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CJam, 20 bytes

Laq~{'a+_@f*\f+z}/N*

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La                      Push [[]]
  q~                    Push input n
    {           }/      For i in 0..n-1 ...
     'a+                  Add to char 'a to give current char
        _@f*              Join each row by char
            \f+           Add char to the end of each row as well
               z          Zip to transpose
                  N*    Join result by newlines

20/21-byte alternatives:

Laaq~{'a+aff+:sz}/N*
Laaq~{'a+\Laf+f*z}/N*
Laq~{'a+f{_@*\+}z}/N*

MATL, 23 bytes

97tiq:+"TFX*tXa~@wZ(!]c

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This uses repeated Kronecker tensor product for extending the array, followed by transposition. At each iteration, new columns containing zeros are interleaved with the old; those zeros are then replaced by the appropriate new value (which increases at each iteration); and the matrix is transposed.

(One byte wasted because Octave's Kronecker product doesn't allow char input. This will be fixed for next release).

EXplanation

97       % Push 97 (ASCII for 'a')
t        % Duplicate
iq:      % Take input n. Range [1 2 ... n-1]
+        % Add. Gives [98 99 ... 97+n-1] (letters to be filled)
"        % For each
  TFX*   %   Kronecker product with [1 0]. This interleaves new columns with zeros
  tXa~   %   Duplicate. Logical index for the new columns
  @wZ(   %   Assign letter to those columns
  !      %   Transpose (zip)
]        % End if
c        % Convert to chat. Implicitly display

Perl, 110 104 100 99 91 89 87 + 1 (-p flag) = 88 bytes

#!perl -p
$==$_/2;$_=a.$/;$"=b;s/\w/$&.$"/ge,$"++,s/\n/$&.$"x2**$%.$&/ge,$"++until$=<++$%;$\=$_}{

Using:

> echo 5 | perl -pe '$==$_/2;$_=a.$/;$"=b;s/\w/$&.$"/ge,$"++,s/\n/$&.$"x2**$%.$&/ge,$"++until$=<++$%;$\=$_}{'

Ungolfed:

while (<>) {
# code above added by -p
    # $_ has input value
    # $/ = "\n" by default
    # $% = 0 by default
    my $n = $_ / 2;   # input
    my $s = "a" . $/; # "a\n"
    my $c = "b";      # "b"
    my $i = $%;       # 0
    while (++$i <= $n) {
        $s =~ s/(\w)/$1 . $c/ge;
        $c++;
        $s =~ s/(\n)/$1 . ($с x 2**$i) . $1/ge;
        $c++;
    }
    $\ = $s;
} {
# code below added by -p
    print;  # prints $_ (undef here) and $\
}

Ideone.