lectures 1&2. introduction to wbg semiconductors ... · this power module will cover an...

Prof. Petru Andrei, EEL 4351 Lectures 1&2 (Power Module)

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This power module will cover an introduction to WBG semiconductors and the simulation of power devices. The module is split into four lectures that cover the following topics:

Lectures 1&2. Introduction to WBG semiconductors; presentation of the structure and principle of operation of heterojunction transistors including GaN/AlGaN and SiC transistors; discussion of energy diagrams and current characteristics; analytical derivation of the current characteristics and discussion of the breakdown voltage and on-state resistance.

Lectures 3&4. Summary of the basic semiconductor equations; discretization and meshing techniques in finite element simulations; introduction to power device simulations using Sentaurus (including building geometry, meshing 2-D and 3-D structures, defining the material parameters of WBG semiconductors, and running time-dependent simulations); basic optimization techniques.

1. Introduction

Wide-bandgap (WBG) semiconductors are semiconductor with bandgaps of the order of 2-4 eV. The relatively large bandgap permit devices made of WBG semiconductors to operate at much higher voltages, frequencies and temperatures than conventional semiconductors that usually have badgaps of the order of 1-1.5 eV. This makes WBG semiconductors highly attractive in aerospace applications, military applications, or in applications that requires large power systems (switches, converters, etc.). The US Department of Energy also predicts that WBG semiconductors will play an important role in new electrical grid and alternative energy devices, as well as the robust and efficient power components used in high energy vehicles from electric trains to plug-in electric vehicles. [1, 2]

Table 1. Bandgap of common WBG semiconductors compared to Si

Material Eg (eV) Si 1.12

GaAs 1.43 CdSe 1.73 GaP 2.26 SiC 2.86

GaN 3.2 CeO2 3.2 ZnO 3.3 TiO2 3.4

InSnO 3.5-4.06 Y2O3 5.6 ZrO2 5-7 AlN 6.01-6.05

Table 2. Physical characteristics of Si and few WBG semiconductors (from L. M. Tolbert et al, http://web.eecs.utk.edu/~tolbert/publications/iasted_2003_wide_bandgap.pdf)


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Property Si GaAs 6H-SiC GaN 4H-SiC Diamond Eg (eV) 1.12 1.43 3.03 3.2 3.26 5.45 Dielectric constant 11.9 13.1 9.66 9 10.1 5.5 Electric breakdown field (kV/cm) 300 400 2500 2000 2200 10000 Electron mobility (cm2/Vs) 1500 8500 500 1250 1000 2200 Hole mobility (cm2/Vs) 600 400 101 850 115 850 Thermal conductivity (W/cm K) 1.5 0.46 4.9 1.3 4.9 22 Saturated electron drift velocity (107 cm/s) 1 1 2 2.2 2 2.7

A. Optical properties of WBG semiconductors a) Visible light has wavelengths between approximately 450 nm and 650 nm. This corresponds

to photons with energies between 3.1 eV and 1.8 eV, respectively. Hence, WIBG semiconductors are often used to build LEDs and semiconductor laser in the visible spectrum.

b) Why is Si black and shiny? (Answer: Eg = 1.12 eV so all visible light will be absorbed and Si appears black. It is also shiny because there are many delocalized photons in the conductions band which scatter photons)

c) Why GaP is yellow (Answer: Eg = 2.26 eV corresponds to l = 549 nm. Hence photons with E > 2.26 eV (i.e. green, blue, violet) are absorbed; photons with E < 2.26eV (i.e. yellow, orange, red) are transmitted; the sensitivity of the human eye is greater for yellow than for red, so GaP Appears Yellow/Orange)

d) Why is glass transparent (Answer: Eg > 5 eV. Hence all colored photons are transmitted, with no absorption, hence the light is transmitted & the material is transparent)

e) Why many insulators or wide band gap semiconductors are transparent to visible light, whereas narrow band semiconductors (Si, GaAs) are not?

B. Saturation velocity of WBG semiconductors

Low band curvature high effective masses of charge carriers low mobilities. The fast response times of devices with WBG semiconductors is due to the high saturation velocity.

C. Breakdown field of WBG semiconductors

The breakdown electric field in a few wide bandgap semiconductors is presented in Table 2.

D. Polarization of WBG semiconductors

Most WBG semiconductors have a wurtzite and zincblende structure. Wurtzite phases result ina a spontaneous polarization in the (0001) direction the polar surfaces of these materials have higher sheet carrier density than the bulk strong electric fields which creates high interface charge densities

E. Thermal properties of WBG semiconductors a) Higher energy gaps higher temperatures of operation higher powers (remember that

e gE kTc vnp N N −= and 2e gE kT

i c vn N N −= )


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b) Si and other semiconductor with a bandgap on the order of 1-1.5 eV are readily activated by thermal energy. This limits the operational temperatures below approximately 100o Celsius

c) WBG semiconductor can operate at temperatures of the order of 300o Celsius (SiC devices can operate up to 600°C)

d) WBG devices have a higher thermal conductivity (4.9 W/cm-K for SiC and 22 W/cm-K for diamond, as opposed to 1.5 W/cm-K for Si). Therefore, heat is more easily transferred out of the device, and thus the device temperature increase is slower. GaN is an exception in this case [].

e) The I-V characteristics of WBG semiconductor devices vary only slightly with temperature; hence, they are more reliable.

F. Switching losses a) WBG semiconductor-based bipolar devices have excellent reverse recovery characteristics.

With less reverse recovery current, switching losses and electromagnetic interference are reduced, and there is less or no need for snubbers.

b) Because of low switching losses, WBG semiconductor-based devices can operate at higher frequencies (>20 kHz) not possible with Si-based devices in power levels of more than a few tens of kilowatts.

G. Main figures of merit for WBG semiconductors (summary)

Table 3. Main figures of merit for WBG semiconductors compared with Si (from L. M. Tolbert et al, http://web.eecs.utk.edu/~tolbert/publications/iasted_2003_wide_bandgap.pdf)

Si GaAs 6H-SiC 4H-SiC GaN Diamond JFM 1.0 1.8 277.8 215.1 215.1 81,000

BFM 1.0 14.8 125.3 223.1 186.7 25,106

FSFM 1.0 11.4 30.5 61.2 65.0 3,595

BSFM 1.0 1.6 13.1 12.9 52.5 2,402

FPFM 1.0 3.6 48.3 56.0 30.4 1,476

FTFM 1.0 40.7 1,470.5 3,424.8 1,973.6 5,304,459

BPFM 1.0 0.9 57.3 35.4 10.7 594

BTFM 1.0 1.4 748.9 458.1 560.5 1,426,711

JFM: Johnson’s figure of merit, a measure of the ultimate high-frequency capability of the material BFM: Baliga’s figure of merit, a measure of the specific on-resistance of the drift region of a vertical field effect transistor (FET) FSFM: FET switching speed figure of merit BSFM: Bipolar switching speed figure of merit FPFM: FET power-handling-capacity figure of merit FTFM: FET power-switching product BPFM: Bipolar power handling capacity figure of merit BTFM: Bipolar power switching product


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2. Heterojunction and other power devices

a. Breakdown Voltage: P-N junctions

In the case of an ideal p-n junction with abrupt doping profile we can solve the Poisson equation and get the relation between the breakdown voltage and doping concentration.

The Poisson equation:

( ) ( ) ( ) ( )2

2 d aSi

d V q p x n x N x N xd x ε

+ − = − − + − (1)

The width of the depletion region:

( )2 1 1Sbi app

a d

W V Vq N Nε

= + −

(2)

The maximum value of the electric field

The maximum value of the electric field:

( ) ( )20 a d a dp n bi app

Si Si S a d

qN qN N NqE x x V VN Nε ε ε

= = = −+

(3)

Built-in potential

2ln a dbi T

i

N NV Vn

= (4)

Fig. 10 p-n junction diode: (a) equivalent circuit, (b) structure, (c) electric field as a function of position.


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At breakdown (see Fig. 1):

app bi appBV V V V= − ≈ − (5)

( ) max0E E= , DW W= and we obtain the following equations

max

2D

BVWE

= (6)

The on-state resistance (assuming that the contact resistances are negligible):

2 DON D

WR cm Wq N

ρm

W = = (7)

where m and N are the majority carrier mobility and doping concentration in the drift region (i.e. nm

and nN if the drift region is an n-type semiconductor or pm and pN if the drift region is an n-type

semiconductor). From eqs. (2), (5), and (6) we obtain:

2

2max

24 SBV BVE qN

ε=

2max

2S ENqBV

ε= (8)

Substituting (6) and (8) into (7) we obtain:

3max

4ON

S

BVREε m

= (9)

Factor 3maxS Eε m is an indicator of the impact of the semiconductor material properties on the resistance

of the drift region and is commonly referred as Baliga’s figure of merit for power devices [].

b. Heterojunction BJT


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Fig. 10 Heterojunction BJT: (a) single heterojunction structure, (b) double heterojunction structure, (c) energy bands of a single heterojunction BJT.

Using the Ebers-Moll model:

0 0 01 1CBEB EB

T T T

VV VV V V

E F R R FI I e I e I eα

= − − − ≈

(in the active region) (10)

0 0 01 1CBEB EB

T T T

VV VV V V

C F F R F FI I e I e I eα α

= − − − ≈

(in the active region) (11)

where

00 0

sinh

BBF F R R

B

B

pDI I qAL W

L

α α= =

(12)

0 0 0 cothE BF E B

E B B

D D WI qA n pL L L

= +

and 0 0 0 cothC BR C B

C B B

D D WI qA n pL L L

= +

(13)

we derive the following equations for the case when the transistor is operating in the active region:

• Emitter efficiency:


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0

2,

0 0 2,

coth1

coth 1 tanh

BB

Ep B B

E B B i EE E BE B

E B B B E E i B B

D WpI L L

D L N nI D D W Wn pL L L D L N n L

γ

= ≈ ≈ =

+ +

(14)

• Common base dc current gain ( 0C dc E CBI I Iα= + ):

0

2,

0 0 2,

sinh1

coth cosh sinh

BB

B

BCdc F

E B B i EE E BE B

E B B B B E E i B B

pDL W

LID L N nI D D W W Wn p

L L L L D L N n L

α α

≈ ≈ = ≈ =

+ +

, ,, ,

1 1

1 expcosh exp sinh G B G EG B G E E B BE B B

B E E BB B E E B

E EE E D L N WD L NW WD L N L kTL D L N kT L

= ≈−− ++

(15)

• Common emitter dc current gain ( 0C dc B CEI I Iβ= + ):

, ,exp1

Ep G B G Edc E B Bdc

E dc B E E B

I E ED L N WI D L N L kT

αβα

− ≈ = ≈ −

(16)

In deriving the previous equations we have assumed the density of states are the same in the emitter,

base and collector. How does the , ,exp G B G EE EkT−

term affect the emitter efficiency, common base

dc current gain, and common emitter dc current gain when the transistor is operating in the active region?

c. Power MOSFETs The principle of operation of power MOSFETs discussed in this section is very similar to the principle of operation of “lateral” MOSFETs presented in the previous lectures.


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Fig. 1 V-groove MOSFET structure Fig. 2 U- MOSFET structure

Fig. 3 VD-MOSFET structure Fig. 4 Lateral MOSFET structure

Fig. 5 IE- MOSFET (Implantation epitaxial MOSFET, https://unit.aist.go.jp/adperc/cie/teams/s-pdt.html)

https://unit.aist.go.jp/adperc/cie/teams/s-pdt.html


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Fig. 6 SiC-BGSIT(SiC Buried Gate Static Induction Transistor, https://unit.aist.go.jp/adperc/cie/teams/s-pdt.html)

d. SiC PiN Diode

It is anticipated that SiC-PiN diodes will be used as ultra-low-loss power diodes in high-voltage applications (electric power distribution systems, rapid transit railways, etc.) exceeding 3 kV.

Fig. 7 SiC-PiN diode (https://unit.aist.go.jp/adperc/cie/teams/s-pdt.html)

e. Thyristor

Discuss the structure and principle of operation of thyristors.





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Fig. 8 Thyristor: (a) structure, (b)-(c) two coupled thyristor equivalent circuit, (c) output characteristic.

Fig. 9 IGBT: (a) structure of a symmetric IGBT, (b) structure of an asymmetric IGBT, (c) equivalent circuit, (d) large voltage characteristics showing device breakdown, and (e) low voltage characteristics showing the on-state resistance.


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References

[1] http://www1.eere.energy.gov/manufacturing/rd/pdfs/wide_bandgap_semiconductors_factsheet.pdf

[2] https://en.wikipedia.org/wiki/Wide-bandgap_semiconductor

[3] B.J. Baliga, Fundamentals of Power Semiconductor Devices, Springer, 2008.

http://www1.eere.energy.gov/manufacturing/rd/pdfs/wide_bandgap_semiconductors_factsheet.pdf

https://en.wikipedia.org/wiki/Wide-bandgap_semiconductor


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This power module will cover an introduction to WBG semiconductors and the simulation of power devices. The module is split into four lectures that cover the following topics:

Lectures 1&2. Introduction to WBG semiconductors; presentation of the structure and principle of operation of heterojunction transistors including GaN/AlGaN and SiC transistors; discussion of energy diagrams and current characteristics; analytical derivation of the current characteristics and discussion of the breakdown voltage and on-state resistance.

Lectures 3&4. Summary of the basic semiconductor equations; discretization and meshing techniques in finite element simulations; introduction to power device simulations using Sentaurus (including building geometry, meshing 2-D and 3-D structures, defining the material parameters of WBG semiconductors, and running time-dependent simulations).

3. Basic semiconductor equations

Poisson equation:

( )2a d

qV n p N Nε

− +∇ = − + − (1)

Hole current continuity equation:

1

p p pp G Rt q

∂= − ∇⋅ + −

∂J (2)

Electron current continuity equation:

1

n n nn G Rt q

∂= ∇ ⋅ + −

∂J (3)

Hole current density:

p p pq p V qD pµ= − ∇ ∇J - . (4)

Electron current density:

n n nq n V qD nµ= − ∇ + ∇J . (5)

In the first order approximation the generation-recombination rates are 0p p L

p

p pG R Gτ−

− = − and

0n n L

n

n nG R Gτ−

− = − , where 0n and 0p are the electron and hole concentrations at equilibrium.


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However, in a more detailed analysis, the detailed generation-recombination mechanism should be considered. In the next 5 sections, we describe the main generation-recombination mechanisms that appear in WBG semiconductors. Notice that, depending on the WBG material, one or more mechanisms might dominate.

A. Impact ionization

Impact ionization is a process in which one energetic carrier can lose energy by creating other charge carriers. In power devices, impact ionization is usually responsive for the breakdown of the devices; therefore, this process needs to be modelled accurately in order to predict the I-V characteristics of the device correctly. In general, impact ionization can be modeled by using a net generation term in the electron and hole current continuity equations.

pnII II II II

n p n pR G G Gq q

α α− = − − = − −jj

(6)

where nα and pα are the ionization rates for electrons and holes defined as the number of generated

electron-hole pairs per unit length of travel.

B. Direct generation-recombination

At non-equilibrium:

1 2Direct DirectR G C np C− = − (7)

At equilibrium:

1 0 0 20 Direct DirectR G C n p C= − = − (8)

2 1 0 0C C n p= (9)

Hence 1 1 0 0Direct DirectR G C np C n p− = − and:


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( )1 0 0Direct DirectR G C np n p− = − (10)

C. Auger generation-recombination (non-radiative)

At non-equilibrium (everything that recombines minus everything that is generated):

2 21 3 2 4

Au Au Au Au Au Aun p n pR G R R G G C n p C n C np C p− = + − − = − + − (11)

At equilibrium (we need both conditions because otherwise the electrons or the holes will gain energy):

21 30 Au Au

n nR G C n p C n= − = − 23 1 0 0 1 iC C n p C n= = (12)

22 40 Au Au

p pR G C np C p= − = − 24 2 0 0 2 iC C n p C n= = (13)

Hence

( ) ( )2 2 2 2 2 21 1 2 2 1 2

Au Aui i i iR G C n p C nn C np C pn C n np n C p np n− = − + − = − + −

( )( )21 2

Au AuiR G C n C p np n− = + − (14)

D. Shockley-Read-Hall generation-recombination

Let tf be the fraction of occupied trap levels and 0tf the fraction of occupied trap levels at equilibrium.


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At non-equilibrium (electrons and holes always act in pairs):

SRH SRH SRH SRH SRH SRHn n p pR G R G R G− = − = − (15)

At -equilibrium:

( )1 0 31 t tC n f C f− = 03 1 0 1 1

0

1 t

t

fC C n C nf−

= 3

( )2 0 4 1t tC p f C f= − 04 2 0 2 1

01t

t

fC C p C pf

=−

3

Hence:

( )( )

1 1 1

2 2 1

1

1

SRH SRHn n t t

SRH SRHp p t t

R G C n f C n f

R G C pf C p f

− = − −

− = − −

Setting the two rates equal we get:

( ) ( )

1 2 1

1 1 2 1t

C n C pfC n n C p p

+=

+ + + (16)

and the net generation/recombination is

( ) ( )

2 2

1 1 1 1

2 1

SRH SRH i i

p n

np n np nR G n n p p n n p pC C

τ τ− −

− = =+ + + + ++

(17)

where we have used Boltzmann statistics 21 1in n p= and introduced pτ and pτ as the reciprocals of the

corresponding capture rates per single carriers. If we introduce the corresponding capturing cross-sections:

1 p th tC v Nσ= and 2 p th tC v Nσ= (18)

Since (at equilibrium) 01

1t Ft E EkT

fe

−=+

we have:


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01 0 0

0

01 0 0

0

1

1

t F

t F

E Et kT

tE E

t kT

t

fn n n ef

fp p p ef

−

−−

−= =

= = −

(19)

Using the last 2 equations we get:

( )2

0 0

t F t F

n p th t iSRH SRHE E E E

kT kTn p

v N np nR G

n n e p p e

σ σ

σ σ− −

−

−− =

+ + +

(20)

Since 0

t F F i t F t iE E E E E E E EkT kT kT kT

i in e n e e n e− − − −

= = and 0

t F i F t F t iE E E E E E E EkT kT kT kT

i ip e n e e n e− − − −

− − −= = we get

( )2

t i t i

n p th t iSRH SRHE E E E

kT kTn i p i

v N np nR G

n n e p n e

σ σ

σ σ− −

−

−− =

+ + +

(21)

E. Band to band tunneling: Schenk model

Under large electric fields, tunneling carriers can recombine through band to band tunneling. This term is important in abrupt junction with doping levels of 1019 cm-3 or when the field exceeds 8x105 V/cm.

( )

( )( )

272

iBB BB

i i

np nR G C

n n p p−

− =+ +

E (22)

Band to band tunneling is usually negligible under the normal operation of power semiconductor devices.

F. Mobility of WBG materials

It is very important for the modeling and simulation of WBG semiconductors to carefully specify the electron and hole motilities as a function of temperature and doping concentration. In device simulation, carrier mobilities are usually specified analytically as a function of temperature and acceptor and donor concentrations. The next plots show the experimental values of the electron an hole mobilities in a few common WBG semiconductors.


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Fig. 1 Electron Hall mobility vs temperature and electron and hole Hall mobility vs doping concentration in SiC [http://www.ioffe.ru/SVA/NSM/Semicond/SiC/hall.html]

Fig. 2 Electron Hall mobility vs temperature (for two samples) and hole Hall mobility vs temperature in GaN [http://www.ioffe.ru/SVA/NSM/Semicond/GaN/hall.html]

Fig. 3 Electron Hall mobility for different doping levels ranging from 2×1015 cm-3 to 1.2×1017 cm-3 and degrees of compensation in GaAs [http://www.ioffe.ru/SVA/NSM/Semicond/GaAs/electric.html]


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4. Discretization and meshing techniques

Three techniques are currently used to discretize the semiconductor transport equations presented at the beginning of Lecture 3.

1. Finite differences (RandFlux). This technique has the advantage that it can be easily implemented numerically and can be used on computers with relatively small memory. It is quite robust but it can be used to simulate only structures that are relatively rectangular (i.e. do not have round edges). Unfortunately, unless some special techniques such as the terminating mesh-points technique is used, this discretization method often introduces extra mesh points which can increase the total computation time.

2. Finite volume discretization (RandFlux, Comsol, Sentaurus). This technique has the big advantage that it inherently conserves current. For this reason, it is the method of choice in most semiconductor device simulators and provides the most accurate current carrier distributions. It can simulate round boundaries which makes this technique ideal for a multitude of applications. Unfortunately, the numerical implementation of the method is slightly more difficult than the finite differences method.

3. Finite element discretization (Comsol). This technique conserves the energy, however, does not inherently conserve current. Since, current conservation is essential in semiconductor device simulation, this technique is the least used among the three techniques presented in this section. To obtain accurate results, one needs to use very refined meshes and decrease the error margins when solving the semiconductor transport equations.

5. Power device simulations in Sentaurus

The attached tutorial gives an introduction on how to simulate a power MOSFET in Sentaurus. After the students read the attached tutorial they have to:

1. Compute the on-state resistance (at a drain-to-source voltage of 3V) 2. Simulate the same device under large applied voltages and compute the breakdown voltage 3. Change the material to SiC and re-compute the on-state resistance and break down voltage 4. Change the material to GaN and re-compute the on-state resistance and break down voltage 5. Finally the students need to design a GaN/AlGaN heterojunction transistor to simulate the I-V

characteristics. For this purpose they can start with the example that comes by default in Sentaurus

Simulation of power MOSFETs in Sentaurus

These slides a part of Lecture 4 on the modeling of WBG semiconductor devices.

Petru Andrei, Florida State University

Follow the steps presented on these slides to open, mesh, and compute the I-C characteristics of a power MOSFET in Sentaurus.

lectures 1&2. introduction to wbg semiconductors ... · this power module will cover an...

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