Page 192 - Principles and Applications of NanoMEMS Physics

180 Chapter 4

the separation required between adjacent donors. In the presence of a
magnetic field B z , and assuming a donor nucleus with I = 2 / 1 embedded
in a silicon host, the interaction in question, namely, the nuclear-spin
interaction, is given by the Hamiltonian [202],
N
e
H e− N = µ B Bı − g N µ N Bı + Aı e ı ⋅ N , (3 )
7
z
z
where µ is the nuclear magneton, σ are the Pauli spin matrices, g is the
N N
nuclear g-factor, and A = 8 ʌµ g µ Ȍ () 0 2 is the contact hyperfine
3 B N N
interaction energy when the probability density of the electron wavefunction,
Ȍ () 0 2 is evaluated at the nucleus. Clearly, examination of Eq. (3 )
7
indicates that the interaction energy is a directly proportional to the magnetic
field and is a strong function of the wave function probability density at the
nucleus. A trade-off exists, however, because for electrons in their ground
state the frequency separation between nuclear levels is [202],

hȞ = 2g µ B + 2A + 2A 2 , (3 )
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A N N
µ B
B
which, for fields B < 3.5T is dominated by the second term. Thus, in this
regime the magnitudes of the nuclear magneton and the wavefunction
probability density at the nucleus take on a dominant character.
To perform arbitrary rotations on the nuclear spin, Kane indicates that it is
necessary to alter its precession frequency in comparison with that resulting
from the applied magnetic field B [202]. This is accomplished by
ac
exploiting the fact that the proximity of the donor-nuclear spin system to the
A gate allows the hyperfine interaction to be reduced by shifting the
envelope of the electron-donor wavefunction away from the nucleus, i.e., by
2
reducing () 0Ȍ . In essence, such a shifting alters the frequency, Eq. (35),
and causes the nuclear spin-donor system to behave as a voltage-controlled
oscillator producing, for a donor placed 200 Å under the gate, a tuning
parameter of the order of 30 MHz/V [202].
In addition to the single-qubit rotation, the two-qubit C-NOT operation
must be implemented in order to enable general quantum computations. In
the context of the nuclear spin-donor system, accomplishing this requires
developing the ability to induce nuclear-spin exchanges between two
nucleus-electron spin systems. The interaction between two such systems is
captured by the Hamiltonian [202],

H = H () AB + ı 1N ı ⋅ 2e + A ı 2N ı ⋅ 2e + Jı 1e ı ⋅ 2e , (3 )
9
1 2