Power considerations in submicron digital CMOS
2.2. Fundamental limits
Fundamental limits for power can be derived from the basic principles of electromagnetics, thermodynamics and quantum mechanics [1], [2], [3]. They are asymptotic limits and give the minimum possible power.
2.2.1. Thermodynamic limit
The thermodynamic limit can be derived from the following model. The resistor R with a mean square noise voltage given in fig.2.2 has been applied to a digital circuit modeled with an equivalent resistance R. The available noise power is P_{AV} =kTDf (see reference [4]) where k is Boltzmann’s constant, T is absolute temperature and Df is the noise bandwidth. Since we want to transfer energy to the circuit in a digital form, the average signal power P_{S }transferred, should be greater than the available noise power P_{AV} by a factor g ³ 1. Therefore, the switching energy E_{s} transferred to the digital circuit should be:
( 2.1)
Although, the constant factor g is larger than one we do not know yet how large has to
be, in order to be able to have the probability of error in data transmission sufficiently small. For comparison sake, the necessary energy to move a single electron through a potential difference of about 0.1V it is in the order of 4kT=0.1eV at room temperature.
The probability of errors due to the thermal noise can be found by using Poisson distribution and it is proportional to:
(2.2)

Fig.2.2: Thermodynamic limit 
The bit error rate in a modern digital transmission BER should be better than 10^{10}. As a conclusion, in order to get reliable results, we have to use switching energies with a factor g=10^{7} larger than the above mentioned value.
2.2.2. Quantum mechanics limit
The uncertainty principle derived by Heisenberg provides the second fundamental limit [5]. An energy change D E associated with a switching transition must satisfy the following condition to give a measurable outcome:
(2.3)
where D t is the switching time and h is Plank's constant. During a switching transition of a single electron wave packet, the required average power P can be deducted from (2.3) as:
(2.4)
The point where the thermodynamic limit reaches the same value as the quantum limit
represents the boundary between the region where the electron can be treated as a classical particle and the region where it must be treated as a wave packet. Hence, the switching time D t can be computed as a function of absolute temperature:
Fig.2.3: Fundamental limits of power in digital 
(2.5)
Fig.2.3 illustrates the fundamental limits of power in digital signal processing from thermodynamics and quantum mechanics considerations in the power Pdelay Dt plane. The value of g is taken 4 for the sake of comparison. Switching transitions at the left of their loci are forbidden. The zone for low power lies to the right of those limits.
The physical limits shown above are not the only limits where power can be addressed. Material limits, circuit limits and system limitations can bring also clues about power. The fundamental limits are asymptotic limits and show how far one can go in the direction of power reduction. For realistic comparisons between different possible implementations of the same digital function we have to address the practical limits as explained in the following section.
