JAVA APPLETS FOR PHYSICAL ELECTRONICS Final Report Submitted to Dr.Stephen Saddow ECE 6243:Physical Electronics Department of Electrical and Computer Engineering Mississippi State University Mississippi State, MS-39762

November 24,2000

Metal

Semiconductor

Metal

Semiconductor

Depletion Region Submitted by Ramya Chandrasekaran Pujita Pinnamaneni Beta Team ECE 6243:Physical Electronics Department of Electrical and Computer Engineering Mississippi State University Mississippi State, MS-39762

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ABSTRACT

We present here two computer programs, one that simulates the working of a schottky diodemetal-semiconductor junction and the other that graphically depicts the variation of concentration with temperature of a PN junction diode.

This program enables the students to create a scho ttky diode with a suitable metal and a semiconductor with appropriate doping. While creating such diodes the students can observe the changes in the energy bands and fermi level as a response of doping. The students can also determine the direction of current flow in the schottky diode for different metal-semiconductor junctions. The second program is programmed so as to realize the variations of concentration with temperature for the PN junctions of various semiconductor materials.The flexibility of the program allows it to be used by students over a wide range of academic level.

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INTRODUCTION The “Java applets for Physical Electronics” project is mainly designed to help students understand some of the concepts of Physical Electronics in a more efficient way. The primary audiences for this material are undergraduate and the graduate students who need to understand the behavior of various metal-semiconductor junctions and to study the variations of concentrations with temperature.

Since Internet has the potential to assist the educational process in a more efficient way than any other means, we have designed the project using Java programming. Java applets can be easily uploaded and is easily accessible. With the help of this project, students can effectively analyze the behavior of an Schottky diode. They can also understand the energy band diagrams and the fermi level changes for different concentration levels. Students can also predict the performance of the diodes for various combinations of metals and semiconductors. Here we have taken into account three semiconductors Si, SiC and GaAs and four metals Al, Cr, Mo and Pt. With the help of the band bending in the energy band diagram, one can realize the direction of electron flow and hence the direction of current.Our project also effectively depicts the variation of free carrier concentration with temperature for different semiconductors such as Si, SiC, and GaAs.

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ANALYSIS Semiconductors have a peculiar behavior with varying temperatures compared to conductors and semiconductors. Semiconductors can act as both conductors and insulators at different temperatures. Semiconductors are classified into two types based on their impurity level. They are intrinsic or pure semiconductors and extrinsic.

Intrinsic Semiconductors A perfect semiconductor crystal with no impurity or lattice defects is called an intrinsic semiconductor. In such materials there are no charge carriers at 0°K,since the valence band is filled with electrons and the conduction band is empty. At higher temperatures, the valence band electrons are excited thermally across the band gap to the conduction band producing electronhole pairs. These electron hole pairs, which are thermally generated, are the only charge carriers in an intrinsic material. Since the electrons and holes are created in pairs the conduction band electron concentration is given by n(electron/cm3 ) is equal to the concentration of holes in the valence band p(hole/cm3 ) n=p=ni (intrinsic concentration)

If a steady state concentration is maintained, the rate of generation of electron-hole pairs is equal to the rate of recombination. Recombination occurs when an electrons in the conduction band makes a transition direct or indirect to an empty state (hole) in the valence band, thus annihilating the pair. Generation rate of electron-hole pair =gi

Recombination rate = ri At equilibrium, gi = ri

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These rates are temperature dependent. At any temperature these rate of recombination of electrons and holes ri

is

proportional to the equilibrium concentration of electrons no and

concentration of holes po. So ri = α rno po = α rni = gi α r is the constant of proportionality which depends on the mechanism by which recombination takes place.

Extrinsic Semiconductors In addition to the intrinsic carrier generated thermally its possible to create carrier in semiconductors by properly introducing impurities into the crystal. This process is called doping and is the most common technique for much conductivity in semiconductors. There are two types of doped semiconductors N-type (mostly electrons) and P-type (Mostly holes). When a crystal is doped such that the equilibrium carrier concentration no and po are different from the intrinsic carrier concentration ni, the material is called extrinsic. A pure semiconductor behaves as an insulator at low temperature. With increase in temperature the conductivity Ec Ed

Ev

Ec Ed

Ev 1.1 Donation of electrons from a donor level to conduction band

of the semiconductor increases as the covalent bonds are broken in a process called thermal generation.

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The band diagrams shown above and below show the behavior of dopants. Fig 1.1 shows the situation where the energy level of the donor impurity Ed is very close to the conduction-band Ec.If the temperature is very low the impurity atoms are ionized. However, if the thermal energy exceeds the small ionization energy of the donor atoms, electrons will be removed from the donor atoms. The resulting electrons will move into the conduction band. The concentration of these electrons depends on the concentration of the donor impurity. Therefore, the carrier concentration can be varied in the controlled way by varying the concentration of the donors.

The situation regarding acceptors can be analyzed in a similar way. In this case the acceptor level Ea due to the impurity is close to the valence –band Ev. At very low temperatures, electrons from the filled states in the valence band cannot access the impurity level. However, at higher temperatures this transition happens, leading to the formation of holes in the valence band.

In addition to the doping- induced carriers, thermally generated electron hole pairs are always present in the extrinsic semiconductor. The concentration of these electron hole pairs is extremely low, the ramification being that a small number of the holes will coexist with the electrons in an n-type semiconductor, or vice versa. Electrons and holes in an n-type semiconductor are referred to as majority and minority carriers.

In the case of elemental semiconductors, i.e. Si several impurities such as As, Sb and P serve as n-type dopants, whereas p-type dopants are B, Al, Ga, In and so on.the behavior of these impurities may be qualitatively understood in terms of bonding. Both Si, Ge are covalent bonded solids. Each Si or Ge atom has 4- valance electron and is tetrahedrally coordinated to 4 other atoms. When a group five donor atom replaces the host lattice atom one of the impurity electron is not involved in bonding. This extra electron is donated to the conduction band. On the other hand, when the impurity has three valence electrons, it can tetrahedrally coordinate with lattice atoms by pulling an electron from the top of the valence band. This produces a hole in the valence band.

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The donor and acceptor levels in a semiconductor are generally very shallow. The column V donor levels lie approx. .01 eV below the conduction band and group III acceptor levels lie about 0.02 to 0.04 eV above the valence band. Ec

Ec

Ea

Ea

Ev

Ev 1.2 Acceptance of valence band electrons by an acceptor level, the resulting creation of holes

Fermi Level Electrons in solids obey Fermi-Dirac statistics. The distribution of electrons over a range of allowed energy levels at thermal equilibrium is given by, F (E) = 1/[1+e (E- Ef)/kt]

………(1)

Where k is the Boltzmann’s constant. The function f (E), the fermi- Dirac distribution function, gives the probability that an electron at absolute temperature T will occupy an available energy at E. The quantity Ef is called the fermi level, and it represents an important quantity in the analysis of semiconductor behavior. For an energy E equal to the fermi level energy Ef , the occupation probability is F(Ef) = 1/[1+e(Ef-Ef)/kt ] ……………(2) = 1/[1+1] = ½

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Thus the energy state at the fermi level has the probability of ½ of being occupied by an electron.

For intrinsic material the concentration of holes in the valence band is equal to the concentration of electrons in the conduction band. Therefore, the fermi level Ef must lie at the middle of the band gap in the intrinsic material. In n-type material there is a high concentration of electrons in the conduction band compared with the hole concentration in the valence band. Thus in an n-type material, the distribution function F (E) must lie above its intrinsic position. Correspondingly the fermi level lies near the valence band for a p-type material.

Electron and Hole Concentration at Equilibrium The fermi distribution is very useful in calculating the concentration of electrons and holes in a semiconductor, provided the densities of the available states in the valence and conduction band are known. For example, the concentration of electrons in the conduction band is ∝

ηo = ∫ f(E)N(E) dE

………….(3)

Ec

where N (E) is the density of states in the energy range state dE. The number of electrons per unit volume in the energy range dE is the product of the density of states and the probability of occupancy f (E). Thus the total electron concentration is the integral over the entire conduction band. The density of states in the conduction band increases with the electron energy. On the other hand, the fermi function becomes very small for large energies. Therefore the product f (E) N (E) decreases rapidly above Ec and only few electrons occupy energy states far above the conduction band. Similarly the probability of the empty state (hole) in the valence band (1 – f (E)) falls down rapidly below the Ev and most holes occupy the states near the top of the valence band.

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As result of integration in eq (1) is the same as that obtained if we represent all of the distributed electrons states in the conduction band by an effective density of states Nc located at the conduction band Ec. Hence the conduction band electron concentration is given by the effective density of states at Ec times the probability of occupancy at Ec. ηo = Ncf(Ec). ………………(4)

Here we assume that the fermi level Ef lies at least several kt below the conduction band. This exponential term is greater than unity and the fermi function f (Ec) is given by F (Ec) = 1/[1+e(Ec- Ef) / kt ] ≈ e- (Ec- Ef)/kt …………….(5) For this condition the concentration of electrons in the conduction band is no = Nce- (Ec- Ef) / kt ……………….(6) The effective density of states Nc is shown in Appendix IV to be Nc = 2((Πmn *kt)/h2 )3/2 …………………….(7)

Since the quantities in the above equation are known, values of Nc can be tabulated as a function of temperature. As equation (6) indicates, the electron concentration increases as EF moves closer to the conduction band.

By similar arguments, the concentration of holes in the valence band is

p0 = Nv [1 – f(Ev )]

………………………….. (8)

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Where Nv is the effective density of states in the valence band. The probability of finding an empty state at Ev is 1 – f (Ev ) = 1 –(1/(1+e(Ev -EF)/kt ) ≈ e-(EF – Ev )/kt

………..(9)

for EF larger than Ev by several kt. From these equations, the concentration of holes in the valence band is p0 = Nv e-(EF – Ev )/kt

……………………….. (10)

The effective density of states in the valence band reduced to the band edge is Nv = 2((2Πmp *kt)/h2 )3/2

………………(11)

As expected hole concentration increases as EF moves closer to the valence band. The electron and hole concentrations predicted by equation (6) and (10) are valid whether the material is intrinsic or doped, provided thermal equilibrium is maintained. Thus for intrinsic material, EF lies at some intrinsic level Ei near the middle of the band gap, and the intrinsic electron and hole concentrations are ni=Nce-(Ec- Ei)/kt ,

pi = Nv e-(Ei – Ev )/kt …………(12)

The product of n0 and p0 at equilibrium is a constant for a particular material and temperature, even if the doping is varied: n0 p0 = (N ce-(Ec- EF)/kt )(N v e-(EF- Ev )/kt ) = NcNv e-(Ec- Ev )/kt = NcNv e-Eg/kt ………….(13) nipi = (N ce-(Ec- Ei)/kt )(Nv e-(Ei- Ev )/kt ) = NcNv e-Eg/ kt …………….(14)

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The intrinsic electron and hole concentrations are equal (since the carriers are created in pairs), ni = pi ; thus the intrinsic concentration is ni = √NcNv e-Eg/

2kt

………………………………….(15)

The constant product of electron and hole concentration in equations (13) and (14) can be written conveniently as n0 p0 = ………………….(16) ni 2

This is an important relation, and we shall use it extensively in later calculations. The intrinsic concentration for Si at room temperature is approximately ni = 1.5 * 1010 cm-3 . Comparing equations (12) and (15), we note that the intrinsic level Ei is the middle of the band gap (Ec – Ei = Eg / 2), if the effective densities of states Nc and Nv are equal. There is usually some difference in effective mass for electrons and holes, however, and Nc and Nv are slightly different as equations (7) and (11) indicate. The intrinsic level Ei is displaced from the middle of the band gap, more for GaAs than for Ge or Si.Another convenient way of writing equations (6) and (10) is

n0 = ni e (EF – Ei ) /kt

……………………….. (17)

…………………… ….(18)

p0 = ni e (Ei – EF) / kt

obtained by the application of equation (12). This form of the equations indicates directly that the electron concentration is ni when EF is at the intrinsic level Ei, and that n0 increases exponentially as the Fermi level moves away from Ei toward the conduction band. Similarly, the hole

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concentration p0 varies from ni to larger values as EF moves from Ei toward the valence band. Since these equations reveal the qualitative features of carrier concentration so directly, they are particularly convenient to remember.

Temperature Vs Concentration We know that carrier concentrations depend on temperature. These dependences are described in the graph plot shown below. Fig 1.3

n/ND

Freeze out Region

Extrinsic region

Intrinsic region ni/N D

T (K) In the plot, there are three distinct regions present (1) carrier freeze-out, (2) intrinsic region and (3) extrinsic. The freeze-out region characterizes unionized dopants. The extrinsic region represents ionized dopants and control conductivity. The intrinsic region represents the broken covalent bonds, which exceeds doping, where the material becomes electrically neutral. The two breakdown temperatures in the plot are because of higher temperature, the concentration of covalent bonds exceeds the doping density as a result of which the net carrier concentration increases. Thus the material looks intrinsic. The lower end of the curve is due to the finite energy required for the dopants to give up electrons or acceptors to take in electrons which leads to the presence of extra electron (donor) which are still loosely bound to the parent even when the dopant atom has satisfied the covalent condition.

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METAL – SEMICONDUCTOR JUNCTIONS Schottky diodes are the rectifying metal semiconductor junctions. An appropriate metal semiconductor contact achieves many of the useful properties of PN junction diodes. This metal semiconductor contact is attractive because of its simplicity in fabrication. The role of Schottky diodes is firmly established in the semiconductor technology because they perform functions that no other junctions can accomplish. Ohmic contacts are used in a very large number of devices such as Field Effect Transistors, Bipolar junction transistors, Photo detectors, Light emitting diodes, Double heterostructure lasers, solar cells and mainly used when high speed rectification is required. These are attractive because of their high breakdown field, wide band gap, much high temperature operations, and low power dissipation.

An Schottky diode is a special type of diode with a very low forward voltage drop. When current flows through a diode, it has some internal resistance to that current flow, which causes a small voltage drop across the diode terminals. In a normal diode the voltage drop is between 0.7 – 1.7 volts, while in a schottky diode it is approximately between 0.15 - 0.45 volts. This lower voltage drop results in higher system efficiency.

A Schottky barrier diode, being a majority carrier device, does not have the minority carrier storage effect and therefore is ideal for high frequency applications due to fast switching speed and low switching power dissipation. In Schottky barrier junctions the carriers for the electrical conduction are the minority carriers of the semiconductor. Since electrons have higher mobility, the n-type Schottky barrier contact is more common than p-type schottky barrier junction.

A metal – semiconductor junction can be formed with either n-type or p-type semiconductor material. The energy band diagram of a metal deposited on a n-type semiconductor is shown in figure 1.4

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Metal

Semiconductor + + +

Surface charge

n-type

W Depletion Width Metal

Semiconductor

Vaccum Level qVi= q(φ M - φ s )

qφ S qφ M

qqx ECB(n) Ef(n)

qφ M qVi

qφ B

EVB(n)

qx

W

Ef(M)

(a)

(b) Fig 1.4

(a) Two materials isolated from each other (b) At thermal equilibrium after the contact is made.

Here qφ m and qφ s refers to the work functions of the metal and semiconductor respectively and these represent the energy required to remove electrons at the respective fermi levels to the vacuum level. If we assume the φ m is greater than φ s then the fermi level in the metal is at a lower position than that in the semiconductor. qχ is the electron affinity of the semiconductor and is the energy difference between the conduction edge and vacuum level. When a junction is formed between a metal and a n-type semiconductor, the electron from the conduction band state from the semiconductor flow into the metal because they have a higher energy. The electrons flow across until the fermi level is aligned. The electrons that flow from the semiconductor to the metal leaves behind positively charged donor ions in a thickness W in the semiconductor. This thickness W is called the depletion region. As a result of this electron flow, the surface 14

region near the fermi level moves deeper into the semiconductor, leading to upward bending of energy bands. Because of the electron transfer from semiconductor to metal, the negative charge develops on the metal side and is contained within an atomic distance from the surface. The presence of these two types of charges on either side of the junction establishes an electric field that is directed from semiconductor to metal. The equilibrium contact potential of the junction Vo prevents further flow of electrons from the semiconductor into the metal. This represents the difference between the work function potentials φ m and φ s . The barrier height qφ b for the injection of electrons from the metal to the semiconductor is given by qφ b = - qχ.This type of metal – semiconductor junction is called schottky barrier. Metal

Semiconductor + + +

Metal φ M< φ S

_ _ _

p_type

Semiconductor

qx qφ M

ECB(p)

qφ S ECV(p)

Ef(M) Ef(p) EVB(p)

Ef(p) EVB(p)

q(φ S - φ M) = qVo

p-type

W

(a)

(b)

fig 1.5 (a) Two materials isolated from each other (b) At thermal equilibrium after the contact is made.

When a metal is deposited on a p-type semiconductor also a Schottky barrier is formed. Here we assume that φ m < φ s. This is illustrated in fig 1.5. Here in order to align the fermi level, the electrons must flow from the metal to the semiconductor, which results in a positive surface

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charge in the metal and a negative charge in the semiconductor. The negative charge exits within a depletion region W in which acceptor ions are left uncompensated by the holes. The potential barrier Vo regarding the diffusion of holes from the semiconductor to the metal is φ s- φ m .The Schottky barrier heights qφ b for different metals should be different because of the changes in the metal work function qφ m.. This ideal situation is not always realized. Very often the Schottky barrier heights for various metals on a particular semiconductor is the same. This behavior is attributed to the presence of a large number of interface states in the band gap of the surface region of the semiconductors. These states arise from the surface dangling bonds and impurities. As a result, the addition or removal of electrons from the semiconductor does not alter the position of fermi level at the surface and the fermi level is said to be pinned. At the equilibrium, the electrons with greater energies than qVo thermionically emitted over the barrier into the metal. Alternately if the surface region of the semiconductor is heavily doped such that the depletion region becomes thin, electrons will tunnel into the metal through the barrier. These electrons create a current Ims that flows from metal to semiconductors. Since at equilibrium the net current is zero, this current is exactly balanced by an equal and opposite current Ism . When the semiconductor is biased negative with respect to metal by voltage Vf (ie.) forward bias, the barrier to the flow of electrons from the semiconductor to the metal decreases from qVo to q (Vo – Vf) so more electrons flow from semiconductors to the metal and the current Ims increases above the equilibrium value. But Ism does not change because qφ b remains almost unaltered on biasing. Therefore net current flow occurs between metal to semiconductors occurs. Application of reverse bias causes the electron to flow from the semiconductor to metal because the barrier increases to q(Vo + Vf). As a result, Ims is reduced below its equilibrium value, whereas Ism remains almost unchanged. Thus a small current flows.

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Conclusion Analyzing the carrier concentration Vs. temperature graph we observe three different regions namely ionization, extrinsic and intrinsic. As carrier concentration increases with temperature more and more donor impurities are ionized, that is the valence electrons are removed from the impurities. In the extrinsic region the carrier concentration is essentially constant. In the intrinsic region the carrier concentration increases dramatically with temperature because of thermally generated electron hole pairs.

Based on the preceding discussion, the position of the fermi level changes with increasing temperature. At low temperatures when impurities are fully ionized and Ef will be close to the conduction band in an n-type material. At higher temperatures it will shift toward the middle of the gap.

This project also helps us to realize the simulation of energy band diagrams of schottky diodes for various metal – semiconductor junctions. Here we illustrated the energy band diagrams for three semiconductor materials and four metals for three doping concentrations.

This simulation can also be extended to Ohmic contacts and also to double heterojunctions.

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APPENDIX A

References 1. The potential of fast high voltage Sic Diodes-Heinz Mitlehner, Wolfgang Bartsch, Manfred Bruckmann, Karl Otto Dohnke, Ulrich Weinert 2.

The Effect of Heat treatment on Au Schottky E.Ioannou, Nick A.Papanicolaou, Paul E.Nordquist, JR.

contacts

on

B-SiC-Dimitris

3. Au-SiC Schottky Barrier Diodes-S.Y.Wu, R.B.Campbell 4. Pt and Pt SiC Schottky contacts on n-type B-SiC-N.A.Papanicolaou, A.Christou, and M.L.Gipe 5. High-Voltage Ni-and Pt-Sic Schottky Diodes utilizing Termination-Vik Saxena, Jian Nong (Jim) Su, and Andrew J.Steckl

Metal

field

plate

6. Schottky Barrier Diodes-Dallas Morisette, Mitch McGlothlin, J.A.Cooper, Jr., and M.R.Melloch 7. Planar Terminations in 4H-SiC Schottky Diodes with low leakage and high yieldsR.Singh and J.W.Palmour 8. Silicon Carbide High-Power Devices-Charles E.Weitzel, John W.Palmour, Calvin H.Carter, Jr., Karen Moore, Kevin J.Nordquist, Scott Allen, Christine Thero, and Mohit Bhatnagar 9. A 3 kV Schottky barrier diode in 4H-SiC-Q.Wahab,T.Kimoto, A.Ellison, C.Hallin, M.Tuominen, R.Yakimova, A.Henry, J.P.Bergman, and E.Janzen 10. Al/Ti Schottky Barrier Diodes with the Gaurd -Ring Termination for 6H-SiCKatsunori Ueno, Tatsuo Urushidani, Kouichi Hashimoto, and Yasukazu Seki 11. Surface Barrier Diodes on Silicon Carbide-S.H.Hagen 12. Efficient Power Schottky Rectifiers of 4H-SiC-A.Itoh,T.Kimoto and H.Matsunami 13. Schottky and S.Gonda

Barrier

diodes

on

3C-SiC-S.Yoshida,K.Sasaki,

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E.Sakuma,

S.Misawa,

14. Solid State Electronic Devices- Ben G.Streetman 15. Principles of growth and Processing of Semiconductors- Subash Mahajan and K.S.Sree Harsha

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TYPE

ENERGY BAND GAP (eV)

METALS

SEMICONDUCTORS

Al

4.28

Pt

5.65

Cr

4.5

Mo

4.6

Si

1.12

SiC

2.86

GaAs

1.43

Appendix B SEMICONDUCTOR

CARRIER CONCENTRATION (carriers/cm-3)

Si

1.18 E 10

GaAs

2.25 E 06

SiC

1 E 06

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