Motion of the Piezoelectric Polaron at Zero Temperature

Abstract

We analyze a series of theories which are used to obtain the energy-momentum relation for the piezoelectric polaron. Perturbation theory cannot be trusted, because there is a degeneracy in the unperturbed energy levels. The Tamm-Dancoff one-quantum cutoff in this case diagonalizes the degenerate states exactly, but has other shortcomings. The intermediate coupling theory gives what we believe is a reasonable energymomentum relation, which starts out quadratically and becomes approximately linear as the velocity approaches the speed of sound. WHITFIELD,GERSTNER, AND THARMALINGAM

Energy–momentum relations in polaron problems with arbitrary coupling

“…I n its turn, E ( p ) =: p a t large p as was mentioned earlier in weak [8 to 111 and strong [12] coupling limits. Comparison of our numerical results with those of [8,9,121 shows that our energy values are always lower. Comparison with [ 131, where only the large-momenta cases were examined, leads to practically complete agreement.…”

“…Therefore we shall proceed from the theoretical assumption that phonons are in their ground state. So let us consider the following expression that resembles the partition function: (9) where try means the trace over electron coordinates, and 10) represents the vacuum state of the phonon.…”

“…After separating from the energy that part which corresponds to the weak coupling limit (v = 1) we obt,ain from (19) the piezoelectric polaron with arbitrary coupling strength. Before passing to numerical calculations let us note that the cases of weak [8] and intermediate [9] coupling are contained in them. Particularly, when 8 = U,, and v = 1 the energy and momentum totally coincide with [9].…”

“…Before passing to numerical calculations let us note that the cases of weak [8] and intermediate [9] coupling are contained in them. Particularly, when 8 = U,, and v = 1 the energy and momentum totally coincide with [9]. These equations also contain the strong coupling limit [12] (at Y, D > 1).…”

Absence of binding in the Nelson and piezoelectric polaron models

“…where now Λ is kept fixed at a positive value, the so-called Debye wave number [16,Page 430,Footnote 6]. α is defined in terms of quantities that depend on the crystal in question (such as the speed of sound); see [27,29] for a precise definition of this constant, and also [26,30] to gain a better understanding of the model. We will study here the case of two electrons in a piezoelectric crystal, whose Hamiltonian follows directly from (1.3),…”