Inelastic electron tunneling through adatoms and molecular nanomagnets

We discuss a theoretical description of the inelastic electron tunneling spectra (IETS) of a magnetic nanosystem (an atom or a molecule) adsorbed on a solid surface measured in a scanning tunneling microscope (STM). We represent the nanosystem by means of a cluster Hubbard model, which allows us to study scenarios when the tunneling electrons sequentially interact with several magnetic centers inside the nanosystem or when the magnetic centers are made out of heavy atoms with a strong spin-orbit coupling and large orbital moments. The sequential tunneling through multiple centers is illustrated on an adatom probed by an STM tip with a nickelocene molecule attached to it. For atoms with large orbital moments, we find that the exchange interaction between the atom and the spin <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="bold">s</a:mi> </a:math> of the tunneling electron is richer than the usually assumed Heisenberg form <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi mathvariant="bold">S</c:mi> <c:mo>·</c:mo> <c:mi mathvariant="bold">s</c:mi> </c:mrow> </c:math> or <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi mathvariant="bold">J</f:mi> <f:mo>·</f:mo> <f:mi mathvariant="bold">s</f:mi> </f:mrow> </f:math> , where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi mathvariant="bold">S</i:mi> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi mathvariant="bold">J</k:mi> </k:math> are the spin and total orbital moments of the atom. For atoms in axially symmetric environments, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="bold">J</m:mi> <m:mo>·</m:mo> <m:mi mathvariant="bold">s</m:mi> </m:mrow> </m:math> exchange would restrict the transitions accessible by IETS to those fulfilling <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mrow> <p:mo>|</p:mo> <p:mi mathvariant="normal">Δ</p:mi> </p:mrow> <p:msub> <p:mi>J</p:mi> <p:mi>z</p:mi> </p:msub> <p:mrow> <p:mo>|</p:mo> <p:mo>≤</p:mo> <p:mn>1</p:mn> </p:mrow> </p:mrow> </p:math> , where <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:msub> <r:mi>J</r:mi> <r:mi>z</r:mi> </r:msub> </r:math> is the projection of the total angular momentum to the symmetry axis, whereas we arrive at a more permissive selection rule <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:mrow> <s:mrow> <s:mo>|</s:mo> <s:mi mathvariant="normal">Δ</s:mi> </s:mrow> <s:msub> <s:mi>J</s:mi> <s:mi>z</s:mi> </s:msub> <s:mrow> <s:mo>|</s:mo> <s:mo>≤</s:mo> <s:mn>2</s:mn> <s:mi>ℓ</s:mi> <s:mo>+</s:mo> <s:mn>1</s:mn> </s:mrow> </s:mrow> </s:math> , where <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:mi>ℓ</u:mi> </u:math> is the orbital momentum quantum number of the partially filled atomic shell carrying the magnetic moment.