the Metal/Semiconductor contact the (20) – (25) relations are still valid (the contact

field being considered the external field), but additionally the dielectric constant of the

semiconductor r must be considered.

11

2x

electronImage particle

(a)

Wmax

-q2/160x

rmax

W0

r0 x

W(x)

EF

(c)

x r0

W(x)

W0

M

EF

-q2/160x

(b)

Fig. 7. Diagrams showing Schottky effect

xmax

’MS

MS

EF

EV

EC

SnM

Fig. 8. Schottky effect at M/Sn interface

Therefore the Schottky barrier, Fig. 8, will be:

(26)

Assuming the Schottky effect within the isothermal thermionic emission theory, the

saturation current density becomes:

(27)

This relation shows that under high reverse biasing, the current grows with the applied voltage.

The slope of the line ln(J’S) = f(V1/2) enables the determination of the depletion layer thickness w,

if the Schottky coefficient S is known (or r for the semiconductor). Using the crossection of this

line with the V = 0 axis, the barrier height could be obtained taking for Richardson’s constant the

value 120 A/cm2 K2.

This is OK, in the case of thermionic emission in vacuum, but in the case of M/Semiconductor

contact this is a big mistake, due to the fact that A*=120 x m*/m A/cm2 K2. Unfortunately, many

times appear this mistake in the papers which were published in prestigious journal ( see:Fu-

Ren Fan and Lary R. Faulkner, J. Chem. Phys. 69(7), 3334-3339, 1978; S. B. JUNG, S. Y.

YOO, Y. S. KWON, E. PARK and C. KIM, Journ. Of Korean Physical Society, 40(1) 132-

135, 2002)

To avoid this wrong approach, a family of I-V reverse characteristics for different temperatures

must be draw. In this case, using the ln(JS/T2) = f(103/T) plot, the height of the Schottky barrier

MS, could be obtained from the slope of the straight line, and Richardson constant A* from the

intersection with 103/T axis, and then the effective mass m* for the charge carriers in the

semiconductor could be obtained.

12

Such a procedure was used by us every

time to characterize different electronic and optoelectronic devices containing a blocking M/S

contact, [3 – 10].

For example the reverse characteristic of ITO/TPyP/Al, at different temperatures ranging from

295 to 360 K are shown in Fig. 9. The current levels are lower than the forward bias and show a

linear dependence of ln Js vs. U1/2, according with equation (27). This behaviour may be

interpreted either in terms of the Schottky effect ( field-lowering of the interfacial barrier at the

blocking electrode) or the Poole-Frenkel effect (field-assisted thermal detrapping of carriers).

The J-U characteristics for these processes are [11]:

(i)

for the Schottky effect and

(ii) for the Pool-Frenkel effect,

Fig. 9. Reverse bias characteristics at different temperatures ranging from 295 to 360 KS. Antohe, Phys. Stat. Sol. (a) 136, 401-410, (1993)

13

where: A* is the Richardson constant, Ф the Schottky barrier height at the blocking electrode, s

the Schottky coefficient, PF the Pool-Frenkel coefficient, and w is the width of the depletion

region. Theoretical values of these coefficients are related by the expression : s = PF = (q3/

)1/2. For = 1.77x10-11 F/m the following values were obtained, respectively : 2.68x10-5 (Vm)1/2.

From the slope of the straight line 1n Js versus U1/2, at room temperature (Fig. 9) for d = 2x10-

5cm, an experimental value of 2.23x10-5 (Vm)1/2

was obtained. This value is close to the theoretical

value of the Schottky coefficient leading to the

conclusion that the Schottky effect is dominant and

most of the applied voltage was dropped across the

Schottky depletion region. Following this

interpretation the above data yield values of w

varying between 193 nm for T = 295 K and 50nm

for T = 360K. It thus appears that annealing reduces

the thickness of the depletion region at the

ITO/TPyP interface. This feature is consistent with

the theory of the depletion layer in a Schottky

barrier diode. On the other hand, in accord with (27)

or (i) , plotting 1n(Jso/T2) versus 1/T a straight line

was obtained (Fig. 10) and from the slope = 0.68

eV resulted. Jso is the intercept of a line ln J versus

U1/2 (Fig. 9) with the current axis.

These results obtained from the analysis of dark electrical characteristics lead to the conclusion

that the TPyP is an n-type organic semiconductor which form an Ohmic contact with Al (metal

with low work function) and a blocking contact with ITO (electrode with high work function of -

4.70 eV [12]). The conduction- and valence-band edges are bending upwards at the interface

ITO/TPyP under short-circuit conditions. The energy diagram proposed for the ITO/TPyP/Al

structure is presented in Fig. 11, where the TPyP layer is assumed to be an extrinsic n-type

Fig. 10. Dependence of ln(JS0/T2) on reciprocal temperature 103/T for the reverse biased ITO/TPyP(200nm)/Al cellS. Antohe, Phys. Stat. Sol. (a) 136, 401-410, (1993)

14

organic semiconductor with the energy levels estimated from the result previously described and

those reported in [12].

3.2. The simple isothermal diffusion theory by Schottky

This theory is appropriate when calculating I-V characteristics for the rectifying M/S

contacts, meant for semiconductors with low carrier concentrations and motilities (i.e. AlSb,

Cu2O, some organic semiconductors). The theory uses the following assumptions:

a) The barrier height at the M/S contact is higher than kT at room temperature;

b) Since the mean free path for the charge carriers is shorter than the depletion region thickness

w ( < w), the effect of electron collisions within the depletion region is included;

c) The carrier concentrations at x = 0 and x = w, are unaffected by the current flow (they have

constant values equal to their equilibrium values);

d) The impurity concentration of the semiconductor is nondegenerate.

Thus the current density has a drift and a diffusion component. For the one-dimensional

case the total current density is:

(28)

Using the relationship between the field and the potential and also Einstein’s relation, (28)

becomes:

Fig.11 Schematic representation of energy bands at the ITO/TPyP (200 nm)/Al cell. is the barrier height at the ITO/TPyP interfaceS. Antohe, Phys. Stat. Sol. (a) 136, 401-410, (1993)

15

(29)

Under the steady-state condition, (the current density J is independent of x within the depletion

region), equation (29) can be integrated using “exp[-qV(x)/kT]” as an integrating factor, as

follows:

Thus, the current density within the depletion region is given by:

(30)

Considering the edge of the conduction band as a reference level for measuring the potential

energy, the boundary conditions (31), Fig. 12, are obtained.

(31)

16

n

MS

n0

q(Vbn-V)

x=0 w

MSn

x

MS

x=0 x w

V(x)

n

qVbn

M Sn

n0

qV

Ei

Fig. 12. Diagrams showing the boundary conditions for the diffusion theory

These boundary conditions take in account weak external fields (at low injection level), so (Vbn –

V) > kT/q (the depletion layer is present), meaning that n(0) is independent of the applied voltage

V considered to be low. Moreover, the most significant contribution in calculating the integral

from the following relation, belongs to the region from the close vicinity of x = 0, where V(x) is

maximum and where the function “exp(-qV(x)/kT)” changes quickly, because for x > 0, V(x)

lowers rapidly Fig. 6, while the term “dV/dx” changes very slowly as compare with the

exponential term. Thus, nearby the M/S interface, ES-1= (dV/dx)-1 comes out as a common factor

in front of the integral at denominator.

Finally, neglecting the unit in the denominator of the last term as compares to the exponential

term (from the condition (Vbn – V) > kT/q), the current density becomes:

(32)

It is obvious that unlike the thermionic emission theory, in the diffusion theory the reverse

current JS depends on the applied voltage, because of the field inside the contact region. If biasing

the structure, the depletion layer thickness and further the contact field depend on the voltage as it

follows:

(33)

Thus the current density can be written as (34), where the JS ~ V1/2 dependence is obvious.

(34)

Furthermore, the plot of (JS)2=f(V), under reverse biasing, allows the determination of the doping

profile ND and the built-in potential Vbn. The analysis of the I-V characteristics family for

different temperatures enables the determination of the Schottky barrier height.

17

3.3. The thermionic emission-diffusion theory by Crowell and Sze

This theory is a synthesis of the thermionic emission and diffusion approaches. The

previous theories neglected the effect of the surface states, image force and other factors such as

electron-phonon scattering or the quantum-reflection and quantum tunnelling at the M/S

interface.

This approach, combining the thermionic emission and diffusion mechanisms, will

incorporate these effects and will settle on what conditions one of the two is dominant.

Including the image force effect, the barrier height at the M/S interface becomes

, and reaches the maximum value at xm distance from the metal.

18

Fig. 13. Potential energy over the depletion layer

Within the depletion region (0 and w), the potential energy of the electron, due to the applied

voltage is qV(x). The drift component of the current density in this region is: .

In the above statements V(x)

is the potential and is the electric field in the depletion region (caused by the

applied voltage on the structure), therefore the current density could be rewritten as:

(35)

The density of electrons at x distance from the metal is:

(36)

It is assumed that within the depletion region, between xm and w, the temperature is

constant. Under these circumstances the equations (35) and (36) are valid only between xm and w.

Between x = 0 and x = xm they do not apply since there the potential energy of electron

changes rapidly in distances comparable to the electron mean free path. In this region the

distribution of electrons cannot be described by a function nor be associated with

an effective density of states in the Conduction Band (CB).

Because of the surface states present here the depletion region between x = 0 and x = xm acts like

a sink for electrons (or a potential hole). On these conditions the current density at x = xm (at the

maximum of the potential energy) can be written in terms of an effective recombination velocity

vR:

(37)

where nm is the electron density for x=xm at nonequilibrium (under the applied voltage V, that

causes the current flow through the structure)

(38)

and n0 is the electron density for x=xm at equilibrium, without disturbing the location of the

potential energy maximum or its height.

19

(39)

Regarding V(x):

Thus the current density at x=xm will be:

(40)

This relation stands as a boundary condition when solving equation (35) as follows:

(41)

Replacing from equation (41) in equation (40) leads to:

(42)

20

If vD is the diffusion velocity, expressed by (43)

(43)

then the current density becomes:

(44)

The effective diffusion velocity vD is associated with the transport of electrons from the edge of

the depletion layer w to xm where the potential energy maximum is located. The I – V

characteristic (equation 44) expresses the current density versus voltage, for a Schottky diode, as

a result of the two combined effects.

At the limit it will change in one form or another (thermionic emission or diffusion).

In fact:

1) If electrons between (xm, w) have a Maxwell distribution and if there are no reflected

electrons at the M/S interface (electrons are not crossing backwards from the metal into

semiconductor), the semiconductor acts as a thermionic emitter and then:

where is the Richardson constant.

Example: For n-type GaAs with relatively high mobility, the effective recombination velocity is

vR =107cm/s at T = 300K and this is obvious smaller than vD

Because , considering w ≈ 60Å , and

, then:

21

but , so can be neglected in equation (44) which becomes:

and

(45)

which is Bethe’s thermionic emission outcome

2) If then

(46)

which is the diffusion theory outcome.

In summary equation (44) gives a result which is a synthesis of Schottky’s diffusion

theory and Bethe’s thermionic emission theory and which predicts currents in essential agreement

with the thermionic emission theory if . This criterion is more rigorous than Bethe’s

condition where is the carrier mean free path.

22

111 2*2*

''

kT

qV

kTkT

qV

kT

cc

kT

qV

kTRc eeTAee

qN

TAqNeevqNj

msmsms

12* kT

qV

kTT eeTAj

ms

In these circumstances it can be indicated an inferior limit for the field strength the

thermionic emission starts from, and a superior limit for the field strength to which this

mechanism is valid. For GaAs these limits are and .

References[1]. S. M. Sze, Physics of Semiconductor Devices; Wiley-Interscience, New York. London. Sydney. Toronto (1967) [2]. Roderick, Metal Semiconductor Contact;[3]. S. Antohe, "Organic Materials and Electronic Devices", Pgs. 1-227, Ed. University of Bucharest (1996), ISBN 973-575-102-X[4]. S. Antohe and A. Vonsovici, Phys. Stat. Sol. (a) 124, 583, (1991)[5]. S. Antohe, N. Tomozeiu and S. Gogonea, Phys. Stat. Sol. (a) 125, 397, (1991)[6]. S. Antohe, I Munteanu, I. Dima, Rev. Roum. Phys., 34 (6), 665-671, (1989)[7]. S. Antohe, Phys. Stat. Sol. (a) 136, 401-410, (1993)[9]. S. Antohe, Romanian Reports in Physics, 47 (2), 211-228, (1995)[10] S. Antohe and V. Ruxandra, Romanian Reports in Physics, 46, (7-8), 703-710, (1994)[11]. J. G. Simmons, J. Phys. D 4, 613 (1971)[12] K. Yamashita, Y. Harima and Y. Matsumura, Bull. Chem. Soc. Japan 58, 1761 (1985) [13] Fu-Ren Fan and Larry R. Faulkner, J. Chem. Phys. 69(7), 3334, 1978[14] S. B. Yung, S. Y. Yoo, Y. S. Kwon, E. Park and C. Kim, Journal of the Korean Physical Society, 40(1) 132-135, 2002

23