Constant Voltage Scaling and Constant Field Scaling

Constant Voltage Scaling vs Constant Field Scaling

Constant Voltage Scaling and Constant Field Scaling

1. Why Scaling Matters

In semiconductor device design, especially MOSFET scaling, transistor dimensions are reduced to improve speed and density.

Suppose we shrink all linear dimensions by a factor S > 1.

L' = L / S
W' = W / S
t_ox' = t_ox / S

Where:

L = channel length
W = transistor width
t_ox = oxide thickness

The major question becomes: should voltage also scale?

2. Constant Voltage Scaling

In constant voltage scaling, device dimensions shrink but supply voltage remains fixed.

V' = V

L' = L / S

Electric Field

Electric field is approximately:

E = V / L

After scaling:

E' = V' / L'

E' = V / (L/S)

E' = S E

Thus electric field increases by factor S.

Consequences

Higher carrier velocity
Faster switching
Hot carrier effects
Oxide breakdown
Reliability problems

MOSFET Current Scaling

I_D ∝ μ C_ox (W/L) (V_GS - V_T)²

Oxide capacitance:

C_ox = ε_ox / t_ox

Since:

t_ox' = t_ox / S

Then:

C_ox' = S C_ox

Current scales approximately as:

I_D' = S I_D

Delay Scaling

τ ≈ CV / I

Capacitance scales as:

C' = C / S

Therefore:

τ' = τ / S²

Power Density

P = CV²f

Power density rises sharply in constant voltage scaling.

3. Constant Field Scaling (Dennard Scaling)

In constant field scaling, both dimensions and voltage scale together.

L' = L / S

V' = V / S

Electric Field

E = V / L

After scaling:

E' = (V/S) / (L/S)

E' = E

Therefore electric field remains constant.

Geometry Scaling

Quantity	Scaling
Length L	1/S
Width W	1/S
Oxide Thickness	1/S
Area	1/S²

Gate Capacitance

C_g = C_ox W L

After scaling:

C_g' = C_g / S

Drain Current

I_D' = I_D / S

Delay

τ' = τ / S

Power Per Transistor

P' = P / S²

Power Density

Since transistor density increases as S² and power per transistor decreases as 1/S², overall power density remains approximately constant.

4. Why Dennard Scaling Broke Down

Threshold voltage stopped scaling
Leakage current increased
Oxide tunneling effects appeared
Short-channel effects became severe
Power density became difficult to manage

Modern processors now rely on:

Multicore architectures
FinFETs and GAAFETs
Dynamic voltage/frequency scaling
Specialized accelerators

5. Comparison Table

Parameter	Constant Voltage Scaling	Constant Field Scaling
Length	1/S	1/S
Voltage	Constant	1/S
Electric Field	Increases by S	Constant
Current	Increases by S	Decreases by 1/S
Delay	1/S²	1/S
Power Density	Increases sharply	Approximately constant
Reliability	Worsens	Maintained

6. Summary