For an
n
n
-component link, Milnor’s isotopy invariants are defined for each multi-index
I
=
i
1
i
2
.
.
.
i
m
(
i
j
∈
{
1
,
.
.
.
,
n
}
)
I=i_1i_2...i_m~(i_j\in \{1,...,n\})
. Here
m
m
is called the length. Let
r
(
I
)
r(I)
denote the maximum number of times that any index appears in
I
I
. It is known that Milnor invariants with
r
=
1
r=1
, i.e., Milnor invariants for all multi-indices
I
I
with
r
(
I
)
=
1
r(I)=1
, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with
r
=
1
r=1
coincide. This gives us that a link in
S
3
S^3
is link-homotopic to a trivial link if and only if all Milnor invariants of the link with
r
=
1
r=1
vanish. Although Milnor invariants with
r
=
2
r=2
are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with
r
≤
2
r\leq 2
are self
Δ
\Delta
-equivalence invariants. In this paper, we give a self
Δ
\Delta
-equivalence classification of the set of
n
n
-component links in
S
3
S^3
whose Milnor invariants with length
≤
2
n
−
1
\leq 2n-1
and
r
≤
2
r\leq 2
vanish. As a corollary, we have that a link is self
Δ
\Delta
-equivalent to a trivial link if and only if all Milnor invariants of the link with
r
≤
2
r\leq 2
vanish. This is a geometric characterization for links whose Milnor invariants with
r
≤
2
r\leq 2
vanish. The chief ingredient in our proof is Habiro’s clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory.