The whole topic is one promise and its collapse: if you shrink a transistor and lower its voltage by the SAME factor, the chip runs faster and packs more devices while burning the same heat per square millimetre. This note builds every letter — L V C I τ k V t h τ = C V / I see the triangle of charge, current, and time behind it.
Before any symbol, look at the object we keep shrinking.
A MOSFET (the switch our chips are built from) is a tiny sandwich. Current wants to flow between two terminals called the source and the drain . Between them sits a strip of silicon called the channel . Sitting on top — separated by a thin glassy insulator — is the gate , the control knob. Put voltage on the gate and it invites charge into the channel, opening a path for current; remove it and the path closes.
Everything in Dennard scaling is about making this sandwich smaller while keeping it working. See MOSFET Operation and Square-Law Current for how the current actually flows.
L horizontal distance the charge must travel from source to drain — the length of the red path in the figure above.
Picture: the gap the current crosses.Why we need it: shorter L L point of every new chip generation.
t o x thickness of the thin insulating glass (o x
Picture: the gap between the two plates of a capacitor (the gate is one plate, the channel the other).Why we need it: a thinner insulator lets the gate grip the channel more tightly — a stronger capacitor. This is why thinning t o x
V "push" we apply — how hard we drive electricity through the device. Measured in volts.
Picture: the height of a waterfall; higher V Why we need it: more V V down along with size.
E E ≈ V / L force per unit charge felt inside the channel. Squeeze a voltage V L E ≈ L V .
Picture: the steepness of the slope the charge rolls down. Same drop V L Why the topic needs it: if you shrink L V k E constant — "constant-field scaling" — is the whole trick. See Constant-Field vs Constant-Voltage Scaling .
k k > 1 how much smaller we make things this generation. Length becomes L / k V / k
Picture: a shrink-ray dial. Set k = 1.4 Why we need it: it lets us track one dial through the whole derivation and read off how speed, power, and heat respond.
k k ompresses. Anything that gets smaller is divided by k more numerous or faster is multiplied by k
C charge the gate must store to switch the channel on. The gate + oxide + channel form a parallel-plate capacitor : C o x = t o x ε o x W L .
Picture: a bucket. Bigger plates (W L t o x ε o x material constant of the oxide — how good that particular glass is at storing charge. It does not change when we shrink, so it rides along untouched.Why we need it: switching = filling and emptying this bucket. A bigger bucket is slower to fill, so C
I rate of charge flow — how many electrons per second cross the channel. Measured in amps (coulombs per second).
Picture: litres of water per second down the highway.Why we need it: current is what fills the capacitor bucket . More current = faster switching but more power. The parent's square-law I ≈ μ C o x ′′ L W ( V − V t h ) 2 I
τ time to switch one gate : fill the bucket C V I τ = I C V .
Picture: time = (water needed) ÷ (flow rate). Charge needed is C × V I Why we need it: τ inverse speed . Halve τ
f clock speed (GHz). It is simply the inverse of delay: faster switching (smaller τ f f ∝ 1/ τ
Picture: heartbeats per second of the chip.Why we need it: "the chip got faster" means f f
P P / A Power is energy burned per second — the heat the device makes. Two useful forms: P = I V P d y n = 2 1 C V 2 f Power density P / A area A = W L
Picture: P P / A panel packed with bulbs gets.Why we need it: the single quantity Dennard scaling holds constant is P / A power wall .
V t h minimum gate voltage that turns the transistor ON. Below V t h
Picture: the lip of a dam . Raise the water (gate voltage) past the lip and it flows.Why we need it: the drive term is ( V − V t h ) 2 headroom between how hard you push and the lip. When V t h ≈ 26 V Subthreshold Leakage and Static Power .
k T / q ≈ 26 temperature (T k not the scaling factor!), and electron charge (q
Picture: the fuzzy waterline of a jittering pond — you can't build a dam lip below the ripples or water spills over anyway.Why we need it: it is the hard floor under V t h V t h subthreshold leakage I l e ak ∝ e − V t h / ( n k T / q ) static power 's villain.
k Why it confuses: the same letter k scaling factor (k > 1 Boltzmann's constant (in k T / q Fix: context decides. If it multiplies dimensions it's the shrink dial. If it sits next to T q
Electric field E = V over L
Thermal voltage kT over q
Read it top-down: the geometry and voltage feed the field, capacitance, and current; those feed delay (hence speed) and power (hence power density); and the thermal-voltage floor under V t h is what eventually breaks the chain.
Cover the right side and test yourself — you are ready for the parent note when each is instant.
What do L W t o x Channel length (source-to-drain distance), channel width (into the page), and oxide-insulator thickness.
Why is the field E ≈ V / L Voltage V L L V
What is k k The shrink factor ( k > 1 ) k k 2
Why does capacitance C = ε o x W L / t o x 1/ k 1/ k 2 Area W L 1/ k 2 t o x / k k 1/ k
What does τ = C V / I Time to switch = charge to move ( C V ) ( I )
How are τ f Inversely — smaller delay means higher clock frequency, f ∝ 1/ τ
Give the two forms of power used in the topic. P = I V P d y n = 2 1 C V 2 f
What is V t h ( V − V t h ) The gate voltage that turns the transistor on; drive current depends on ( V − V t h ) 2 V t h
What is k T / q Thermal voltage ≈ 26 V t h V t h I l e ak ∝ e − V t h / ( n k T / q )
Ready? Head back to the parent note and the derivation will read like sentences, not symbols.
