Friedel oscillations of the magnetic field penetration in systems with spatial quantization

The magnetic field, applied to a size-quantized system produces equilibrium persistent current non-uniformly distributed across the system. The distributions of dia-and para-magnetic currents and magnetic field in a quantum well is found. We discuss the possibility of observation of field distribution by means of NMR. Traditionally, the magnetic field, penetrating into the system with spatial quan-tization is considered as uniform and coinciding with the external field. The mag-netization of such system and diamagnetic currents are weak and the corrections to the external field are rather small. Nevertheless diamagnetic currents in quantum systems are essential in such phenomena as NMR. It is well known that the magnetic field acting on the atomic nucleus is partially screened by electron shells that results in the chemical shift of NMR line. This shift is measurable due to very narrow width of NMR line as compared with the typical electron relaxation rates. In a spatially quantized system non-uniform electronic currents of magnetization also produce the screening of the external field resulting in the change of effective magnetic field acting on nuclei. The orbital magnetism in systems with spatial quantization was studied in a number of papers (see, e.g., [1]-[3]). These works consider the total magnetization of small systems. The purpose of the present paper is to find the current and magnetic field distribution in a quantum well. Let the magnetic field B to be directed along x axis in the film plane (x, y). The magnetic field in the film is determined by the Maxwell equation ∂B/∂z = 4πj(z)/c. The diamagnetic current density j has the only y component. Since diamagnetism is weak we shall neglect corrections to the uniform external field in the expression for diamagnetic current. We shall consider diamagnetic current in linear in external magnetic field approximation. The equilibrium density of current can be found from the expression j y (z) = Sp(ˆ j y (z)f (ˆ H)), (1) wherê H = (ˆ p + eA/c) 2 /2m + U (z) is the electron Hamiltonian, A = (0, −B 0 z, 0) is the vector potential of the external magnetic field B 0 , U (z) is the confining potential, ˆ j y (z) = −e{ˆv y , δ(z − ˆ z)}/S is the orbital current density operator, ˆ v = (ˆ p + eA/c)/m is the electron velocity operator, {...} stand for the operation of symmetrization, f (E) = (exp ((E …