  We present a theory of electronic properties and the spin blockade phenomena
in a gated linear triple quantum dot. Quadruple points where four different
charge configurations are on resonance, particularly involving (1,1,1)
configuration, are considered. In the symmetric case, the central dot is biased
to higher energy and a single electron tunnels through the device when (1,1,1)
configuration is resonant with (1,0,1),(2,0,1),(1,0,2) configurations. The
electronic properties of a triple quantum dot are described by a Hubbard model
containing two orbitals in the two unbiased dots and a single orbital in the
biased dot. The transport through the triple quantum dot molecule involves both
singly and doubly occupied configurations and necessitates the description of
the (1,1,1) configuration beyond the Heisenberg model. Exact eigenstates of the
triple quantum dot molecule with up to three electrons are used to compute
current assuming weak coupling to the leads and non-equilibrium occupation of
quantum molecule states obtained from the rate equation. The intra-molecular
relaxation processes due to acoustic phonons and cotunneling with the leads are
included, and are shown to play a crucial role in the spin blockade effect. We
find a quantum interference-based spin blockade phenomenon at low source-drain
bias and a distinct spin blockade due to a trap state at higher bias. We also
show that, for an asymmetric quadruple point with
(0,1,1),(1,1,1,),(0,2,1),(0,1,2) configurations on resonance, the spin blockade
is analogous to the spin blockade in a double quantum dot.
