Flutter Search Delegate Taking Full Width

Question

I made a master-detail type view app using flutter, running on flutter desktop. I implemented the SearchDelegate on appBar, but it takes the full width of the app.

Here are some screenshots.

When the search is active

Here is some code:

main_page.dart

class _MainPageState extends State<MainPage> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: LayoutBuilder(
        builder: (context, constraints){
          if(!(constraints.maxWidth > 768)){
              return MasterTab();
          }
          return Row(
            children: [
              Container(
                width: constraints.maxWidth / 4,
                child: MasterTab(),
              ),
              Expanded(
                child: DetailTab(),
              )
            ],
          );
        },
      ),
    );
  }
}

master_tab.dart (left side, which is I want the search view to show)

  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text("Primary"),
        actions: [
          IconButton(
              onPressed: () {
                showSearch(context: context, delegate: SearchView());
              },
              icon: Icon(Icons.search))
        ],
      ),
      body: ListView.builder(
        itemCount: _notes.length,
        itemBuilder: (context, index){
          return ItemNote(noteEntity: _notes[index]);
        },
      ),
      floatingActionButton: FloatingActionButton(
        onPressed: _onAddNoteClick,
        child: Icon(Icons.create),
      ),
    );
  }

detail_tab.dart (right side, the red)

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Column(
        children: [
          Align(
            alignment: Alignment.topRight,
            child: DetailAction(),
          ),
          Expanded(child: Container(
            color: Colors.red,

          ))
        ],
      )
    );
  }

search_delegate.dart

class SearchView extends SearchDelegate {
  @override
  List<Widget> buildActions(BuildContext context) {
    return [
      IconButton(onPressed: () {
        query = "";
      }, icon: Icon(Icons.close))
    ];
  }

  @override
  Widget buildLeading(BuildContext context) {
    return IconButton(onPressed: () {
      Navigator.pop(context);
    }, icon: Icon(Icons.arrow_back));
  }

  @override
  Widget buildResults(BuildContext context) {
    return Container(
      color: Colors.amber,
      child: Center(
        child: Text("Search query"),
      ),
    );
  }

  @override
  Widget buildSuggestions(BuildContext context) {
    return Container(
      child: Center(
        child: Text("Suggestion"),
      ),
    );
  }
  
}

I want to make the search view only works on the left side only (not taking the fullwidth), how to achieve this?