Demo. the App Bar Tabs.

TabBar.dart

import 'package:flutter/material.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  MyApp({Key key}) : super(key: key);
  @override
  Widget build(BuildContext context) {
    final title = 'Silver AppBar With ToolBar';
    return MaterialApp(
      title: title,
      home: Scaffold(
        // No appbar provided to the Scaffold, only a body with a
        // CustomScrollView
        body: SilverAppBarWithTabBarScreen(title: title),
      ),
    );
  }
}

class SilverAppBarWithTabBarScreen extends StatefulWidget {
  final String title;
  SilverAppBarWithTabBarScreen({Key key, this.title}):super(key: key);
  @override
  _SilverAppBarWithTabBarState createState() => _SilverAppBarWithTabBarState();
}

class _SilverAppBarWithTabBarState extends State<SilverAppBarWithTabBarScreen>
    with SingleTickerProviderStateMixin {
  TabController controller;
  @override
  void initState() {
    super.initState();
    controller = TabController(length: 3, vsync: this);
  }
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: CustomScrollView(
        slivers: <Widget>[
          SliverAppBar(
            title: Text(widget.title),
            pinned: true,
            expandedHeight: 160.0,
            bottom: TabBar(
              tabs: [
                Tab(text: 'Tab 1'),
                Tab(text: 'Tab 2'),
                Tab(text: 'Tab 3'),
              ],
              controller: controller,
            ),
          ),
          SliverFillRemaining(
            child: TabBarView(
              controller: controller,
              children: <Widget>[
                Text("Content for Tab 1"),
                Text("Content for ab 2"),
                Text("Content for Tab 3"),
              ],
            ),
          ),
        ],
      ),
    );
  }
}