A Conformal-Mapping

Treatment of the Effect of a

Semi-Infinite Gate on a

Two-Dimensional Electron Gas

ARL-TR-2153 August 2000

Frank J. Crowne 

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return it to the originator.
ARL-TR-2153 August 2000

Army Research Laboratory

Adelphi, MD 20783-1197

A Conformal-Mapping

Treatment of the Effect of a

Semi-Infinite Gate on a

Two-Dimensional Electron Gas

Frank J. Crowne 

Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.
Abstract

A pinched-off high-electron-mobility transistor containing a perfectly conducting two-dimensional electron gas (2DEG) is described mathematically

within an idealized two-dimensional geometry, so that conformal mapping

techniques can be used to compute internal fields at the transistor drain.

The field and charge distribution at the drain end of the 2DEG calculated

in this way suggest that the charge is a nonmonotonic function of position

in this region.

ii
Contents

1 Introduction 1

2 Schwartz-Christoffel Transformation 3

3 Perfectly Conducting 2DEG 8

4 Conclusions 10

References 11

Distribution 13

Report Documentation Page 15

Figures

1 Cross section of FET geometry on drain side of 2DEF FET .... 2

2 Physical (z) plane for Schwartz-Christoffel mapping ....... 2

3 Schwartz-Christoffel plane ...................... 3

4 Branch cuts for w(z) ......................... 4

5 Graphical solution to equation for c, which locates mapping point

q that terminates 2DEG ........................ 6

6 Density distribution of perfectly conducting 2DEG ........ 9

iii
1. Introduction

The physics of field-effect transistors (FETs) fundamentally differs from

that of bipolar transistors and heterojunction-bipolar transistors (HBTs),

in which currents flow perpendicular to layers of various materials. This

difference arises from the lateral geometry that underlies the operation of

FETs, i.e., the modulation of a “horizontal” current that arises from a drainto-source voltage by “vertical” electric fields generated by a gate electrode.

The FET geometry gives rise to internal electric fields that are complicated and intrinsically two-dimensional (2D), so that the device terminal

characteristics (source, drain, and gate voltages and currents) become difficult to predict and dependent on many internal parameters. The geometry

makes the physics of 2D electron-gas (2DEG) FETs (high-electron-mobility

transistors—HEMTs) especially difficult to unravel, since the quantummechanical nature of the electron dynamics in the 2D channel cannot easily

be disentangled from the electrostatic problem of finding the fields in the

neighboring dielectric materials.

A problem of particular interest to device designers is to determine what

happens to the 2DEG as it emerges from under the gate region. Because

the electric fields are no longer screened by the gate, the capacitance of the

electron gas per unit length with respect to the gate must change, since

field lines can now escape to infinity. This problem is made even more interesting by the existence of plasma oscillations predicted by Dyakonov

and Shur in 1993 [1], which are profoundly different for a gated and an

ungated electron gas [2]. In this report, conformal mapping is used to compute the change in field configuration of the electron gas in this situation,

with the extreme assumption of a perfectly conducting electron gas. The

finite-conductivity effects needed for the study of plasma oscillations will

be the subject of a subsequent report.

Figure 1 is a cross section of the device geometry near the drain. The 2DEG

is assumed to be a vertical distance h below the gate and to terminate

at a horizontal distance q beyond the drain end of the gate. Let us begin

with the symmetrized version of this geometry shown in figure 2 (resembling that of a junction FET (JFET) [3]) and tailored for a treatment based

on complex-variable theory. The new geometry, which is embedded in the

complex plane z = x + iy, has the following idealized features:

1. There are two gates separated by a distance h, with the 2DEG between

them along the x = Re z axis and ending at a point x = q > 0 on the

real axis.

2. The 2DEG is infinitely extended to the left.

3. The gates are both assumed to be infinitely thick, with vertical boundaries along the y = Im z-axis.

1
4. All dielectric constants are set equal to one.

This idealized geometry will allow us to set up the conformal-mapping

problem discussed in the next section.

Figure 1. Cross section

of FET geometry on

drain side of 2DEF FET:

(a) real and (b)

idealized.

Gate

AlGaAs

GaAs

Ohmic

h

Gate

Gate

(a) X = 0 X = q

(b)

Figure 2. Physical (z)

plane for

Schwartz-Christoffel

mapping.

R

R

V W

U

T S

z = ih

z = –ih

z = q

z-plane

–π/2

–π/2

+π

+π

–π

2
2. Schwartz-Christoffel Transformation

Following standard procedures [4], I seek a complex-variable function that

will map the shaded area in figure 2 (the z-plane) into the upper half of a

new complex-variable plane, the w-plane. The special points R, S, T, U, V,

and W identify the right-hand half-space (R), the lower gate edge (S),

points deep inside the gated channel and below the 2DEG (T), the end of

the 2DEG (U), points deep inside the gated channel and above the 2DEG

(V ), and the upper gate edge (W). The Schwartz-Christoffel recipe treats

this geometry as a seven-sided polygon with the point R at infinity. Because of the symmetry, we can assume that the points in the complex zplane other than R pair symmetrically in the w-plane; this results in the

following differential equation for the Schwartz-Christoffel mapping:

dz

dw = Aw

√

w2 − c2

w2 − 1 , (1)

where A and c are to be determined. The function w(z) maps the points

R, S, T, U, V, and W in the z-plane into the points in figure 3 on the real axis

of the complex w-plane. The problem now reduces to

1. solving this equation,

2. finding values of A and c,

3. solving the Laplace equation in the w-plane, and

4. inverting the transformation to return to the z-plane.

Figure 4 shows the proper de

√

finition of the branch cuts for the function

w2 − c2 : arg(w − c) ∈ [−π, π], arg(w + c) ∈ [−π, π].

Figure 3.

Schwartz-Christoffel

plane.

w-plane

W = –C W = –1 W = 1 W = C

RW VU T S R

3
Figure 4. Branch cuts for

w(z).

• •

w-plane

–c +c

The differential equation is trivially solved for the function z(w):

z(w) = z0 + A

w

c

ξ2 − c2

ξ2 − 1 ξ dξ = z0 + A







w2 − c2

c2 − 1 − tan−1



w2 − c2

c2 − 1



 , (2)

which we can rewrite as

z(w) = z0 + A







w2 − c2

c2 − 1 − 12i

ln







1 + i

w2−c2

c2−1

1 − i

w2−c2

c2−1









 . (3)

Now, consider the action of this mapping from various regions of Re w to

the z-plane:

1. w ∈ [−∞, −c] (segment RW): for large w, z ≈ z0 + A √ w

c2−1 → +i∞,

so that A = iB with B < 0. As w → c, the term in brackets vanishes;

since w = c must correspond to z = +ih in the physical z-plane, we

have z0 = ih. Then

z = ih + iB







w2 − c2

c2 − 1 − tan−1



w2 − c2

c2 − 1



 . (4)

2. w ∈ [−c, −1] (segment W V ): On this segment of Re w, we have w2−c2

c2−1 =

i

c2−w2

c2−1 , with c2−w2c2−1 < 1, according to our choice of branch cut, so

we can rewrite z(w) as

z = ih + iB



i



c2 − w2

c2 − 1 − 12i

ln







1 −

c2−w2

c2−1

1 + c2−w2

c2−1











= ih − B







c2 − w2

c2 − 1 − 12

ln







1 + c2−w2

c2−1

1 −

c2−w2

c2−1











= ih − B







c2 − w2

c2 − 1 − tanh−1



c2 − w2

c2 − 1



 . (5)

4
From this we see that w → −1 ⇒ z → ih − ∞, so long as B < 0 and

real. This takes care of segment W V .

3. w ∈ [−1, 0] (segment V U): On this segment of Re w, we have c2−w2

c2−1 >

1, so that the hyperbolic tangent is affected. Write

1

2

ln







1 + c2−w2

c2−1

1 −

c2−w2

c2−1





 = 1

2

ln







c2−w2

c2−1 + 1

(−1) c2−w2

c2−1 − 1









(6)

and pick the log branch cut such that ln(−x) = −iπ + ln(x). Then

z = 1

2

ln







1 +  c2−1

c2−w2

1 −

 c2−1

c2−w2







+

iπ

2

= tanh−1







c2 − 1

c2 − w2



 +

iπ

2 (7)

and so

z = ih − B







c2 − w2

c2 − 1 − tanh−1







c2 − 1

c2 − w2



 − iπ

2



 . (8)

Now, let us choose the point z = q, where q is real, to correspond to

w = 0. This will be true if B = −2h

π and

q = 2h

π





c

√

c2 − 1 − tanh−1







c2 − 1

c







 . (9)

This equation now determines c in terms of the real parameter q. Then

z = 2h

π







c2 − w2

c2 − 1 − tanh−1







c2 − 1

c2 − w2







 (10)

and w → −1 ⇒ z → −∞ (from the inverse hyperbolic tangent), while

w → 0 ⇒ z → q.

Let x =

√c2−1

c . Then equation (9) becomes

1

x − πq2h = tanh−1(x). (11)

The plot in figure 5 shows that there is an acceptable root x for every

q ∈ [−∞,∞], where the root x is always in the range [0, 1].

4. w ∈ [0, 1] (segment UT): This is the same as w ∈ [−1, 0], since the

function depends on w2, but it runs backwards, and w → 1 ⇒ z →

−∞, w → 0 ⇒ z → q once more.

5. w ∈ [1, c] (segment T S): Here we have

z = 2h

π







c2 − w2

c2 − 1 − tanh−1







c2 − 1

c2 − w2







 . (12)

5
Figure 5. Graphical

solution to equation for

c, which locates

mapping point q that

terminates 2DEG; x> is

root for πq/2h > 0, and

x< is root for πq/2h < 0.

x

1 −

0 2h >

πq

< 0

x< x>

x-axis

x = 1

tanh–1(x)

2h

2h

πq

πq

Since w > 1 ⇒

 c2−1

c2−w2 > 1, we have

− tanh−1







c2 − 1

c2 − w2



 = −1

2

ln







1 +  c2−1

c2−w2

1 −

 c2−1

c2−w2







= 1

2

ln







(−1)  c2−1

c2−w2 − 1



 c2−1

c2−w2 + 1







= 1

2

ln







1 −

c2−w2

c2−1

1 + c2−w2

c2−1





 − iπ

2

= − tanh−1







c2 − w2

c2 − 1



 − iπ

2 . (13)

Then

z = 2h

π







c2 − w2

c2 − 1 − tanh−1







c2 − w2

c2 − 1



 − iπ

2





= −ih +

2h

π







c2 − w2

c2 − 1 − tanh−1







c2 − w2

c2 − 1







 (14)

on this segment. Note that w → 1 ⇒ z → −ih − ∞, as it should.

6. w ∈ [c,∞] (segment SR): On this segment we have c2−w2

c2−1 = −i

w2−c2

c2−1 ,

so that

tanh−1



−i



w2 − c2

c2 − 1



 = 1

2

ln





1 − i

w2−c2

c2−1

1 + i

w2−c2

c2−1





= −1

2

ln





1 + i

w2−c2

c2−1

1 − i

w2−c2

c2−1





6
= −i

1

2i

ln





1 + i

w2−c2

c2−1

1 − i

w2−c2

c2−1





= −itan−1







w2 − c2

c2 − 1



 (15)

and

z = −ih +

2h

π



−i



w2 − c2

c2 − 1 + itan−1







w2 − c2

c2 − 1









= −ih − 2ih

π







w2 − c2

c2 − 1 − tan−1







w2 − c2

c2 − 1







 . (16)

Clearly, for large real w > 0, we have z ≈ −ih − 2ih

π √ w

c2−1 → −i∞, which is

physically correct. Note that the Schwartz reflection principle implies that

if w = u + iv and z is a function of w2, then

z({−u + iv}2) = z({w∗}2) = z({w2}∗) = z∗(w2). (17)

Hence, moving to Re w < 0 in the upper half of the w-plane takes you to

the lower half of the z-plane.

7
3. Perfectly Conducting 2DEG

Here V = 0 on the electrodes, and V = V0 on the 2DEG between them.

Then the complex potential is

V (z) = V0

π

Im 

ln w(z) − 1

w(z)+1 ≡ Im U(w(z)), (18)

which reproduces the expression given in Churchill and Brown [5]. The

electric field components are

Ex = ∂V

∂x = Im 

∂U

∂x 

= Im 

∂U

∂z 

= Im U(w)w(z)

 (19)

and

Ey = ∂V

∂y = Im 

∂U

∂y 

= Im i

∂U

∂z 

= Re U(w)w(z)



. (20)

But

dw

dz =

 dz

dw−1

= iπ

2h

w2 − 1

w

√

w2 − c2 (21)

and

∂U

∂z = 2V0π

1

w2 − 1

, (22)

so that

U

(w)w(z) = 2V0

π

1

w2 − 1

iπ

2h

w2 − 1

w

√

w2 − c2

= iV0

h

1

w

√

w2 − c2 . (23)

Since |w| < c everywhere on the z-axis, we can write

Ex = V0

h Im 1

w

√

c2 − w2 ,

Ey = V0

h Re 1

w

√

c2 − w2 , and (24)

|E| = V0

h









1

w

√

c2 − w2







 .

A few things to note:

1. Ex = 0 for real w; i.e., the vector E is normal to the Re z axis, as it

should be for a perfect conductor.

2. The field has a |z|

−1/2 singularity at the end of the 2DEG.

3. The field has |c ± z|

−1/3 singularities at the electrode corners.

8
4. At large z, Re z > 0 (or large w, Re w > 0), the field goes as |z|

−2, i.e.,

like a dipole. Hence, the electrode charges completely screen out the

2DEG charge at large distances.

5. For large z, Re z < 0, and Im z[−ih, ih] (or w → ±1), the field approaches a constant:

Ey = ±V0

h

1

c(q)2 − 1

, (25)

where the sign is the same as that of Imz. This is also correct, since the

problem is x-independent in this limit. Since C = Q

V0 and Q = 2Ey

by Gauss’s law, we obtain the following nontrivial fringing capacitance/length,

C = 2

h

1

c(q)2 − 1

, (26)

associated with the termination of the gate.

Figure 6 shows a numerical calculation of the 2D density of a perfectly conducting 2DEG using the expressions derived in this report and the parameter values q = 4, h = 1 (only their ratio enters in, so the units are arbitrary). Note that in a plasma-active device structure, the potential and density would be coupled in a complicated way, with both acting as degrees of

freedom, rather than having the potential as a constant. Despite the oversimplification, however, the variation of the 2DEG density is highly nontrivial, with a minimum about 80 percent of the way to the end of the gas.

This is presumably because the image fields from the sharp corners are trying to hold the electrons within the region between the electrodes, which

competes with their natural tendency to accumulate at the end of the gas.

The fringing capacitance here is about 12.6 /h for this geometry.

Figure 6. Density

distribution of perfectly

conducting 2DEG. Note

divergence at endpoint

(z = 4).

14

12

10

8

6

4

2

Distance from electrode end plane

–10 1 2 3 4

Normalized change density

9
4. Conclusions

The predicted form of the charge density near the end of the gate suggests

that plasma oscillations that propagate to this point will see a depletion

region. This corresponds to a region of decreased Dyakonov-Shur plasma

velocity, which in turn suggests that plasma waves will see an impedance

mismatch between the gated and ungated regions over and above what

would be expected from their intrinsic dissimilarity. Because the central

assumption of the Dyakonov-Shur plasma model of a 2DEG FET is a special set of boundary conditions at the gate and source, the existence of this

“natural” boundary condition may require that their assumptions be reexamined, and could perhaps explain why plasma oscillations have yet to be

observed in standard HEMTs.

10
References

1. M. Dyakonov and M. Shur, “Shallow Water Analogy for a Ballistic

Field Effect Transistor: New Mechanism of Plasma Wave Generation

by dc Current,” Phys. Rev. Lett. 71 (1993), 2465.

2. F. Crowne, “Contact Boundary Conditions and the Dyakonov-Shur Instability in High Electron Mobility Transistors,” J. Appl. Phys. 82 (1997),

1242.

3. S. M. Sze, Physics of Semiconductor Devices, Wiley & Sons, New York

(1981), ch 6.

4. G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable, McGraw-Hill, New York (1966), §4-4.

5. R. V. Churchill and J. W. Brown, Complex Variables and Applications,

McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York

(1989), p 237.

11
13

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

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298-102

A Conformal-Mapping Treatment of the Effect of a

Semi-Infinite Gate on a Two-Dimensional Electron Gas

August 2000 Interim, 6/99–12/99

A pinched-off high-electron-mobility transistor containing a perfectly conducting

two-dimensional electron gas (2DEG) is described mathematically within an idealized

two-dimensional geometry, so that conformal mapping techniques can be used to compute

internal fields at the transistor drain. The field and charge distribution at the drain end of the

2DEG calculated in this way suggest that the charge is a nonmonotonic function of position in

this region.

Conformal mapping, two-dimensional electron gas

Unclassified

ARL-TR-2153

ONE6K1

622705.H94

AH94

62120.A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release;

distribution unlimited.

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Unclassified Unclassified UL

2800 Powder Mill Road

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U.S. Army Research Laboratory

Attn: AMSRL-SE-RE fcrowne@arl.mil

U.S. Army Research Laboratory

15

Frank J. Crowne

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197