Model of MOSFET in Delphi

Technical report, IDE1066, February 2011

Master’s Thesis in Microelectronics and Photonics

Andrey Prokhorov Olesya Gerzheva

School of Information Science, Computer and Electrical Engineering

Acknowledgment

We want to thank our supervisors: Docent Ying Fu from the Royal institute of Technology for his valuable suggestions which helped us greatly during the process of preparing and writing this thesis; and we also want to thank Doctor Lars Landin who supervised us in the beginning of work. Thanks to Håkan Pettersson, the Head of the Dept. of Mathematics, Physics and Electrical Engineering, and main teacher in semiconductor physics, for his help and advice. Thanks to all members of our families for their support and encouragement.

2 Abstract

In modern times the increasing complexity of transistors and their constant decreasing size require more effective techniques to display and interpret the processes that are inside of devices. In this work, we are modeling a two‐dimensional n‐MOSFET with a long channel and uniformly doped substrate. We assume that this device is a large geometry device so that short‐channel and narrow‐width effects can be neglected. As a result of the thesis, a demonstration program was built. In this executable file, the user can choose parameters of the MOSFET‐model: drain and gate voltage, and different geometrical parameters of the device (junction depth and effective channel length). In the advanced regime of the program, the user can also specify the model re‐calculation parameter, doping concentration in n+ and bulk regions. The program shows the channel between the source and drain region with surface diagrams of carrier density and potential energy as an output. It is possible to save all calculated results to a file and process it in any other program, for example, plot graphics in Matlab or Matematica. The model can be used in lectures that are related to semiconductor physics in order to explain the basic working mechanisms of MOSFETs as well as for further detailed analysis of the processes in MOSFETs. It is possible to use our modeling techniques to rebuild the model in another computer language, or even to build other models of transistors, performing similar calculations and approximations.

It is possible to download the executable file of the model here: http://studentdevelop.com/projects/MOSFET_model.zip

Keywords

MOSFET, semiconductor, model, Delphi, channel, carrier concentration, potential energy.

3 Table of Contents

Acknowledgment ...... 2

Abstract ...... 3

1. Basic theory of semiconductors ...... 5

1.1 Energy bands ...... 5

1.2 Intrinsic semiconductor ...... 5

1.2 Carrier concentration and Fermi level ...... 6

2. The MOSFET ...... 8

2.1 Introduction...... 8

2.2 Energy band diagrams ...... 9

2.3 The MOSFET ...... 12

2.4 Characteristics of the MOSFET ...... 13

2.5 Operating regions of the MOSFET ...... 14

3. Modeling of MOSFET ...... 17

3.1 Numerical methods ...... 17

3.2 Description of the model...... 19

3.2.1 Programming model ...... 19

3.2.2 Mathematical model ...... 21

4. Development environment Delphi ...... 23

5. Results ...... 24

6. Conclusion ...... 29

References ...... 30

Appendix ...... 31

Program code of the model in Delphi ...... 31

Download the model ...... 35

4 1. Basic theory of semiconductors

1.1 Energy bands

Semiconductor materials are the basis of modern electronics. Different kinds of transistors, diodes and solar cells are made of semiconductors. The most popular semiconductor material is silicon. Atoms in a silicon crystal have four valence electrons to share with four nearest neighbors. Electrons of an isolated atom may occupy only certain discrete energy levels. If two atoms in a semiconductor move closer to each other then energy levels split to accommodate all electrons in the system. Usually, the system has a large number of atoms and the higher energy levels tend to unite into two separate bands of allowed energies, called the Conduction band and the Valence band, respectively [1]. The Conduction band – is the upper band, where energy levels are almost empty. Energy level Ec – is the bottom of the conduction band. The Valence band – is the lower band where energy levels are full. Energy level EV – is the top of the valence band. The difference between two of these levels is called the bandgap energy ( Eg )[2]. The bandgap energy of most semiconductors decreases as the temperature increases (Eq.1.1) [1].

⎧1.206 − 2.73⋅10−4 T (for T ≥ 250 K) ⎪ −5 -7 2 E g (T ) = ⎨1.179 − 9.03⋅10 T - 3.05 ⋅10 T (for 300K > T > 170 K) , (1.1) ⎪ −5 -7 2 ⎩1.17 −1.06 ⋅10 T - 6.05 ⋅10 T (for T ≤ 170 K)

where T – is the temperature, (K).

1.2 Intrinsic semiconductor

A pure semiconductor without dopant species added is called undoped or intrinsic semiconductor. Intrinsic semiconductor has the same number of electrons in the conduction band as the number of holes in the valence band, at a given temperature.

p = n = ni , (1.2) Where p – is the free hole concentration, (cm-3); n – is the free electron concentration, (cm-3); -3 ni – is the intrinsic carrier concentration, (cm ). The formula 1.3 shows the intrinsic carrier concentration as a function of the temperature.

5 T 3/ 2 Eg (T) Eg (T0 ) -3 ni (T) = ni (T0 )( ) exp[− + ] , (cm ) (1.3) T0 2kT 2kT0

where T0 – is the nominal temperature (T0 = 300 K) [1].

1.2 Carrier concentration and Fermi level

If we consider an intrinsic case without impurities added to the semiconductor, then the number of electrons (occupied conduction band levels) is given by the total number of states N(E) multiplied by the occupancyF(E ) , integrated over the conduction band:

∞ ∞ n = ∫ n(E)dE = ∫ N(E)F(E)dE, (1.4) EC EC n(E) – is the electron density [(cm-3-eV)-1], -3 -1 N(E) – is the density of allowed energy states [(cm -eV) ], F(E) – is the Fermi-Dirac distribution function.

3 2mn 2 N(E) = 4π ( 2 ) E − EC (1.5) h

where mn – is the effective mass of the electrons; h – is the Planck constant;

EC ‐ is the conduction band edge.

The Fermi-Dirac distribution function F(E ) shows the probability of electron occupation of an electronic state with energy E. 1 F(E) = (E−E ) / kT (1.6) 1+ e F 1 The energy at which the probability of occupation by an electron is called the Fermi 2 energy ( EF ). Large numbers of states are allowed in the conduction and valence band. However, there would not be many electrons in the conduction band for an intrinsic semiconductor because electrons prefer states with lower energy that are in the valence band. Therefore, an electron can occupy one of these upper states with a very small probability. Most of the allowed states in the valence band will be occupied by electrons. Hence, the electron can occupy one of these states with a probability near one. Unoccupied electron states in the valence band are referred to as holes. Substituting Eq. 1.5 and Eq. 1.6 into Eq. 1.7 we can find the carrier concentration [2].

6 2m 3 ∞ E − E dE n = 4π ( n ) 2 C h 2 ∫ 1+ e(E−EF ) / kT EC (1.7)

If x ≡ (E − EC ) / kT , where x – is the carrier energy in units of kT ,

and η ≡ (EF − EC ) / kT , where η – is the Fermi level in units of kT , then Eq.7 becomes

1 2m ⋅ kT 3 ∞ x 2 dx n = 4π ( n ) 2 . (1.8) 2 ∫ h 0 exp(x −η) +1 By collecting up parameters, we can express the electron concentration as:

2 (1.9) n = N C ⋅ F1/ 2 (η), π where 2πm kT N ≡ 2( n ) 3 / 2 C h 2 (1.10)

NC – is the effective density of states in the conduction band; 1 F (η) – is the Fermi – Dirac integral of the order of . 1/ 2 2 We can find the Fermi – Dirac integral for the two cases, when η << −1 andη >> 1:

⎧ π −3 ⎪ exp[η(1− 2 2 )exp(η) + ...] if η << −1 ⎪ 2 F1 (η) = ⎨ (1.11). 2 2 3 ⎪ η 2 (1+ 0.125π 2η −2 + 0.267η − 4 + ...) if η >> 1 ⎩⎪3 1 The approximate Fermi-Dirac integral of the order of for the value of η , over the range 2 ( − ∞ to + ∞ ) will be [7]:

2 π F1 (η) ≈ −3 , (1.12) 2 3 π a 8 + 4exp(−η)

where a = η 4 + 33.6η(1 − 0.68 exp(−0.17(η + 1)2 ))+ 50 . (1.13)

7 2. The MOSFET

2.1 Introduction

A MOS (Metal Oxide Semiconductor) diode is a structure where a thin layer of oxide is grown on top of semiconductor substrate, and after that, a metal layer is deposited on the oxide, as is shown in Fig. 2.1.

Fig.2.1 Cross-section of a MOS diode

In the MOS diode the voltage applied to the gate controls the state of the Si-surface underneath. There are two states of the MOS diode that can be used to make a voltage-controlled switch – accumulation and inversion. The MOS diode is in an accumulation state when a negative gate voltage is applied, that attracts the holes from the p-type silicon to the surface; and in the inversion state when a positive voltage (larger than the threshold voltage) is applied, creating an inverted layer of electrons at the surface. The threshold voltage is the gate voltage when the channel just starts to form at the oxide-substrate interface. There are two modes of the switch: on and off that correspond to the existence or absence of the electron layer (the channel). When the gate voltage is below the threshold voltage there is no channel and the source and drain n+ regions are isolated by the p-type substrate. This is the off-mode of the switch. When the gate voltage is higher than the threshold voltage (on-mode) the current flows through the surface and the channel appears [4].

8 2.2 Energy band diagrams

The energy band diagrams of the three separate components (metal, oxide and semiconductor) of the MOS diode are shown in the Fig. 2.2, where: E0 – is the vacuum energy level of free electrons. The parameter Eox – is the bandgap of SiO2 , typically Eox = (8.0 − 9.0 ) eV .

q ⋅ Φ m – is the work function of the metal.

Figure 2.2 The energy band diagrams of MOS diode components

The work function is the energy that must be given to an electron to pass over the surface energy barrier, in other words, across the energy difference between the vacuum level E0 and the Fermi energy of the metal E fm :

q ⋅ Φ m = E0 − E fm . (2.1)

For aluminum the work function is: q ⋅ Φ m = 4. 1 eV .

Electron affinity (q ⋅ χ ) – is the energy difference between the vacuum level E0 and the conduction band edge Ec at the surface:

q ⋅ χ = E0 − Ec (2.2)

9 The affinity is a property of a material and it is not affected by the presence of impurities or imperfections. q ⋅ χ s – is the electron affinity in the semiconductor, its quantity varies as a function of doping. For silicon, the affinity is q ⋅ χ s = 4. 05 eV .

q ⋅ Φ s – is the work function of the semiconductor: E q ⋅ Φ = q ⋅ χ + g + qϕ (eV) for p-type, (2.3) s s 2 p

where ϕ p – is the Fermi potential for p-type silicon. E q ⋅ Φ = q ⋅ χ + g + qϕ (eV) for n-type, (2.4) s s 2 n

Where: ϕn – is Fermi potential for n-type silicon. For the same doping concentration:

ϕ p = ϕ n = ϕ (2.5)

Nb ϕ = Vt ln( ) (V), (2.6) ni where: kT V = – is the thermal voltage, t q

Nb – is the substrate doping concentration.

15 −3 For p-type silicon with an acceptor concentration N b =10 cm and q ⋅ϕ = 0. 29 eV , the work function will be q ⋅ Φ s = 4. 90 eV . When three components of the MOS structure are connected, the work function can be determined only by the difference between metal and semiconductor parts [1]. The energy band diagram of an ideal p-type MOS diode at V=0 is shown in Fig. 2.3. At zero bias, the energy difference between the metal work function q ⋅ Φ m and the work function

q ⋅ Φ s is zero or, in other words, the work function difference q ⋅ Φ ms is zero. E q ⋅ Φ ≡ (q ⋅ Φ − q ⋅ Φ ) = q ⋅ Φ − (q ⋅ χ + g + q ⋅ψ ) = 0 , (2.7) ms m s m 2 b

where the sum of the three variables in the brackets is equal to q ⋅ Φ s .

10 VACUUM LEVEL

qχs qΦs Eg/2 qΦm Ec qψB Ei Ef Ef

METAL ALUMINIUM d SEMICONDUCTOR (SILICON p-TYPE)

INSULATOR (SILICON DIOXIDE)

Figure 2.3 Energy band diagram of an ideal MOS diode at V=0

11 2.3 The MOSFET

A MOSFET – Metal Oxide Semiconductor Field Effect Transistor (Fig. 2.4) is a transistor based on the MOS diode. On the top of the oxide, a gate electrode is deposited (a conducting layer of metal). Under the oxide and inside the substrate there are two heavily doped regions: source and drain. The source-to-drain electrodes are equivalent to two p-n junctions that are situated back-to-back. The central MOS diode with the inverted channel connects the source and drain junctions. The flow of charge carriers in the channel region between the source and the drain is thus controlled by an electric field, hence the name MOSFET, created by a voltage Vg applied to the gate electrode. The MOSFET may be n-channel or p-channel depending on the type of carriers in the channel region [1]. In our MOSFET model, the channel contains electrons (n-channel), the source and drain regions are heavily n+ doped and the substrate is p-type.

Figure 2.4 N-channel MOSFET diagram

12 When there is no voltage applied to the gate and there is no conduction channel between the drain and source regions, the MOSFET is referred to as a normally-off device (an enhancement-mode device). A certain minimum voltage – called a threshold (turn-on) voltage

Vth, should be applied to the gate to induce a conduction channel. If a conduction channel exists between the source and the drain regions even at zero gate voltage (normally-on device) – then it is called a depletion-mode device. In this case, the current flow is not exactly at the surface, some carriers are in the bulk of the silicon.

2.4 Characteristics of the MOSFET

Under normal operating conditions, the drain- and source- voltages should be applied in a way that the source and drain-to-substrate p-n junctions will be reverse biased (i.e. a negative voltage is applied to the p-side with respect to the n-side). There will be no significant current until the voltage reaches the critical value called “the junction breakdown voltage” after which the current dramatically increases.

In case when the source and bulk regions are grounded (Vb=Vs=Vsb=0), depletion regions are formed around the n+ source and drain region (even when the gate voltage is zero) due to n+- -3 p junctions formed with the p-type substrate of concentration Nb (cm ).

In fig. 2.5, a cross-section of an n-channel MOSFET is illustrated, where Xsd and Xdd are the widths of the depletion region under the source and drain, respectively.

Figure 2.5 Cross-section of n-channel MOSFET

13 2ε 0ε iϕi X sd = X dd = (cm) at Vds=Vbs=0, (2.8) qNb

where ϕi – is the built-in potential between the source/drain to substrate p-n junction.

N sd Nb ϕi = Vt ln( ) (V), (2.9) ni kT where V = – is the thermal voltage, t q

+ -3 Nsd – is the concentration in the n regions (cm )

With positive voltage Vds applied to the drain contact, and positive gate voltage less than the threshold voltage (Vgs < Vth ), applied to the gate, the p-type region will be depleted under the gate oxide. Holes are pushed away from the surface, therefore immobile negative bulk charge Qb appears at the silicon substrate.

If Vgs > Vth is applied to the gate, then a conduction channel with mobile negative charge Qi is formed at the surface. This channel is called inversion layer because the surface layer is inverted from p-type to n-type after formation of the channel. The thickness of this inversion layer depends on the applied bias.

When Vgs = Vth , the concentration of the minority carriers (electrons) at the surface equals that of the majority carriers (holes) in the bulk but the higher the Vgs is than Vth the higher will be minority charge density Qi (inversion charge) will be.

Qg=Qi+Qb,, (2.10)

where Qg – is the gate charge.

If there is a voltage difference between the source and the drain, then the current Ids will flow through the channel, due to the drift of carriers from the source to the drain [1].

2.5 Operating regions of the MOSFET

Linear region

The linear region is a region where Ids increases linearly with Vds for a given Vgs, which is higher than Vth. If a small drain voltage is applied, electrons will flow from the source to the drain, therefore current will flow in the reverse direction (from the drain to the source) through the conduction channel (see Fig. 2.6 a).

14 Saturation region

This is a region where Ids no longer increases with Vds, Ids is saturated. When the drain voltage increases, eventually it reaches VDsat; the thickness of the inversion layer xi near y=L is reduced to zero. This is called the Pinch-off point (see Fig. 2.6 b). At this point, the drain current remains the same, because for Vd>VDsat, at point P the voltage VDsat remains the same. Therefore, the number of carriers arriving to point P from the source remains the same (this is the same with current arriving from the drain) [2]. The pinched-off portion of the channel moves towards the source end due to the widening of the drain depletion region. If the voltage VDsat increases beyond pinch-off, the pinch-off region between the channel pinch-off point and drain region causes the effective channel length to decrease from L to L’ (see Fig. 2.6 c).

Breakdown region

When Vds increases even more, the transistor enters a region where Ids suddenly increases until breakdown of the drain-to-substrate p-n junction occurs. The breakdown is caused by the high electric field in the drain end. In short devices, this is called hot-carrier effect, due to the high electric field at the drain end, and it can also result in device breakdown. [1].

Cut-off region

This is the region where Vgs < Vth for which no channel exist between the source and the drain, and Ids=0.

Figure 2.6 Operating regions in the MOSFET a) Linear region b) Pinch-off point c) Saturation region

16 3. Modeling of MOSFET

In order to describe the model that was built, we need to explain it from two different points of view – practical and theoretical. That will be done in the following paragraphs Programming model and Mathematical model respectively. Since all the mathematical calculations were made by a program, we used numerical methods in our model. Numerical methods will be described in the next section.

3.1 Numerical methods

First derivative The derivative of a function f(x) is the limit where an increment of the function is divided by an independent variable that goes to zero (vanish):

∂y Δy f ′(x) = = lim , Δx → 0 (3.1) ∂x Δx Δy We should replace the ratio of infinitely small increments by the ratio of finite Δx differences in order to perform numerical solution of the derivative. Then, the smaller increment of the argument we take, the more precise numerical value of the derivative we will get. In the two-point method for calculating derivatives two points are used that are obtained by adding and subtracting Δx from the desirable point x, where the derivative should be determined.

Fig. 3.1 Geometrical illustration of first derivative

17 According to Figure 3.1, we can write:

∂y Δy y(x + Δx) − y(x − Δx) ≈ = (3.2) ∂x Δx 2Δx

Let us write a formula for the case of an array A that has n elements [0..n]. The values of A are written in such a way that the difference between indexes (x) and (x+1) is Δx, so the formula (1) will look like: ∂A A(x +1) − A(x −1) ≈ (3.3) ∂x 2 Example:

If we take function y=x2 and ∆x = 0.001, the first derivative will be:

∂y Δy y(x + 0.001) − y(x − 0.001) y′ = ≈ = , ∂x Δx 0.001⋅ 2

If we want to find derivative of this function at the point x = 1, then:

2 ∂y Δy (1.001)2 − (0.999) y′(1) = ≈ = = 2 ∂x Δx 0.002

That could be proved by analytical solving: y′ = (x2 )′ = 2x; y′(1) = 2 ⋅1 = 2

Second derivative

The second derivative is calculated as a derivative of the first derivative.

′ ⎛ u ⎞ u′v − v′u The rule is ⎜ ⎟ = (3.4). ⎝ v ⎠ v2

′ ′ ∂2 y ⎛ ∂y ⎞ ⎛ y(x + Δx) − y(x − Δx) ⎞ = ⎜ ⎟ = ⎜ ⎟ = ∂x2 ∂x 2Δx ⎝ ⎠ ⎝ ⎠ ′ ()y(x + Δx) − y(x − Δx) ⋅ 2Δx − (2Δx)′⋅(y(x + Δx) − y(x − Δx)) = 4Δx2

Finally we get the second derivative:

∂2 y y(x + Δx) − 2⋅ y(x) + y(x − Δx) = . (3.5) ∂x2 Δx2 18 3.2 Description of the model

3.2.1 Programming model

The programming model of the MOSFET is represented by several matrixes – 2D arrays of main parameters of the MOSFET – Fermi-level Ef, carrier concentration N and potential energy φ. The size of one matrix (number of elements) is limited by the performance of the CPU and the memory size of the computer. We were working with the computer that has following parameters: CPU 2300 Mhz and 2 Gb RAM, that allowed us to operate with matrixes 2000 x 1000 elements. By trying to calculate values with different sizes of matrixes, we determined that the optimal size of array for a demonstration model is 200 x 100 elements, since it is time of working is more important that accuracy. In every matrix, the following zones are described: source, drain, gate, their contacts and oxide layer. Schematically, it is shown in the figure 3.2.

Figure 3.2. Model of MOSFET with geometrical parameters

In our model, every zone of the MOSFET will be represented by an area of cells in a two- dimensional array (figure 3.3). For example, the source contact will be described as rectangular with coordinates (0, 0, 400, 100) – top left corner (x1,y1) and bottom right corner (x2,y2).

Figure 3.3. Representation of the model in computer memory

Each element of the matrix (a cell) represents a small unit inside the MOSFET. For example, for 200 nm-width transistor – the width of one unit will correspond to: 200 nm divided by 2000 – which is 0.1 nm. For each element, the mathematic model is applied, i.e. in every point carrier concentration n and electric field E are calculated. The real MOSFET, as it is shown in the figure 3.4, is fabricated with much bigger contacts and thicker oxide layer than was described in the model. However, for the model, the size of contact and oxide layer do not affect any calculations, since in all points of oxide the carrier density is determined as 0 (no charges) and in all points of metal contacts primitively ε is much larger than in other regions (ε =1000 for metal and ε =12 for n-regions).

Figure 3.4. Real ( fabrication) MOSFET structure

20 3.2.2 Mathematical model

The mathematical model of the MOSFET is described by formulas used to calculate the main parameters: Ef, n and φ. The Fermi level can be calculated from the doping concentration

ND that is given by the user through the program interface (in the bulk region and in n-regions). From the Fermi level, it is possible to calculate the carrier concentration n and from that the potential energy. To calculate the Fermi level, we used an approximation of the Fermi-Dirac integral of the 1 order of (see Eq. 1.12 and Eq. 1.13 in chapter 1): 2 2 π F1 (η) ≈ −3 , where 2 3 π a 8 + 4exp(−η)

4 2 a =η + 33.6η{1 − 0.68exp[− 0.17(η + 1) ]}+ 50 andη ≡ (EF − EC ) / kT . Then, from the Fermi level, we can calculate the carrier concentration using Eq. 1.8 from 2 chapter 1 n = NC ⋅ F1/ 2 (η). π The potential energy can be found from the Poisson equation. It cannot be solved analytically hence numerical methods should be used. In order to calculate the potential energy (ϕ = −eE ), the Laplace operator needs to be calculated:

δ 2 E δ 2 E δ 2 E ΔE = ∇2 E = + + . (3.6) δx2 δy 2 δz 2 If we want to solve Poisson equation in a point (x,y), then we need E(x,y) at this point and calculate the Laplace operator: ∂ 2 E ∂ 2 E ΔE(x, y) = ∇ 2 E(x, y) = + (3.7) ∂ 2 x ∂ 2 y That means we need to find two second derivatives of E by x and by y. If we use the formula (2) for second derivative, we can rewrite the equation (3):

∂ 2 E ∂ 2 E E(x + Δx) − 2 ⋅ E(x) + E(x − Δx) E(y + Δy) − 2 ⋅ E(y) + E(y + Δy) + = + (3.8) ∂ 2 x ∂ 2 y Δx 2 Δy 2

If we have a matrix a(i,j) where each element is a value of ∇ 2 E(x, y) , and ∆x and ∆y is a small value Δ , we finally we get:

21 ∂ 2 E ∂ 2 E E(i + Δ, j) − 2 ⋅ E(i, j) + E(i − Δ, j) ∇ 2 E(i, j) = + = + ∂ 2i ∂ 2 j Δ2 , (3.9) E(i, j + Δ) − 2 ⋅ E(i, j) + E(i, j − Δ) = A ⋅ n(i, j) Δ2 eΔ2 where A = . εε 0

We can simplify the equation (3.9):

E(i + Δ, j) − 4 ⋅ E(i, j) + E(i − Δ, j) + E(i, j + Δ) + E(i, j − Δ) A ⋅ n(i, j) = (3.10) Δ2

We can continue:

4 ⋅ E(i, j) = (E(i + Δ, j) + E(i − Δ, j) + E(i, j + Δ) + E(i, j − Δ)) − A ⋅ n(i, j) ⋅ Δ2

E(i + Δ, j) + E(i − Δ, j) + E(i, j + Δ) + E(i, j − Δ) A ⋅ Δ2 n(i, j) E(i, j) = − 4 4

2 q ∇ E(x, y) = [ p(x, y) − n(x, y) + N D (x, y) − N a (x, y)] ε 0ε

In the bulk region: p(x,y) – Na(x,y) = 0

2 q Eventually: ∇ E(x, y) = [N D (x, y) − n(x, y)] εε 0 In this step, we can find the electric field E(x,y) from the carrier concentration n.

E(i + Δ, j) + E(i − Δ, j) + E(i, j + Δ) + E(i, j − Δ) A⋅ Δ2 (N (i, j) − n(i, j)) E(i, j) = − D (3.11) 4 4⋅εε 0 Since the potential is: − e2Δ2 ϕ = V (i, j) = −eE = −e⋅ AA[n − N D ] = [n − N D ], εε 0

− e2Δ2 AA = . εε 0 Then, finally, the formula to calculate the potential will be: V (i + Δ, j) +V (i − Δ, j) +V (i, j + Δ) +V (i, j − Δ) q 2 ⋅Δ2 (N (i, j) − n(i, j)) V (i, j) = − D . 4 4⋅εε 0

22 4. Development environment Delphi

To build our MOSFET model we used the Borland Delphi 7 program. Delphi is a programming language that originally was developed from the object-oriented Turbo Pascal language. Borland Delphi is a so called development environment, where a programmer can comfortably make Windows-based applications. The main window of Borland Delphi is shown in figure 4.1.

Figure 4.1 Interface of Borland Delphi

We chose this development environment because we have had experience with this program before and it is easy to build applications with it: the user just needs to drag elements to a form. For example, we used lists, buttons, and combo-boxed elements in our MOSFET model.

23 5. Results

The main window of our program is shown in the figure 5.1.

Figure 5.1 Main window of MOSFET model program

The program has two regimes: demonstration and advanced. The demonstration regime can be used in lectures, for explanation of the basic working mechanism of the MOSFET. The advanced regime can be used for detailed investigations of the MOSFET model. In the regime the user can change many parameters of the model (figure 5.2).

Figure 5.2 Window of the MOSFET model program in the advanced regime

24 In the figures 5.3 – 5.5, the results of our calculations are shown for the donor concentration ND, and carrier concentration n and potential energy φ, when n and φ parameters were not recalculated (the first approximation).

Figure 5.3 Donor concentration ND

In figure 5.3, the doping profile is shown for all regions of the MOSFET as a surface diagram. It is clearly seen there that the three contacts have the highest doping concentration (ND 20 -3 18 -3 = 10 cm ), the highly doped source and drain have ND =10 cm , and the bulk region 15 -3 (substrate) – ND =10 cm . In the oxide part, the donor concentration goes down to zero, since it is undoped.

Figure 5.4. First approximation of carrier concentration

Figure 5.5 First approximation of potential energy

The first approximation of the carrier concentration and the potential energy is shown in figures 5.4 and 5.5. In these figures the source and the drain region have rectangular form. This is the initial guess for the model which will be improved later. After 5 000 loops of recalculations (iterations), when neighboring cells are considered, the form of the regions becomes smooth, as shown in figures 5.6 and 5.7.

Figure 5.6 Potential energy after 5000 iterations

In figure 5.6, the result of the potential energy calculation is shown as a surface diagram, with drain voltage VD= -0.15 V and gate voltage VG= -0.2 V. The potential energy at the drain is lower than the energy at the source because of the additional voltage applied to drain.

Figure 5.7 Carrier concentration after 5000 iterations

27 After 5000 recalculations of the initial guess of the carrier concentration, the depletion region appears; there were no depletion region in the first approximation. The result is shown in figure 5.7. The gate voltage moves holes away from the gate region. If the gate-inducted vertical electric field is strong enough, a channel between these two n-regions will appear. We define in the model, that the channel appears in the place where the concentration of electrons is greater 17 -3 15 -3 18 -3 than 10 cm , since the donor concentration in the bulk region is ND=10 cm and n=10 cm in the n-regions.

28 6. Conclusion

We have built a model of the MOSFET that is possible to use as a demonstration program for teaching and for further analysis. In the model it is possible to save the results of the calculation to a file as a two-dimensional array, where parameters are calculated in every point of the device. Large mathematical analysis was made and surface diagrams of the carrier concentration, the potential energy and the donor concentration distributions were presented. The channel distribution in the MOSFET was calculated and plotted. All calculation processes are described in sufficient details to allow reconstitution of the model in another computer language or building other models of transistors performing similar calculations and approximations.

29 References

[1] Narain Arora, MOSFET Modeling for VLSI Simulation: Theory and Practice, ISBN- 13: 978-981-256-862-5, ISBN-10: 981-256-862-X, World Scientific Publishing Co. Pte. Ltd., Singapore, p. 632, 2007. [2] S.M. Sze, Semiconductor devices: Physics and technology, SBN/ISSN: 0-471-33372-7, 2 Uppl., Wiley, New York viii, p. 564. , 2002. [3] S. M. Sze, Kwok K. Ng, Physics of Semiconductor Devices, ISBN-I 3: 978-0-47 1-1 4323- 9, ISBN-10: 0-471-14323-5, John Wiley & Sons, Inc., Hoboken, New Jersey, p. 5, 2007. [4] Sima Dimitrijev, Principles of Semiconductor Devices, ISBN-10: 0195161130, ISBN- 13: 978-0195161137, Oxford University Press, USA, p. 578, 2005. [5] Umesh K. Mishra, Jasprit, Singh Semiconductor device physics and design, ISBN 978-1- 4020-6480-7, Dordrecht, The Netherlands: Springer, p. 560, 2008. [6] E. J. Farrell, S. E. Laux, P. L. Corson, E. M. Buturla Animation and 3D color display of multiple-variable data: Application to semiconductor design. [7] Ying Fu, M. Willander Physical models of semiconductor quantum devices, ISBN 0-7923-

8457-1, USA: Springer, p. 266, 1999.

30 Appendix

Program code of the model in Delphi

program MOSFET; uses SysUtils, Math;

(* Comments: V: the potential energy (eV); ND: doping profile (cm-3) W: dielectric constant, F: Fermi energy cell size=0.1 nm, DM: carrier effective mass *)

const q=1.6E-19; (* C *) delta =1E-7; (* cm, step: 1 nm *) x = 0.1; (* update parameter *) mu= 1450; (* cm^2/V-s *) eps0=8.85E-14; (* permittivity, F/cm *) k = 1.38E-23; (* Boltzmann constant, J/K *) h = 6.62E-34; (* Planck constant , J*s *) m0 = 9.1095E-31; (* effective mass, kg *) var m :integer; //scale-parameter of matrix IX:integer; //width of matrix IY:integer; //height of matrix

Leff, Xj: string; IX1,IX2,recount : integer; ff,ff1,ff2,ff3:Text;

ND_n,NB : Real; eps : array [1..2000,1..1000] of integer; //dialectic constant (permittivity)

V,ND,P,n : array [1..2000,1..1000] of real; //V-the potential energy (eV), ND-doping concentration (cm-3), //P - temp array, n-carrier concentration (cm-3)

A,B,DM,F,T, min_v, breakdown_v : real;

VD, VG : Real; //VD - Drain voltage (V) and VG - Gate voltage (V) k_channel : Integer = 0; //number of point in channel array channel:array[1..2, 0..1000]of integer; procedure FERMI(ND,DM,T:real; var F:Real); var S,X,Y,Z: Extended; j : LongInt; begin T:=300; (* temperature [K] *) T:=T/1.1604E4; (* eV *)

Z:=0; F:=-0.5; //start value

X:=ND/(power((T*DM),1.5)*6.037E21); S:=1.0E-3/T; F:=F/T;

F:=F-S; S:=S*0.5; for j:=1 to 500000 do 31 begin F:=F+S; Y:=0.0; if not(-F>90.0) then begin Y:=Power(F,4)+33.6*F*(1.0-0.68*exp(-0.17*power((1.0+F),2))); Y:=1.0/(exp(-F)+1.32934038675/power((50.0+Y),0.375))-X; if (J=1) then Z:=Y; end;

if (Y*Z<=0) then break;

end;

F:=(F+0.5*S)*T; end;

(* CC ------*)

procedure CC(var A,DM,T,nu: real); //calc carrier concentration var aa,n,FD : real; begin nu:=(nu-A)/T; n:=0;

if (-nu <= 90) then begin aa:=Power(nu,4)+33.6*nu*(1.0-0.68*EXP(-0.17*power((1.0+nu),2))); FD:=1.0/(exp(-F)+1.32934038675/power((50.0+aa),0.375)); n:=FD*power((T*DM),1.5)*6.037E21; end;

A:=n; end; procedure main; var x,y,i,j,k, model:integer; z:real; begin

T:=300; (* temperature [K] *) T:=T/1.1604E4; (* eV *) DM:=1.08; //effective mass for silicon

FERMI(NB,DM,T,A); FERMI(ND_n,DM,T,B);

breakdown_v := -B;

for Y:=1 to IY do for X:=1 to IX do begin

(* oxide part *) eps[X,Y]:=4; ND[X,Y]:=0;

(* n-region *) if (Y < IY-10*m) then eps[X,Y]:=12; ND[X,Y]:=1.0E15; V[X,Y]:=-A;

(* source *) if (Y <= IY-10*m) and (Y >= IY-33*m) and (X >= 0) and (X <= 67*m) then begin eps[X,Y]:=12; ND[X,Y]:=1.0E18; V[X,Y]:=-B; end;

(* source contact *) 32 if (Y >= IY-10*m) and (X >= 0) and (X <= 32*m) then begin eps[X,Y]:=1000; V[X,Y]:=-B; ND[X,Y]:=1.0E20; end;

(* drain *) if (Y <= IY-10*m) and (Y >= IY-33*m) and (X >= 133*m) then begin eps[X,Y]:=12; ND[X,Y]:=1.0E18; V[X,Y]:=-B-VD; end;

(* drain contact *) if (Y >= IY-10*m) and (X >= 168*m) then begin eps[X,Y]:=1000; V[X,Y]:=-B-VD; ND[X,Y]:=1.0E20; end;

(* gate contact *) if (Y > IY-5*m) and (X > 62*m) and (X < 137*m) then begin eps[X,Y]:=500; V[X,Y]:=-VG; ND[X,Y]:=1.0E20; end; end;

for K:=1 to recount do begin

// ------

for j:=1 to IY do //do 300 Y=2,IY-1 for i:=1 to IX do //do 300 X=2,IX-1 begin if (eps[i,j] = 1000) then n[i,j]:=1E20;

if (eps[i,j] = 4) then begin n[i,j]:=0; //no charges in the oxide ND[i,j]:=0; end;

end;

for j:=2 to IY-1 do begin

//search for IX1 - max V from left to right for I:=2 to IX-1 do if (V[I,j] > 0) then begin IX1:=I; break; end;

// search for IX2 - max V from right to left for I:=IX-1 downto 2 do //do 401 I=2,IX begin if (V[I,j] > -VD) then begin IX2:=I; break; end; end;

33 // ------

for i:=2 to IX-1 do begin P[i,j]:=V[i,j];

//calc n if (eps[i,j] = 12) then // for Source, Drain and p-region begin

A:=V[i,j]; F:=0;

if (i <= IX1) then F:=0; if (i > IX1) and (i < IX2) then F:=(i-IX1)*(-VD/(IX2-IX1)); if (i >= IX2) then F:=-VD;

CC(A,DM,T,F); n[i,j]:=A; //n - elect density

if (n[i,j] < 1E15) then n[i,j]:=1E15; end;

//calc potential if (eps[i,j] <= 20) then //all regions except contacts where eps=1000 begin

B:=0.25*(V[i,j+1]+V[i,j-1]+V[i+1,j]+V[i-1,j]- (ND[i,j]-n[i,j]) * 9.05E-23 * 1.0E2/ (eps[i,j]));

P[i,j]:=B; end;

end; end;

// ------

//UPDATE Potential V for j:=2 to IY-1 do //do 310 Y=2,IY-1 for i:=2 to IX-1 do //do 310 X=2,IX-1 if (eps[i,j] <= 20) then //if current point is part of MOSFET begin if k < Round(recount*0.8) then V[i,j]:=0.1*P[i,j]+0.9*V[i,j] else V[i,j]:=0.01*P[i,j]+0.99*V[i,j]; end;

// edges definition ------j:=1; for i:=2 to IX-1 do //do X=2,IX-1 if eps[i,j]<=100 then V[i,j]:= V[i,j+1];

j:=IY; for i:=2 to IX-1 do //do X=2,IX-1 if eps[i,j]<=100 then V[i,j]:=V[i,j-1];

i:=1; for j:=1 to IY do //do Y=1,IY if eps[i,j]<=100 then V[i,j]:=V[i+1,j];

i:=IX; for j:=1 to IY do //do Y=1,IY if eps[i,j]<=100 then V[i,j]:=V[i-1,j]; 34

end;

// ------//checking for device breakdown min_v := 1000; i:=1; for j:=1 to IY do //do 310 X=2,IX-1 if (eps[i,j]=12) and (min_v > V[i,j]) then min_v := V[i,j];

if (breakdown_v > min_v ) then begin writeln('The device has breakdown!'); end;

// ------// CHANNEL

k_channel:=0; for i:=1 to IX do for j:=1 to IY do begin if (n[i,j]>=1E17) and (n[i,j]<1E20) and (i>=75*m) and (i<=135*m) then begin inc(k_channel); channel[1,k_channel] := i; channel[2,k_channel] := j; break; end; end;

// writing results to files could be here end;

end.

//MAIN PROGRAM begin Main; write('finished.'); readln; end.

Download the model

It is possible to download the executable file of the model here: http://studentdevelop.com/projects/MOSFET_model.zip